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So it could be any number in an interval. For example, suppose your ages range between 18 years and 20 years. 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So it's continuous on the variable. Other examples for continuous, thickness of an item. For example, the thickness. 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0.81640625}, {"start": 382.6, "end": 382.92, "word": " the", "probability": 0.904296875}, {"start": 382.92, "end": 383.4, "word": " spread", "probability": 0.9072265625}, {"start": 383.4, "end": 383.96, "word": " of", "probability": 0.6005859375}, {"start": 383.96, "end": 384.08, "word": " the", "probability": 0.8017578125}, {"start": 384.08, "end": 384.24, "word": " data,", "probability": 0.7080078125}, {"start": 384.28, "end": 384.8, "word": " or", "probability": 0.3076171875}, {"start": 384.8, "end": 385.08, "word": " the", "probability": 0.90966796875}, {"start": 385.08, "end": 385.46, "word": " variability", "probability": 0.95703125}, {"start": 385.46, "end": 385.82, "word": " of", "probability": 0.9609375}, {"start": 385.82, "end": 385.98, "word": " the", "probability": 0.9111328125}, {"start": 385.98, "end": 386.3, "word": " data,", "probability": 0.93701171875}, {"start": 386.82, "end": 386.92, "word": " and", "probability": 0.92138671875}, {"start": 386.92, "end": 387.08, "word": " the", "probability": 0.70849609375}, {"start": 387.08, "end": 387.32, "word": " spread", "probability": 0.9091796875}, {"start": 387.32, "end": 387.68, "word": " is", "probability": 0.92919921875}, {"start": 387.68, "end": 388.22, "word": " sigma,", "probability": 0.75732421875}, {"start": 388.82, "end": 389.3, "word": " or", "probability": 0.9306640625}, {"start": 389.3, "end": 389.42, "word": " the", "probability": 0.90380859375}, {"start": 389.42, "end": 389.78, "word": " variation.", "probability": 0.88916015625}, {"start": 390.84, "end": 391.16, "word": " So", "probability": 0.9453125}, {"start": 391.16, "end": 391.3, "word": " we", "probability": 0.91845703125}, {"start": 391.3, "end": 391.54, "word": " have", "probability": 0.943359375}, {"start": 391.54, "end": 391.86, "word": " two", "probability": 0.9306640625}, {"start": 391.86, "end": 392.36, "word": " parameters,", "probability": 0.97021484375}, {"start": 392.88, "end": 393.14, "word": " mu", "probability": 0.73583984375}, {"start": 393.14, "end": 393.94, "word": " and", "probability": 0.91552734375}, {"start": 393.94, "end": 394.56, "word": " sigma.", "probability": 0.92578125}], "temperature": 1.0}, {"id": 17, "seek": 41305, "start": 395.57, "end": 413.05, "text": " The random variable in this case can take any value from minus infinity up to infinity. So random variable in this case continuous ranges from minus infinity all the way up to infinity. I mean from this point here", "tokens": [440, 4974, 7006, 294, 341, 1389, 393, 747, 604, 2158, 490, 3175, 13202, 493, 281, 13202, 13, 407, 4974, 7006, 294, 341, 1389, 10957, 22526, 490, 3175, 13202, 439, 264, 636, 493, 281, 13202, 13, 286, 914, 490, 341, 935, 510], "avg_logprob": -0.20293899093355453, "compression_ratio": 1.7833333333333334, "no_speech_prob": 0.0, "words": [{"start": 395.57, "end": 395.91, "word": " The", "probability": 0.6181640625}, {"start": 395.91, "end": 396.15, "word": " random", "probability": 0.7470703125}, {"start": 396.15, "end": 396.57, "word": " variable", "probability": 0.8837890625}, {"start": 396.57, "end": 396.77, "word": " in", "probability": 0.8310546875}, {"start": 396.77, "end": 396.99, "word": " this", "probability": 0.95068359375}, {"start": 396.99, "end": 397.37, "word": " case", "probability": 0.90576171875}, {"start": 397.37, "end": 397.97, "word": " can", "probability": 0.8564453125}, {"start": 397.97, "end": 398.27, "word": " take", "probability": 0.876953125}, {"start": 398.27, "end": 398.55, "word": " any", "probability": 0.9140625}, {"start": 398.55, "end": 398.85, "word": " value", "probability": 0.9736328125}, {"start": 398.85, "end": 399.09, "word": " from", "probability": 0.90234375}, {"start": 399.09, "end": 399.45, "word": " minus", "probability": 0.87548828125}, {"start": 399.45, "end": 399.93, "word": " infinity", "probability": 0.89599609375}, {"start": 399.93, "end": 400.49, "word": " up", "probability": 0.9228515625}, {"start": 400.49, "end": 400.65, "word": " to", "probability": 0.96435546875}, {"start": 400.65, "end": 401.17, "word": " infinity.", "probability": 0.88134765625}, {"start": 402.19, "end": 402.79, "word": " So", "probability": 0.71435546875}, {"start": 402.79, "end": 403.01, "word": " random", "probability": 0.658203125}, {"start": 403.01, "end": 403.51, "word": " variable", "probability": 0.88427734375}, {"start": 403.51, "end": 403.87, "word": " in", "probability": 0.66796875}, {"start": 403.87, "end": 404.05, "word": " this", "probability": 0.94775390625}, {"start": 404.05, "end": 404.27, "word": " case", "probability": 0.9208984375}, {"start": 404.27, "end": 404.95, "word": " continuous", "probability": 0.369140625}, {"start": 404.95, "end": 406.83, "word": " ranges", "probability": 0.363037109375}, {"start": 406.83, "end": 408.33, "word": " from", "probability": 0.88671875}, {"start": 408.33, "end": 408.73, "word": " minus", "probability": 0.9833984375}, {"start": 408.73, "end": 409.29, "word": " infinity", "probability": 0.90966796875}, {"start": 409.29, "end": 409.87, "word": " all", "probability": 0.9443359375}, {"start": 409.87, "end": 410.05, "word": " the", "probability": 0.9228515625}, {"start": 410.05, "end": 410.31, "word": " way", "probability": 0.9521484375}, {"start": 410.31, "end": 410.67, "word": " up", "probability": 0.9619140625}, {"start": 410.67, "end": 410.83, "word": " to", "probability": 0.97021484375}, {"start": 410.83, "end": 411.23, "word": " infinity.", "probability": 0.86865234375}, {"start": 411.33, "end": 411.41, "word": " I", "probability": 0.9736328125}, {"start": 411.41, "end": 411.59, "word": " mean", "probability": 0.9580078125}, {"start": 411.59, "end": 411.87, "word": " from", "probability": 0.81689453125}, {"start": 411.87, "end": 412.17, "word": " this", "probability": 0.94873046875}, {"start": 412.17, "end": 412.51, "word": " point", "probability": 0.96533203125}, {"start": 412.51, "end": 413.05, "word": " here", "probability": 0.8115234375}], "temperature": 1.0}, {"id": 18, "seek": 44122, "start": 414.6, "end": 441.22, "text": " up to infinity. So the values range from minus infinity up to infinity. And if you look here, the mean is located nearly in the middle. And mean and median are all approximately equal. That's the features or the characteristics of the normal distribution. Now, how can we compute the probabilities under the normal killer?", "tokens": [493, 281, 13202, 13, 407, 264, 4190, 3613, 490, 3175, 13202, 493, 281, 13202, 13, 400, 498, 291, 574, 510, 11, 264, 914, 307, 6870, 6217, 294, 264, 2808, 13, 400, 914, 293, 26779, 366, 439, 10447, 2681, 13, 663, 311, 264, 4122, 420, 264, 10891, 295, 264, 2710, 7316, 13, 823, 11, 577, 393, 321, 14722, 264, 33783, 833, 264, 2710, 13364, 30], "avg_logprob": -0.19242788461538463, "compression_ratio": 1.599009900990099, "no_speech_prob": 0.0, "words": [{"start": 414.6, "end": 414.94, "word": " up", "probability": 0.38623046875}, {"start": 414.94, "end": 415.1, "word": " to", "probability": 0.96044921875}, {"start": 415.1, "end": 415.54, "word": " infinity.", "probability": 0.7744140625}, {"start": 415.76, "end": 415.86, "word": " So", "probability": 0.89453125}, {"start": 415.86, "end": 416.16, "word": " the", "probability": 0.62646484375}, {"start": 416.16, "end": 416.98, "word": " values", "probability": 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{"start": 434.18, "end": 434.74, "word": " distribution.", "probability": 0.87548828125}, {"start": 436.46, "end": 436.82, "word": " Now,", "probability": 0.95556640625}, {"start": 436.94, "end": 437.16, "word": " how", "probability": 0.9365234375}, {"start": 437.16, "end": 437.4, "word": " can", "probability": 0.9404296875}, {"start": 437.4, "end": 437.66, "word": " we", "probability": 0.9033203125}, {"start": 437.66, "end": 438.38, "word": " compute", "probability": 0.89892578125}, {"start": 438.38, "end": 439.44, "word": " the", "probability": 0.91845703125}, {"start": 439.44, "end": 439.94, "word": " probabilities", "probability": 0.9033203125}, {"start": 439.94, "end": 440.36, "word": " under", "probability": 0.89892578125}, {"start": 440.36, "end": 440.6, "word": " the", "probability": 0.9091796875}, {"start": 440.6, "end": 440.86, "word": " normal", "probability": 0.86328125}, {"start": 440.86, "end": 441.22, "word": " killer?", "probability": 0.202392578125}], "temperature": 1.0}, {"id": 19, "seek": 46670, "start": 443.46, "end": 466.7, "text": " The formula that is used to compute the probabilities is given by this one. It looks complicated formula because we have to use calculus in order to determine the area underneath the cube. So we are looking for something else. So this formula is it seems to be complicated. It's not hard but it's", "tokens": [440, 8513, 300, 307, 1143, 281, 14722, 264, 33783, 307, 2212, 538, 341, 472, 13, 467, 1542, 6179, 8513, 570, 321, 362, 281, 764, 33400, 294, 1668, 281, 6997, 264, 1859, 7223, 264, 13728, 13, 407, 321, 366, 1237, 337, 746, 1646, 13, 407, 341, 8513, 307, 309, 2544, 281, 312, 6179, 13, 467, 311, 406, 1152, 457, 309, 311], "avg_logprob": -0.1880122989904685, "compression_ratio": 1.6141304347826086, "no_speech_prob": 0.0, "words": [{"start": 443.46, "end": 443.74, "word": " The", "probability": 0.72705078125}, {"start": 443.74, "end": 444.16, "word": " formula", "probability": 0.92529296875}, {"start": 444.16, "end": 445.1, "word": " that", "probability": 0.9140625}, {"start": 445.1, "end": 445.32, "word": " is", "probability": 0.94140625}, {"start": 445.32, "end": 445.62, "word": " used", "probability": 0.9111328125}, {"start": 445.62, "end": 445.84, "word": " to", "probability": 0.97021484375}, {"start": 445.84, "end": 446.08, "word": " compute", "probability": 0.919921875}, {"start": 446.08, "end": 446.24, "word": " the", "probability": 0.87646484375}, {"start": 446.24, "end": 446.74, "word": " probabilities", "probability": 0.8681640625}, {"start": 446.74, "end": 447.5, "word": " is", "probability": 0.576171875}, {"start": 447.5, "end": 448.06, "word": " given", "probability": 0.87939453125}, {"start": 448.06, "end": 448.32, "word": " by", "probability": 0.97021484375}, {"start": 448.32, "end": 448.58, "word": " this", "probability": 0.9521484375}, {"start": 448.58, "end": 448.82, "word": " one.", "probability": 0.84814453125}, {"start": 449.06, "end": 449.22, "word": " It", "probability": 0.90380859375}, {"start": 449.22, "end": 449.5, "word": " looks", "probability": 0.429931640625}, {"start": 449.5, "end": 450.44, "word": " complicated", "probability": 0.8427734375}, {"start": 450.44, "end": 450.9, "word": " formula", "probability": 0.7578125}, {"start": 450.9, "end": 452.66, "word": " because", "probability": 0.53466796875}, {"start": 452.66, "end": 452.96, "word": " we", "probability": 0.94970703125}, {"start": 452.96, "end": 453.18, "word": " have", "probability": 0.94873046875}, {"start": 453.18, "end": 453.32, "word": " to", "probability": 0.9677734375}, {"start": 453.32, "end": 453.56, "word": " use", "probability": 0.8798828125}, {"start": 453.56, "end": 453.94, "word": " calculus", "probability": 0.94287109375}, {"start": 453.94, "end": 454.28, "word": " in", "probability": 0.93994140625}, {"start": 454.28, "end": 454.46, "word": " order", "probability": 0.91845703125}, {"start": 454.46, "end": 454.68, "word": " to", "probability": 0.96435546875}, {"start": 454.68, "end": 455.08, "word": " determine", "probability": 0.9033203125}, {"start": 455.08, "end": 455.34, "word": " the", "probability": 0.91357421875}, {"start": 455.34, "end": 455.62, "word": " area", "probability": 0.90283203125}, {"start": 455.62, "end": 456.04, "word": " underneath", "probability": 0.921875}, {"start": 456.04, "end": 456.4, "word": " the", "probability": 0.916015625}, {"start": 456.4, "end": 456.66, "word": " cube.", "probability": 0.248779296875}, {"start": 457.3, "end": 457.82, "word": " So", "probability": 0.9453125}, {"start": 457.82, "end": 458.04, "word": " we", "probability": 0.640625}, {"start": 458.04, "end": 458.2, "word": " are", "probability": 0.93798828125}, {"start": 458.2, "end": 458.46, "word": " looking", "probability": 0.9140625}, {"start": 458.46, "end": 458.7, "word": " for", "probability": 0.95166015625}, {"start": 458.7, "end": 459.02, "word": " something", "probability": 0.86376953125}, {"start": 459.02, "end": 459.46, "word": " else.", "probability": 0.908203125}, {"start": 459.92, "end": 460.12, "word": " So", "probability": 0.919921875}, {"start": 460.12, "end": 460.4, "word": " this", "probability": 0.91064453125}, {"start": 460.4, "end": 460.92, "word": " formula", "probability": 0.91162109375}, {"start": 460.92, "end": 461.2, "word": " is", "probability": 0.7412109375}, {"start": 461.2, "end": 462.62, "word": " it", "probability": 0.3291015625}, {"start": 462.62, "end": 462.98, "word": " seems", "probability": 0.791015625}, {"start": 462.98, "end": 463.16, "word": " to", "probability": 0.9677734375}, {"start": 463.16, "end": 463.34, "word": " be", "probability": 0.95166015625}, {"start": 463.34, "end": 464.04, "word": " complicated.", "probability": 0.91015625}, {"start": 464.98, "end": 465.3, "word": " It's", "probability": 0.95556640625}, {"start": 465.3, "end": 465.52, "word": " not", "probability": 0.9482421875}, {"start": 465.52, "end": 465.88, "word": " hard", "probability": 0.89794921875}, {"start": 465.88, "end": 466.18, "word": " but", "probability": 0.525390625}, {"start": 466.18, "end": 466.7, "word": " it's", "probability": 0.954833984375}], "temperature": 1.0}, {"id": 20, "seek": 49016, "start": 468.04, "end": 490.16, "text": " complicated one, but we can use it. If we know calculus very well, we can use integration to create the probabilities underneath the curve. But for our course, we are going to skip this formula because this formula depends actually on mu and sigma. 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Now, for the standardized normal distribution, the mean is fixed value. The mean is zero, Sigma is one. So, the question is, how can we actually transform from X, which has normal distribution, to Z, which has standardized normal with mean zero and Sigma of one. 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So the z distribution always has mean of zero and sigma of one. So that's the story of standardizing the normal value. Now the", "tokens": [337, 31677, 2710, 7316, 293, 1009, 710, 575, 914, 4018, 293, 439, 293, 12771, 420, 3832, 25163, 13, 407, 264, 710, 7316, 1009, 575, 914, 295, 4018, 293, 12771, 295, 472, 13, 407, 300, 311, 264, 1657, 295, 3832, 3319, 264, 2710, 2158, 13, 823, 264], "avg_logprob": -0.1883311113144489, "compression_ratio": 1.7185185185185186, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 728.73, "end": 729.09, "word": " for", "probability": 0.466552734375}, {"start": 729.09, "end": 729.47, "word": " standardized", "probability": 0.74755859375}, {"start": 729.47, "end": 730.01, "word": " normal", "probability": 0.83447265625}, {"start": 730.01, "end": 730.65, "word": " distribution", "probability": 0.85302734375}, {"start": 730.65, "end": 731.41, "word": " and", "probability": 0.69775390625}, {"start": 731.41, "end": 731.77, "word": " always", "probability": 0.89208984375}, {"start": 731.77, "end": 731.99, "word": " z", "probability": 0.69140625}, {"start": 731.99, "end": 732.23, "word": " has", "probability": 0.93017578125}, {"start": 732.23, "end": 732.43, "word": " mean", "probability": 0.90380859375}, {"start": 732.43, "end": 732.77, "word": " zero", "probability": 0.6708984375}, {"start": 732.77, "end": 733.01, "word": " and", "probability": 0.9306640625}, {"start": 733.01, "end": 733.31, "word": " all", "probability": 0.76220703125}, {"start": 733.31, "end": 733.73, "word": " and", "probability": 0.880859375}, {"start": 733.73, "end": 734.09, "word": " sigma", "probability": 0.85595703125}, {"start": 734.09, "end": 734.75, "word": " or", "probability": 0.5234375}, {"start": 734.75, "end": 735.07, "word": " standard", "probability": 0.90966796875}, {"start": 735.07, "end": 735.37, "word": " deviation.", "probability": 0.8388671875}, {"start": 736.25, "end": 736.97, "word": " So", "probability": 0.84228515625}, {"start": 736.97, "end": 737.19, "word": " the", "probability": 0.73974609375}, {"start": 737.19, "end": 737.37, "word": " z", "probability": 0.953125}, {"start": 737.37, "end": 737.95, "word": " distribution", "probability": 0.81005859375}, {"start": 737.95, "end": 738.69, "word": " always", "probability": 0.900390625}, {"start": 738.69, "end": 739.09, "word": " has", "probability": 0.9423828125}, {"start": 739.09, "end": 739.37, "word": " mean", "probability": 0.94775390625}, {"start": 739.37, "end": 739.53, "word": " of", "probability": 0.970703125}, {"start": 739.53, "end": 739.81, "word": " zero", "probability": 0.88427734375}, {"start": 739.81, "end": 740.17, "word": " and", "probability": 0.94677734375}, {"start": 740.17, "end": 740.57, "word": " sigma", "probability": 0.91357421875}, {"start": 740.57, "end": 741.23, "word": " of", "probability": 0.87939453125}, {"start": 741.23, "end": 741.43, "word": " one.", "probability": 0.90673828125}, {"start": 741.97, "end": 742.21, "word": " So", "probability": 0.93212890625}, {"start": 742.21, "end": 742.51, "word": " that's", "probability": 0.937744140625}, {"start": 742.51, "end": 742.75, "word": " the", "probability": 0.9228515625}, {"start": 742.75, "end": 743.09, "word": " story", "probability": 0.93359375}, {"start": 743.09, "end": 743.67, "word": " of", "probability": 0.96923828125}, {"start": 743.67, "end": 745.49, "word": " standardizing", "probability": 0.955078125}, {"start": 745.49, "end": 746.83, "word": " the", "probability": 0.88916015625}, {"start": 746.83, "end": 747.23, "word": " normal", "probability": 0.85986328125}, {"start": 747.23, "end": 748.31, "word": " value.", "probability": 0.96337890625}, {"start": 749.81, "end": 750.47, "word": " Now", "probability": 0.94140625}, {"start": 750.47, "end": 750.93, "word": " the", "probability": 0.7900390625}], "temperature": 1.0}, {"id": 31, "seek": 76979, "start": 751.51, "end": 769.79, "text": " Formula for this score becomes better than the first one, but still we have to use calculus in order to determine the probabilities under the standardized normal k. But this distribution has mean of zero and sigma of one.", "tokens": [35872, 337, 341, 6175, 3643, 1101, 813, 264, 700, 472, 11, 457, 920, 321, 362, 281, 764, 33400, 294, 1668, 281, 6997, 264, 33783, 833, 264, 31677, 2710, 350, 13, 583, 341, 7316, 575, 914, 295, 4018, 293, 12771, 295, 472, 13], "avg_logprob": -0.2570857502693354, "compression_ratio": 1.4322580645161291, "no_speech_prob": 0.0, "words": [{"start": 751.51, "end": 752.13, "word": " Formula", "probability": 0.331298828125}, {"start": 752.13, "end": 752.47, "word": " for", "probability": 0.9091796875}, {"start": 752.47, "end": 752.71, "word": " this", "probability": 0.39501953125}, {"start": 752.71, "end": 753.07, "word": " score", "probability": 0.85400390625}, {"start": 753.07, "end": 753.81, "word": " becomes", "probability": 0.8408203125}, {"start": 753.81, "end": 754.61, "word": " better", "probability": 0.88232421875}, {"start": 754.61, "end": 754.91, "word": " than", "probability": 0.93994140625}, {"start": 754.91, "end": 755.13, "word": " the", "probability": 0.90087890625}, {"start": 755.13, "end": 755.45, "word": " first", "probability": 0.84716796875}, {"start": 755.45, "end": 755.77, "word": " one,", "probability": 0.9208984375}, {"start": 756.13, "end": 756.21, "word": " but", "probability": 0.88720703125}, {"start": 756.21, "end": 756.61, "word": " still", "probability": 0.93359375}, {"start": 756.61, "end": 757.57, "word": " we", "probability": 0.5576171875}, {"start": 757.57, "end": 757.77, "word": " have", "probability": 0.9365234375}, {"start": 757.77, "end": 757.91, "word": " to", "probability": 0.97021484375}, {"start": 757.91, "end": 758.09, "word": " use", "probability": 0.87744140625}, {"start": 758.09, "end": 758.45, "word": " calculus", "probability": 0.939453125}, {"start": 758.45, "end": 758.85, "word": " in", "probability": 0.91259765625}, {"start": 758.85, "end": 759.11, "word": " order", "probability": 0.93017578125}, {"start": 759.11, "end": 759.35, "word": " to", "probability": 0.9638671875}, {"start": 759.35, "end": 759.77, "word": " determine", "probability": 0.9140625}, {"start": 759.77, "end": 760.57, "word": " the", "probability": 0.78857421875}, {"start": 760.57, "end": 761.13, "word": " probabilities", "probability": 0.8603515625}, {"start": 761.13, "end": 761.53, "word": " under", "probability": 0.9072265625}, {"start": 761.53, "end": 761.79, "word": " the", "probability": 0.82080078125}, {"start": 761.79, "end": 762.23, "word": " standardized", "probability": 0.861328125}, {"start": 762.23, "end": 762.53, "word": " normal", "probability": 0.7080078125}, {"start": 762.53, "end": 762.81, "word": " k.", "probability": 0.1971435546875}, {"start": 765.05, "end": 765.71, "word": " But", "probability": 0.90673828125}, {"start": 765.71, "end": 766.21, "word": " this", "probability": 0.82666015625}, {"start": 766.21, "end": 766.85, "word": " distribution", "probability": 0.8564453125}, {"start": 766.85, "end": 768.21, "word": " has", "probability": 0.88623046875}, {"start": 768.21, "end": 768.37, "word": " mean", "probability": 0.904296875}, {"start": 768.37, "end": 768.51, "word": " of", "probability": 0.9580078125}, {"start": 768.51, "end": 768.73, "word": " zero", "probability": 0.51123046875}, {"start": 768.73, "end": 768.89, "word": " and", "probability": 0.9345703125}, {"start": 768.89, "end": 769.13, "word": " sigma", "probability": 0.87890625}, {"start": 769.13, "end": 769.47, "word": " of", "probability": 0.9111328125}, {"start": 769.47, "end": 769.79, "word": " one.", "probability": 0.77099609375}], "temperature": 1.0}, {"id": 32, "seek": 79547, "start": 770.87, "end": 795.47, "text": " So we have a table on page 570. Look at page 570. We have table or actually there are two tables. One for negative value of Z and the other for positive value of Z. So we have two tables for positive and negative values of Z on page 570 and 571.", "tokens": [407, 321, 362, 257, 3199, 322, 3028, 1025, 5867, 13, 2053, 412, 3028, 1025, 5867, 13, 492, 362, 3199, 420, 767, 456, 366, 732, 8020, 13, 1485, 337, 3671, 2158, 295, 1176, 293, 264, 661, 337, 3353, 2158, 295, 1176, 13, 407, 321, 362, 732, 8020, 337, 3353, 293, 3671, 4190, 295, 1176, 322, 3028, 1025, 5867, 293, 21423, 16, 13], "avg_logprob": -0.15612399457923828, "compression_ratio": 1.7697841726618706, "no_speech_prob": 0.0, "words": [{"start": 770.87, "end": 771.19, "word": " So", "probability": 0.92431640625}, {"start": 771.19, "end": 771.39, "word": " we", "probability": 0.7548828125}, {"start": 771.39, "end": 771.65, "word": " have", "probability": 0.9453125}, {"start": 771.65, "end": 771.93, "word": " a", "probability": 0.9853515625}, {"start": 771.93, "end": 772.23, "word": " table", "probability": 0.89501953125}, {"start": 772.23, "end": 773.17, "word": " on", "probability": 0.90966796875}, {"start": 773.17, "end": 773.45, "word": " page", "probability": 0.896484375}, {"start": 773.45, "end": 774.95, "word": " 570.", "probability": 0.924072265625}, {"start": 775.89, "end": 776.35, "word": " Look", "probability": 0.7822265625}, {"start": 776.35, "end": 776.59, "word": " at", "probability": 0.93798828125}, {"start": 776.59, "end": 776.91, "word": " page", "probability": 0.8974609375}, {"start": 776.91, "end": 777.65, "word": " 570.", "probability": 0.944091796875}, {"start": 778.25, "end": 778.53, "word": " We", "probability": 0.927734375}, {"start": 778.53, "end": 778.69, "word": " have", "probability": 0.9482421875}, {"start": 778.69, "end": 779.15, "word": " table", "probability": 0.55859375}, {"start": 779.15, "end": 779.87, "word": " or", "probability": 0.3955078125}, {"start": 779.87, "end": 780.43, "word": " actually", "probability": 0.84765625}, {"start": 780.43, "end": 780.63, "word": " there", "probability": 0.71875}, {"start": 780.63, "end": 780.75, "word": " are", "probability": 0.94189453125}, {"start": 780.75, "end": 780.91, "word": " two", "probability": 0.8857421875}, {"start": 780.91, "end": 781.37, "word": " tables.", "probability": 0.83349609375}, {"start": 781.71, "end": 781.99, "word": " One", "probability": 0.9287109375}, {"start": 781.99, "end": 782.29, "word": " for", "probability": 0.94677734375}, {"start": 782.29, "end": 782.87, "word": " negative", "probability": 0.93505859375}, {"start": 782.87, "end": 783.23, "word": " value", "probability": 0.81689453125}, {"start": 783.23, "end": 783.39, "word": " of", "probability": 0.94287109375}, {"start": 783.39, "end": 783.51, "word": " Z", "probability": 0.5078125}, {"start": 783.51, "end": 784.65, "word": " and", "probability": 0.71435546875}, {"start": 784.65, "end": 784.81, "word": " the", "probability": 0.7939453125}, {"start": 784.81, "end": 785.01, "word": " other", "probability": 0.8935546875}, {"start": 785.01, "end": 785.27, "word": " for", "probability": 0.94140625}, {"start": 785.27, "end": 785.67, "word": " positive", "probability": 0.92724609375}, {"start": 785.67, "end": 786.11, "word": " value", "probability": 0.73388671875}, {"start": 786.11, "end": 786.31, "word": " of", "probability": 0.89404296875}, {"start": 786.31, "end": 786.43, "word": " Z.", "probability": 0.98486328125}, {"start": 787.39, "end": 787.61, "word": " So", "probability": 0.947265625}, {"start": 787.61, "end": 787.73, "word": " we", "probability": 0.84619140625}, {"start": 787.73, "end": 787.81, "word": " have", "probability": 0.94677734375}, {"start": 787.81, "end": 787.99, "word": " two", "probability": 0.93115234375}, {"start": 787.99, "end": 788.45, "word": " tables", "probability": 0.837890625}, {"start": 788.45, "end": 788.83, "word": " for", "probability": 0.63671875}, {"start": 788.83, "end": 789.19, "word": " positive", "probability": 0.9287109375}, {"start": 789.19, "end": 789.99, "word": " and", "probability": 0.93408203125}, {"start": 789.99, "end": 790.97, "word": " negative", "probability": 0.93701171875}, {"start": 790.97, "end": 791.61, "word": " values", "probability": 0.95947265625}, {"start": 791.61, "end": 792.63, "word": " of", "probability": 0.96044921875}, {"start": 792.63, "end": 792.91, "word": " Z", "probability": 0.9814453125}, {"start": 792.91, "end": 793.25, "word": " on", "probability": 0.58056640625}, {"start": 793.25, "end": 793.47, "word": " page", "probability": 0.85205078125}, {"start": 793.47, "end": 794.55, "word": " 570", "probability": 0.906494140625}, {"start": 794.55, "end": 794.73, "word": " and", "probability": 0.93505859375}, {"start": 794.73, "end": 795.47, "word": " 571.", "probability": 0.8515625}], "temperature": 1.0}, {"id": 33, "seek": 82601, "start": 797.87, "end": 826.01, "text": " Now the table on page 570 looks like this one. The table you have starts from minus 6, then minus 5, minus 4.5, and so on. Here we start from minus 3.4 all the way down up to 0. Look here, all the way up to 0. So these scores here. Also we have 0.00, 0.01, up to 0.09.", "tokens": [823, 264, 3199, 322, 3028, 1025, 5867, 1542, 411, 341, 472, 13, 440, 3199, 291, 362, 3719, 490, 3175, 1386, 11, 550, 3175, 1025, 11, 3175, 1017, 13, 20, 11, 293, 370, 322, 13, 1692, 321, 722, 490, 3175, 805, 13, 19, 439, 264, 636, 760, 493, 281, 1958, 13, 2053, 510, 11, 439, 264, 636, 493, 281, 1958, 13, 407, 613, 13444, 510, 13, 2743, 321, 362, 1958, 13, 628, 11, 1958, 13, 10607, 11, 493, 281, 1958, 13, 13811, 13], "avg_logprob": -0.17912274311824017, "compression_ratio": 1.5823529411764705, "no_speech_prob": 0.0, "words": [{"start": 797.87, "end": 798.15, "word": " Now", "probability": 0.47900390625}, {"start": 798.15, "end": 798.31, "word": " the", "probability": 0.70703125}, {"start": 798.31, "end": 798.63, "word": " table", "probability": 0.88916015625}, {"start": 798.63, "end": 798.99, "word": " on", "probability": 0.8115234375}, {"start": 798.99, "end": 799.49, "word": " page", "probability": 0.8984375}, {"start": 799.49, "end": 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Here we have 0.0, 0.1, 0.2, all the way down up to 3.4 and you have up to 6. Now let's see how can we use this table to compute the probabilities underneath the normal curve. 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And the values could be positive or negative. Values above the mean, zero, have positive Z values. The other one, values below the mean, have negative Z values. 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Now this value could be positive if x is above the mean, as we mentioned before. It could be a negative if x is smaller than the mean or zero.", "tokens": [823, 341, 307, 264, 8513, 321, 362, 11, 710, 6915, 2031, 3175, 2992, 6666, 538, 2309, 13, 823, 341, 2158, 727, 312, 3353, 498, 2031, 307, 3673, 264, 914, 11, 382, 321, 2835, 949, 13, 467, 727, 312, 257, 3671, 498, 2031, 307, 4356, 813, 264, 914, 420, 4018, 13], "avg_logprob": -0.2182904411764706, "compression_ratio": 1.50354609929078, "no_speech_prob": 0.0, "words": [{"start": 881.97, "end": 882.33, "word": " Now", "probability": 0.72314453125}, {"start": 882.33, "end": 882.67, "word": " this", "probability": 0.58056640625}, {"start": 882.67, "end": 882.77, "word": " is", "probability": 0.9375}, {"start": 882.77, "end": 882.89, "word": " the", "probability": 0.90185546875}, {"start": 882.89, "end": 883.23, "word": " formula", "probability": 0.92919921875}, {"start": 883.23, "end": 883.43, "word": " we", "probability": 0.837890625}, {"start": 883.43, "end": 883.65, "word": " have,", "probability": 0.9130859375}, {"start": 884.05, "end": 884.17, "word": " z", "probability": 0.5703125}, {"start": 884.17, "end": 884.65, "word": " equals", "probability": 0.5419921875}, {"start": 884.65, "end": 885.65, "word": " x", "probability": 0.892578125}, {"start": 885.65, "end": 886.01, "word": " minus", "probability": 0.978515625}, {"start": 886.01, "end": 886.27, "word": " mu", "probability": 0.83154296875}, {"start": 886.27, "end": 886.53, "word": " divided", "probability": 0.7412109375}, {"start": 886.53, "end": 886.73, "word": " by", "probability": 0.974609375}, {"start": 886.73, "end": 886.99, "word": " six.", "probability": 0.4765625}, {"start": 892.81, "end": 893.35, "word": " Now", "probability": 0.916015625}, {"start": 893.35, "end": 893.61, "word": " this", "probability": 0.88427734375}, {"start": 893.61, "end": 893.99, "word": " value", "probability": 0.9658203125}, {"start": 893.99, "end": 895.07, "word": " could", "probability": 0.876953125}, {"start": 895.07, "end": 895.23, "word": " be", "probability": 0.9541015625}, {"start": 895.23, "end": 895.65, "word": " positive", "probability": 0.92333984375}, {"start": 895.65, "end": 897.57, "word": " if", "probability": 0.8388671875}, {"start": 897.57, "end": 899.11, "word": " x", "probability": 0.98486328125}, {"start": 899.11, "end": 900.71, "word": " is", "probability": 0.94873046875}, {"start": 900.71, "end": 900.97, "word": " above", "probability": 0.9541015625}, {"start": 900.97, "end": 901.17, "word": " the", "probability": 0.9326171875}, {"start": 901.17, "end": 901.33, "word": " mean,", "probability": 0.984375}, {"start": 901.95, "end": 902.89, "word": " as", "probability": 0.95263671875}, {"start": 902.89, "end": 903.01, "word": " we", "probability": 0.94677734375}, {"start": 903.01, "end": 903.31, "word": " mentioned", "probability": 0.82568359375}, {"start": 903.31, "end": 903.75, "word": " before.", "probability": 0.875}, {"start": 904.29, "end": 904.45, "word": " It", "probability": 0.6025390625}, {"start": 904.45, "end": 904.57, "word": " could", "probability": 0.88720703125}, {"start": 904.57, "end": 904.71, "word": " be", "probability": 0.95068359375}, {"start": 904.71, "end": 904.81, "word": " a", "probability": 0.59814453125}, {"start": 904.81, "end": 905.07, "word": " negative", "probability": 0.94580078125}, {"start": 905.07, "end": 907.55, "word": " if", "probability": 0.74853515625}, {"start": 907.55, "end": 907.89, "word": " x", "probability": 0.99267578125}, {"start": 907.89, "end": 908.07, "word": " is", "probability": 0.935546875}, {"start": 908.07, "end": 908.35, "word": " smaller", "probability": 0.88134765625}, {"start": 908.35, "end": 908.57, "word": " than", "probability": 0.947265625}, {"start": 908.57, "end": 908.73, "word": " the", "probability": 0.92578125}, {"start": 908.73, "end": 908.95, "word": " mean", "probability": 0.97412109375}, {"start": 908.95, "end": 909.51, "word": " or", "probability": 0.55517578125}, {"start": 909.51, "end": 909.87, "word": " zero.", "probability": 0.8203125}], "temperature": 1.0}, {"id": 37, "seek": 93678, "start": 913.12, "end": 936.78, "text": " Now the table we have gives the area to the right, to the left, I'm sorry, to the left, for positive and negative values of z. Okay, so we have two tables actually, one for negative on page 570, and the other one for positive values of z.", "tokens": [823, 264, 3199, 321, 362, 2709, 264, 1859, 281, 264, 558, 11, 281, 264, 1411, 11, 286, 478, 2597, 11, 281, 264, 1411, 11, 337, 3353, 293, 3671, 4190, 295, 710, 13, 1033, 11, 370, 321, 362, 732, 8020, 767, 11, 472, 337, 3671, 322, 3028, 1025, 5867, 11, 293, 264, 661, 472, 337, 3353, 4190, 295, 710, 13], "avg_logprob": -0.1996093784769376, "compression_ratio": 1.6482758620689655, "no_speech_prob": 0.0, "words": [{"start": 913.12, "end": 913.64, "word": " Now", "probability": 0.485107421875}, {"start": 913.64, "end": 914.16, "word": " the", "probability": 0.595703125}, {"start": 914.16, "end": 914.44, "word": " table", "probability": 0.82421875}, {"start": 914.44, "end": 914.62, "word": " we", "probability": 0.82861328125}, {"start": 914.62, "end": 915.02, "word": " have", "probability": 0.9462890625}, {"start": 915.02, "end": 917.06, "word": " gives", "probability": 0.71435546875}, {"start": 917.06, "end": 917.26, "word": " the", "probability": 0.8994140625}, {"start": 917.26, "end": 917.54, "word": " area", "probability": 0.798828125}, {"start": 917.54, "end": 917.78, "word": " to", "probability": 0.93212890625}, {"start": 917.78, "end": 917.92, "word": " the", "probability": 0.91943359375}, {"start": 917.92, "end": 918.14, "word": " right,", "probability": 0.802734375}, {"start": 918.42, "end": 919.02, "word": " to", "probability": 0.6083984375}, {"start": 919.02, "end": 919.12, "word": " the", "probability": 0.919921875}, {"start": 919.12, "end": 919.26, "word": " left,", "probability": 0.892578125}, {"start": 919.32, "end": 919.42, "word": " I'm", "probability": 0.90478515625}, {"start": 919.42, "end": 919.6, "word": " sorry,", "probability": 0.86865234375}, {"start": 919.72, "end": 919.82, "word": " to", "probability": 0.91015625}, {"start": 919.82, "end": 919.96, "word": " the", "probability": 0.92041015625}, {"start": 919.96, "end": 920.18, "word": " left,", "probability": 0.935546875}, {"start": 920.68, "end": 920.84, "word": " for", "probability": 0.94091796875}, {"start": 920.84, "end": 921.24, "word": " positive", "probability": 0.9169921875}, {"start": 921.24, "end": 921.5, "word": " and", "probability": 0.9384765625}, {"start": 921.5, "end": 921.74, "word": " negative", "probability": 0.93994140625}, {"start": 921.74, "end": 922.34, "word": " values", "probability": 0.96435546875}, {"start": 922.34, "end": 923.32, "word": " of", "probability": 0.96240234375}, {"start": 923.32, "end": 924.02, "word": " z.", "probability": 0.4736328125}, {"start": 924.86, "end": 925.38, "word": " Okay,", "probability": 0.65576171875}, {"start": 925.42, "end": 925.64, "word": " so", "probability": 0.94921875}, {"start": 925.64, "end": 925.86, "word": " we", "probability": 0.91845703125}, {"start": 925.86, "end": 926.04, "word": " have", "probability": 0.94482421875}, {"start": 926.04, "end": 926.22, "word": " two", "probability": 0.91845703125}, {"start": 926.22, "end": 926.56, "word": " tables", "probability": 0.81005859375}, {"start": 926.56, "end": 927.0, "word": " actually,", "probability": 0.79736328125}, {"start": 927.16, "end": 927.32, "word": " one", "probability": 0.923828125}, {"start": 927.32, "end": 927.56, "word": " for", "probability": 0.947265625}, {"start": 927.56, "end": 928.0, "word": " negative", "probability": 0.9296875}, {"start": 928.0, "end": 929.44, "word": " on", "probability": 0.673828125}, {"start": 929.44, "end": 929.66, "word": " page", "probability": 0.74560546875}, {"start": 929.66, "end": 930.38, "word": " 570,", "probability": 0.897705078125}, {"start": 931.04, "end": 932.16, "word": " and", "probability": 0.93310546875}, {"start": 932.16, "end": 932.34, "word": " the", "probability": 0.77587890625}, {"start": 932.34, "end": 932.58, "word": " other", "probability": 0.8896484375}, {"start": 932.58, "end": 932.9, "word": " one", "probability": 0.919921875}, {"start": 932.9, "end": 934.92, "word": " for", "probability": 0.8828125}, {"start": 934.92, "end": 935.28, "word": " positive", "probability": 0.93798828125}, {"start": 935.28, "end": 935.88, "word": " values", "probability": 0.966796875}, {"start": 935.88, "end": 936.54, "word": " of", "probability": 0.96044921875}, {"start": 936.54, "end": 936.78, "word": " z.", "probability": 0.97802734375}], "temperature": 1.0}, {"id": 38, "seek": 96424, "start": 937.74, "end": 964.24, "text": " I think we discussed that before when we talked about these scores. We have the same formula. Now let's look at this, the next slide. Suppose x is distributed normally with mean of 100. 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So sigma is 50. Now let's see how can we compute the z-score for x equals 200. Again the formula is just x minus mu divided by sigma x 200 minus 100 divided by 50 that will give 2. 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Now, what's the meaning of 2? What does this value tell you? Yeah, exactly. x equals 200 is two standard deviations above the mean. Because if you look at 200, the x value,", "tokens": [663, 1355, 2031, 307, 5044, 813, 264, 914, 11, 570, 2031, 307, 2331, 13, 823, 11, 437, 311, 264, 3620, 295, 568, 30, 708, 775, 341, 2158, 980, 291, 30, 865, 11, 2293, 13, 2031, 6915, 2331, 307, 732, 3832, 31219, 763, 3673, 264, 914, 13, 1436, 498, 291, 574, 412, 2331, 11, 264, 2031, 2158, 11], "avg_logprob": -0.19544719956044493, "compression_ratio": 1.4197530864197532, "no_speech_prob": 0.0, "words": [{"start": 991.39, "end": 991.77, "word": " That", "probability": 0.65625}, {"start": 991.77, "end": 992.21, "word": " means", "probability": 0.92822265625}, {"start": 992.21, "end": 993.47, "word": " x", "probability": 0.517578125}, {"start": 993.47, "end": 993.91, "word": " is", "probability": 0.904296875}, {"start": 993.91, "end": 994.25, "word": " greater", "probability": 0.82763671875}, {"start": 994.25, "end": 994.77, "word": " than", "probability": 0.953125}, {"start": 994.77, "end": 995.73, "word": " the", "probability": 0.79345703125}, {"start": 995.73, "end": 995.87, "word": " mean,", "probability": 0.97412109375}, {"start": 995.97, "end": 996.27, "word": " because", "probability": 0.89404296875}, {"start": 996.27, "end": 996.51, "word": " x", "probability": 0.88818359375}, {"start": 996.51, "end": 996.59, "word": " is", "probability": 0.84521484375}, {"start": 996.59, "end": 996.89, "word": " 200.", "probability": 0.779296875}, {"start": 997.35, "end": 997.95, "word": " Now,", "probability": 0.91015625}, {"start": 997.99, "end": 998.13, "word": " what's", "probability": 0.8984375}, {"start": 998.13, "end": 998.25, "word": " the", "probability": 0.9208984375}, {"start": 998.25, "end": 998.43, "word": " meaning", "probability": 0.85400390625}, {"start": 998.43, "end": 998.63, "word": " of", "probability": 0.96826171875}, {"start": 998.63, "end": 998.83, "word": " 2?", "probability": 0.58447265625}, {"start": 1000.53, "end": 1001.21, "word": " What", "probability": 0.8798828125}, {"start": 1001.21, "end": 1001.47, "word": " does", "probability": 0.97119140625}, {"start": 1001.47, "end": 1001.73, "word": " this", "probability": 0.94873046875}, {"start": 1001.73, "end": 1002.07, "word": " value", "probability": 0.9716796875}, {"start": 1002.07, "end": 1002.27, "word": " tell", "probability": 0.88525390625}, {"start": 1002.27, "end": 1002.41, "word": " you?", "probability": 0.9609375}, {"start": 1008.23, "end": 1008.91, "word": " Yeah,", "probability": 0.075439453125}, {"start": 1010.91, "end": 1011.39, "word": " exactly.", "probability": 0.87451171875}, {"start": 1011.51, "end": 1011.79, "word": " x", "probability": 0.79443359375}, {"start": 1011.79, "end": 1012.49, "word": " equals", "probability": 0.7626953125}, {"start": 1012.49, "end": 1013.09, "word": " 200", "probability": 0.91064453125}, {"start": 1013.09, "end": 1014.85, "word": " is", "probability": 0.740234375}, {"start": 1014.85, "end": 1015.07, "word": " two", "probability": 0.62939453125}, {"start": 1015.07, "end": 1015.43, "word": " standard", "probability": 0.93505859375}, {"start": 1015.43, "end": 1016.09, "word": " deviations", "probability": 0.9345703125}, {"start": 1016.09, "end": 1016.85, "word": " above", "probability": 0.943359375}, {"start": 1016.85, "end": 1017.05, "word": " the", "probability": 0.9287109375}, {"start": 1017.05, "end": 1017.23, "word": " mean.", "probability": 0.9677734375}, {"start": 1017.85, "end": 1018.19, "word": " Because", "probability": 0.92822265625}, {"start": 1018.19, "end": 1018.35, "word": " if", "probability": 0.87548828125}, {"start": 1018.35, "end": 1018.41, "word": " you", "probability": 0.9130859375}, {"start": 1018.41, "end": 1018.55, "word": " look", "probability": 0.962890625}, {"start": 1018.55, "end": 1018.69, "word": " at", "probability": 0.962890625}, {"start": 1018.69, "end": 1019.07, "word": " 200,", "probability": 0.9248046875}, {"start": 1019.65, "end": 1019.91, "word": " the", "probability": 0.8955078125}, {"start": 1019.91, "end": 1020.27, "word": " x", "probability": 0.98779296875}, {"start": 1020.27, "end": 1020.61, "word": " value,", "probability": 0.82861328125}], "temperature": 1.0}, {"id": 41, "seek": 104765, "start": 1022.03, "end": 1047.65, "text": " The mean is 100, sigma is 50. Now the difference between the score, which is 200, and the mu, which is 100, is equal to standard deviations, because the difference is 100. 2 times 50 is 100. So this says that x equals 200 is 2 standard deviations above the mean.", "tokens": [440, 914, 307, 2319, 11, 12771, 307, 2625, 13, 823, 264, 2649, 1296, 264, 6175, 11, 597, 307, 2331, 11, 293, 264, 2992, 11, 597, 307, 2319, 11, 307, 2681, 281, 3832, 31219, 763, 11, 570, 264, 2649, 307, 2319, 13, 568, 1413, 2625, 307, 2319, 13, 407, 341, 1619, 300, 2031, 6915, 2331, 307, 568, 3832, 31219, 763, 3673, 264, 914, 13], "avg_logprob": -0.22143555292859674, "compression_ratio": 1.6645569620253164, "no_speech_prob": 0.0, "words": [{"start": 1022.03, "end": 1022.33, "word": " The", "probability": 0.587890625}, {"start": 1022.33, "end": 1022.63, "word": " mean", "probability": 0.921875}, {"start": 1022.63, "end": 1023.41, "word": " is", "probability": 0.912109375}, {"start": 1023.41, "end": 1024.03, "word": " 100,", "probability": 0.407470703125}, {"start": 1024.45, "end": 1024.63, "word": " sigma", "probability": 0.62255859375}, {"start": 1024.63, "end": 1024.83, "word": " is", "probability": 0.93603515625}, {"start": 1024.83, "end": 1025.21, "word": " 50.", "probability": 0.9541015625}, {"start": 1025.73, "end": 1025.95, "word": " Now", "probability": 0.90966796875}, {"start": 1025.95, "end": 1026.11, "word": " the", "probability": 0.63427734375}, {"start": 1026.11, "end": 1026.49, "word": " difference", "probability": 0.8681640625}, {"start": 1026.49, "end": 1026.93, "word": " between", "probability": 0.87548828125}, {"start": 1026.93, "end": 1028.27, "word": " the", "probability": 0.7939453125}, {"start": 1028.27, "end": 1028.65, "word": " score,", "probability": 0.81201171875}, {"start": 1029.21, "end": 1029.49, "word": " which", "probability": 0.95361328125}, {"start": 1029.49, "end": 1029.69, "word": " is", "probability": 0.95068359375}, {"start": 1029.69, "end": 1030.09, "word": " 200,", "probability": 0.91015625}, {"start": 1030.37, "end": 1031.51, "word": " and", "probability": 0.9375}, {"start": 1031.51, "end": 1031.67, "word": " the", "probability": 0.8466796875}, {"start": 1031.67, "end": 1031.83, "word": " mu,", "probability": 0.5673828125}, {"start": 1032.07, "end": 1032.11, "word": " which", "probability": 0.955078125}, {"start": 1032.11, "end": 1032.49, "word": " is", "probability": 0.95361328125}, {"start": 1032.49, "end": 1033.25, "word": " 100,", "probability": 0.89697265625}, {"start": 1034.01, "end": 1035.13, "word": " is", "probability": 0.87646484375}, {"start": 1035.13, "end": 1036.15, "word": " equal", "probability": 0.89013671875}, {"start": 1036.15, "end": 1036.81, "word": " to", "probability": 0.7685546875}, {"start": 1036.81, "end": 1037.19, "word": " standard", "probability": 0.5546875}, {"start": 1037.19, "end": 1037.63, "word": " deviations,", "probability": 0.902587890625}, {"start": 1037.81, "end": 1038.07, "word": " because", "probability": 0.89697265625}, {"start": 1038.07, "end": 1038.33, "word": " the", "probability": 0.65576171875}, {"start": 1038.33, "end": 1038.57, "word": " difference", "probability": 0.8818359375}, {"start": 1038.57, "end": 1038.69, "word": " is", "probability": 0.7470703125}, {"start": 1038.69, "end": 1039.03, "word": " 100.", "probability": 0.900390625}, {"start": 1039.67, "end": 1039.91, "word": " 2", "probability": 0.611328125}, {"start": 1039.91, "end": 1040.17, "word": " times", "probability": 0.82958984375}, {"start": 1040.17, "end": 1040.57, "word": " 50", "probability": 0.96826171875}, {"start": 1040.57, "end": 1040.75, "word": " is", "probability": 0.93896484375}, {"start": 1040.75, "end": 1041.03, "word": " 100.", "probability": 0.92041015625}, {"start": 1042.15, "end": 1042.73, "word": " So", "probability": 0.9345703125}, {"start": 1042.73, "end": 1042.97, "word": " this", "probability": 0.8564453125}, {"start": 1042.97, "end": 1043.29, "word": " says", "probability": 0.86865234375}, {"start": 1043.29, "end": 1043.63, "word": " that", "probability": 0.92724609375}, {"start": 1043.63, "end": 1043.93, "word": " x", "probability": 0.77197265625}, {"start": 1043.93, "end": 1044.23, "word": " equals", "probability": 0.80712890625}, {"start": 1044.23, "end": 1044.63, "word": " 200", "probability": 0.7119140625}, {"start": 1044.63, "end": 1044.91, "word": " is", "probability": 0.88037109375}, {"start": 1044.91, "end": 1045.09, "word": " 2", "probability": 0.5576171875}, {"start": 1045.09, "end": 1045.43, "word": " standard", "probability": 0.93212890625}, {"start": 1045.43, "end": 1046.07, "word": " deviations", "probability": 0.958740234375}, {"start": 1046.07, "end": 1047.25, "word": " above", "probability": 0.93115234375}, {"start": 1047.25, "end": 1047.49, "word": " the", "probability": 0.93359375}, {"start": 1047.49, "end": 1047.65, "word": " mean.", "probability": 0.97705078125}], "temperature": 1.0}, {"id": 42, "seek": 107245, "start": 1048.53, "end": 1072.45, "text": " If z is negative, you can say that x is two standard deviations below them. Make sense? So that's how can we compute the z square. Now, when we transform from normal distribution to standardized, still we will have the same shape. I mean the distribution is still normally distributed.", "tokens": [759, 710, 307, 3671, 11, 291, 393, 584, 300, 2031, 307, 732, 3832, 31219, 763, 2507, 552, 13, 4387, 2020, 30, 407, 300, 311, 577, 393, 321, 14722, 264, 710, 3732, 13, 823, 11, 562, 321, 4088, 490, 2710, 7316, 281, 31677, 11, 920, 321, 486, 362, 264, 912, 3909, 13, 286, 914, 264, 7316, 307, 920, 5646, 12631, 13], "avg_logprob": -0.18378585577011108, "compression_ratio": 1.5376344086021505, "no_speech_prob": 0.0, "words": [{"start": 1048.53, "end": 1048.89, "word": " If", "probability": 0.8212890625}, {"start": 1048.89, "end": 1049.07, "word": " z", "probability": 0.6025390625}, {"start": 1049.07, "end": 1049.21, "word": " is", "probability": 0.9482421875}, {"start": 1049.21, "end": 1049.55, "word": " negative,", "probability": 0.94189453125}, {"start": 1049.93, "end": 1050.05, "word": " you", "probability": 0.9423828125}, {"start": 1050.05, "end": 1050.27, "word": " can", "probability": 0.94384765625}, {"start": 1050.27, "end": 1050.53, "word": " say", "probability": 0.9169921875}, {"start": 1050.53, "end": 1051.11, "word": " that", "probability": 0.92529296875}, {"start": 1051.11, "end": 1052.89, "word": " x", "probability": 0.81396484375}, {"start": 1052.89, "end": 1053.33, "word": " is", "probability": 0.79052734375}, {"start": 1053.33, "end": 1053.97, "word": " two", "probability": 0.806640625}, {"start": 1053.97, "end": 1054.33, "word": " standard", "probability": 0.95458984375}, {"start": 1054.33, "end": 1054.87, "word": " deviations", "probability": 0.93701171875}, {"start": 1054.87, "end": 1055.63, "word": " below", "probability": 0.85693359375}, {"start": 1055.63, "end": 1056.19, "word": " them.", "probability": 0.61328125}, {"start": 1056.71, "end": 1056.97, "word": " Make", "probability": 0.65185546875}, {"start": 1056.97, "end": 1057.29, "word": " sense?", "probability": 0.82373046875}, {"start": 1057.65, "end": 1058.31, "word": " So", "probability": 0.92529296875}, {"start": 1058.31, "end": 1058.59, "word": " that's", "probability": 0.873046875}, {"start": 1058.59, "end": 1058.71, "word": " how", "probability": 0.9365234375}, {"start": 1058.71, "end": 1058.89, "word": " can", "probability": 0.5107421875}, {"start": 1058.89, "end": 1059.03, "word": " we", "probability": 0.93603515625}, {"start": 1059.03, "end": 1059.55, "word": " compute", "probability": 0.9052734375}, {"start": 1059.55, "end": 1060.17, "word": " the", "probability": 0.6162109375}, {"start": 1060.17, "end": 1060.35, "word": " z", "probability": 0.439453125}, {"start": 1060.35, "end": 1060.63, "word": " square.", "probability": 0.32763671875}, {"start": 1061.35, "end": 1061.71, "word": " Now,", "probability": 0.93896484375}, {"start": 1062.17, "end": 1062.45, "word": " when", "probability": 0.94091796875}, {"start": 1062.45, "end": 1062.67, "word": " we", "probability": 0.9638671875}, {"start": 1062.67, "end": 1063.35, "word": " transform", "probability": 0.9072265625}, {"start": 1063.35, "end": 1063.67, "word": " from", "probability": 0.8701171875}, {"start": 1063.67, "end": 1064.07, "word": " normal", "probability": 0.85546875}, {"start": 1064.07, "end": 1064.79, "word": " distribution", "probability": 0.83447265625}, {"start": 1064.79, "end": 1065.97, "word": " to", "probability": 0.900390625}, {"start": 1065.97, "end": 1066.39, "word": " standardized,", "probability": 0.74853515625}, {"start": 1067.27, "end": 1067.69, "word": " still", "probability": 0.9267578125}, {"start": 1067.69, "end": 1067.89, "word": " we", "probability": 0.83203125}, {"start": 1067.89, "end": 1068.05, "word": " will", "probability": 0.88037109375}, {"start": 1068.05, "end": 1068.31, "word": " have", "probability": 0.94482421875}, {"start": 1068.31, "end": 1068.49, "word": " the", "probability": 0.91357421875}, {"start": 1068.49, "end": 1068.79, "word": " same", "probability": 0.900390625}, {"start": 1068.79, "end": 1069.17, "word": " shape.", "probability": 0.919921875}, {"start": 1069.47, "end": 1069.49, "word": " I", "probability": 0.982421875}, {"start": 1069.49, "end": 1069.65, "word": " mean", "probability": 0.96728515625}, {"start": 1069.65, "end": 1069.87, "word": " the", "probability": 0.440673828125}, {"start": 1069.87, "end": 1070.45, "word": " distribution", "probability": 0.8662109375}, {"start": 1070.45, "end": 1070.69, "word": " is", "probability": 0.91455078125}, {"start": 1070.69, "end": 1070.91, "word": " still", "probability": 0.95556640625}, {"start": 1070.91, "end": 1071.35, "word": " normally", "probability": 0.90234375}, {"start": 1071.35, "end": 1072.45, "word": " distributed.", "probability": 0.912109375}], "temperature": 1.0}, {"id": 43, "seek": 109716, "start": 1073.92, "end": 1097.16, "text": " So note, the shape of the distribution is the same, only the scale has changed. So we can express the problem in original units, X, or in a standardized unit, Z. So when we have X, just use this equation to transform to this form.", "tokens": [407, 3637, 11, 264, 3909, 295, 264, 7316, 307, 264, 912, 11, 787, 264, 4373, 575, 3105, 13, 407, 321, 393, 5109, 264, 1154, 294, 3380, 6815, 11, 1783, 11, 420, 294, 257, 31677, 4985, 11, 1176, 13, 407, 562, 321, 362, 1783, 11, 445, 764, 341, 5367, 281, 4088, 281, 341, 1254, 13], "avg_logprob": -0.19090909416025334, "compression_ratio": 1.5, "no_speech_prob": 0.0, "words": [{"start": 1073.92, "end": 1074.18, "word": " So", "probability": 0.88232421875}, {"start": 1074.18, "end": 1074.42, "word": " note,", "probability": 0.53125}, {"start": 1075.06, "end": 1075.2, "word": " the", "probability": 0.90869140625}, {"start": 1075.2, "end": 1075.52, "word": " shape", "probability": 0.90625}, {"start": 1075.52, "end": 1075.7, "word": " of", "probability": 0.9658203125}, {"start": 1075.7, "end": 1075.8, "word": " the", "probability": 0.779296875}, {"start": 1075.8, "end": 1076.3, "word": " distribution", "probability": 0.8115234375}, {"start": 1076.3, "end": 1076.52, "word": " is", "probability": 0.88720703125}, {"start": 1076.52, "end": 1076.7, "word": " the", "probability": 0.9140625}, {"start": 1076.7, "end": 1077.0, "word": " same,", "probability": 0.9091796875}, {"start": 1077.6, "end": 1077.9, "word": " only", "probability": 0.9248046875}, {"start": 1077.9, "end": 1078.16, "word": " the", "probability": 0.9169921875}, {"start": 1078.16, "end": 1078.5, "word": " scale", "probability": 0.828125}, {"start": 1078.5, "end": 1078.84, "word": " has", "probability": 0.93017578125}, {"start": 1078.84, "end": 1079.3, "word": " changed.", "probability": 0.90478515625}, {"start": 1081.06, "end": 1081.76, "word": " So", "probability": 0.94677734375}, {"start": 1081.76, "end": 1081.9, "word": " we", "probability": 0.8623046875}, {"start": 1081.9, "end": 1082.08, "word": " can", "probability": 0.939453125}, {"start": 1082.08, "end": 1082.38, "word": " express", "probability": 0.90869140625}, {"start": 1082.38, "end": 1082.74, "word": " the", "probability": 0.91552734375}, {"start": 1082.74, "end": 1083.1, "word": " problem", "probability": 0.87646484375}, {"start": 1083.1, "end": 1083.5, "word": " in", "probability": 0.92626953125}, {"start": 1083.5, "end": 1084.5, "word": " original", "probability": 0.7890625}, {"start": 1084.5, "end": 1085.04, "word": " units,", "probability": 0.87060546875}, {"start": 1085.24, "end": 1085.48, "word": " X,", "probability": 0.56298828125}, {"start": 1086.22, "end": 1086.6, "word": " or", "probability": 0.962890625}, {"start": 1086.6, "end": 1086.82, "word": " in", "probability": 0.9306640625}, {"start": 1086.82, "end": 1086.94, "word": " a", "probability": 0.46826171875}, {"start": 1086.94, "end": 1087.28, "word": " standardized", "probability": 0.91748046875}, {"start": 1087.28, "end": 1088.5, "word": " unit,", "probability": 0.93408203125}, {"start": 1088.94, "end": 1089.22, "word": " Z.", "probability": 0.90771484375}, {"start": 1089.62, "end": 1090.0, "word": " So", "probability": 0.96044921875}, {"start": 1090.0, "end": 1090.36, "word": " when", "probability": 0.84033203125}, {"start": 1090.36, "end": 1090.64, "word": " we", "probability": 0.95947265625}, {"start": 1090.64, "end": 1091.04, "word": " have", "probability": 0.94873046875}, {"start": 1091.04, "end": 1091.44, "word": " X,", "probability": 0.9443359375}, {"start": 1092.06, "end": 1092.46, "word": " just", "probability": 0.912109375}, {"start": 1092.46, "end": 1092.8, "word": " use", "probability": 0.86669921875}, {"start": 1092.8, "end": 1093.12, "word": " this", "probability": 0.94873046875}, {"start": 1093.12, "end": 1093.68, "word": " equation", "probability": 0.966796875}, {"start": 1093.68, "end": 1095.02, "word": " to", "probability": 0.68115234375}, {"start": 1095.02, "end": 1096.38, "word": " transform", "probability": 0.91552734375}, {"start": 1096.38, "end": 1096.62, "word": " to", "probability": 0.75439453125}, {"start": 1096.62, "end": 1096.8, "word": " this", "probability": 0.3095703125}, {"start": 1096.8, "end": 1097.16, "word": " form.", "probability": 0.84814453125}], "temperature": 1.0}, {"id": 44, "seek": 112942, "start": 1101.36, "end": 1129.42, "text": " Now, for example, suppose we have normal distribution and we are interested in the area between A and B. 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But for discrete, it does matter. 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I mean probability of X ranges from minus infinity up to infinity equals one. So probability of X greater than minus infinity up to infinity is one. The total area is one. 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And also you have to keep in mind that the probability always ranges between zero and one. So that means the probability couldn't be negative. It should be positive. It shouldn't be greater than one. So it's between zero and one. 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For negative or positive z's. 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Let's go back to this normal distribution. In the second page, we have positive z-scores. So we ask about the probability of z less than. 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And the table gives the area below. And he asked about here, B of z is smaller than 2. Now 2, if you hear, up all the way down here, 2, 0, 0. So the answer is 9772. So this value, so the probability is 9772.", "tokens": [2709, 3353, 4190, 295, 710, 13, 400, 264, 3199, 2709, 264, 1859, 2507, 13, 400, 415, 2351, 466, 510, 11, 363, 295, 710, 307, 4356, 813, 568, 13, 823, 568, 11, 498, 291, 1568, 11, 493, 439, 264, 636, 760, 510, 11, 568, 11, 1958, 11, 1958, 13, 407, 264, 1867, 307, 1722, 17512, 17, 13, 407, 341, 2158, 11, 370, 264, 8482, 307, 1722, 17512, 17, 13], "avg_logprob": -0.27558876293292944, "compression_ratio": 1.475, "no_speech_prob": 0.0, "words": [{"start": 1355.99, "end": 1356.55, "word": " gives", "probability": 0.325927734375}, {"start": 1356.55, "end": 1357.11, "word": " positive", "probability": 0.89306640625}, {"start": 1357.11, "end": 1357.47, "word": " values", "probability": 0.8857421875}, {"start": 1357.47, "end": 1357.65, "word": " of", "probability": 0.927734375}, {"start": 1357.65, "end": 1357.79, "word": " z.", "probability": 0.71044921875}, {"start": 1360.13, "end": 1360.69, "word": " And", "probability": 0.9189453125}, {"start": 1360.69, "end": 1360.85, "word": " the", "probability": 0.88037109375}, {"start": 1360.85, "end": 1361.11, "word": " table", "probability": 0.830078125}, {"start": 1361.11, "end": 1361.39, "word": " gives", "probability": 0.91748046875}, {"start": 1361.39, "end": 1361.53, "word": " the", "probability": 0.900390625}, {"start": 1361.53, "end": 1361.77, "word": " area", "probability": 0.88671875}, {"start": 1361.77, "end": 1362.13, "word": " below.", "probability": 0.888671875}, {"start": 1363.49, "end": 1363.85, "word": " And", "probability": 0.91162109375}, {"start": 1363.85, "end": 1363.99, "word": " he", "probability": 0.84375}, {"start": 1363.99, "end": 1364.23, "word": " asked", "probability": 0.4853515625}, {"start": 1364.23, "end": 1364.59, "word": " about", "probability": 0.8837890625}, {"start": 1364.59, "end": 1364.99, "word": " here,", "probability": 0.69140625}, {"start": 1365.29, "end": 1365.51, "word": " B", "probability": 0.46923828125}, {"start": 1365.51, "end": 1365.67, "word": " of", "probability": 0.90283203125}, {"start": 1365.67, "end": 1365.83, "word": " z", "probability": 0.83544921875}, {"start": 1365.83, "end": 1366.07, "word": " is", "probability": 0.86083984375}, {"start": 1366.07, "end": 1366.37, "word": " smaller", "probability": 0.87744140625}, {"start": 1366.37, "end": 1366.65, "word": " than", "probability": 0.93896484375}, {"start": 1366.65, "end": 1366.89, "word": " 2.", "probability": 0.7275390625}, {"start": 1368.43, "end": 1368.85, "word": " Now", "probability": 0.923828125}, {"start": 1368.85, "end": 1369.21, "word": " 2,", "probability": 0.50927734375}, {"start": 1369.31, "end": 1369.43, "word": " if", "probability": 0.8544921875}, {"start": 1369.43, "end": 1369.55, "word": " you", "probability": 0.65478515625}, {"start": 1369.55, "end": 1369.89, "word": " hear,", "probability": 0.31298828125}, {"start": 1370.17, "end": 1370.51, "word": " up", "probability": 0.77978515625}, {"start": 1370.51, "end": 1370.81, "word": " all", "probability": 0.91748046875}, {"start": 1370.81, "end": 1370.95, "word": " the", "probability": 0.91650390625}, {"start": 1370.95, "end": 1371.11, "word": " way", "probability": 0.95263671875}, {"start": 1371.11, "end": 1371.41, "word": " down", "probability": 0.8408203125}, {"start": 1371.41, "end": 1371.65, "word": " here,", "probability": 0.82763671875}, {"start": 1371.77, "end": 1371.99, "word": " 2,", "probability": 0.78955078125}, {"start": 1372.33, "end": 1373.31, "word": " 0,", "probability": 0.83984375}, {"start": 1373.39, "end": 1373.67, "word": " 0.", "probability": 0.923828125}, {"start": 1374.31, "end": 1374.77, "word": " So", "probability": 0.951171875}, {"start": 1374.77, "end": 1374.91, "word": " the", "probability": 0.8583984375}, {"start": 1374.91, "end": 1375.17, "word": " answer", "probability": 0.95849609375}, {"start": 1375.17, "end": 1375.37, "word": " is", "probability": 0.94091796875}, {"start": 1375.37, "end": 1376.53, "word": " 9772.", "probability": 0.78564453125}, {"start": 1377.17, "end": 1377.37, "word": " So", "probability": 0.87255859375}, {"start": 1377.37, "end": 1377.61, "word": " this", "probability": 0.81787109375}, {"start": 1377.61, "end": 1377.99, "word": " value,", "probability": 0.9541015625}, {"start": 1379.09, "end": 1379.97, "word": " so", "probability": 0.68603515625}, {"start": 1379.97, "end": 1380.13, "word": " the", "probability": 0.9150390625}, {"start": 1380.13, "end": 1380.53, "word": " probability", "probability": 0.96337890625}, {"start": 1380.53, "end": 1380.95, "word": " is", "probability": 0.94091796875}, {"start": 1380.95, "end": 1382.13, "word": " 9772.", "probability": 0.8429361979166666}], "temperature": 1.0}, {"id": 54, "seek": 141037, "start": 1383.99, "end": 1410.37, "text": " Because it's 2. It's 2, 0, 0. But if you ask about what's the probability of Z less than 2.05? So this is 2. Now under 5, 9, 7, 9, 8. So the answer is 9, 7.", "tokens": [1436, 309, 311, 568, 13, 467, 311, 568, 11, 1958, 11, 1958, 13, 583, 498, 291, 1029, 466, 437, 311, 264, 8482, 295, 1176, 1570, 813, 568, 13, 13328, 30, 407, 341, 307, 568, 13, 823, 833, 1025, 11, 1722, 11, 1614, 11, 1722, 11, 1649, 13, 407, 264, 1867, 307, 1722, 11, 1614, 13], "avg_logprob": -0.29101562553218435, "compression_ratio": 1.2265625, "no_speech_prob": 0.0, "words": [{"start": 1383.99, "end": 1384.47, "word": " Because", "probability": 0.3837890625}, {"start": 1384.47, "end": 1384.95, "word": " it's", "probability": 0.897705078125}, {"start": 1384.95, "end": 1385.39, "word": " 2.", "probability": 0.4619140625}, {"start": 1389.51, "end": 1389.99, "word": " It's", "probability": 0.667236328125}, {"start": 1389.99, "end": 1390.19, "word": " 2,", "probability": 0.61474609375}, {"start": 1390.27, "end": 1390.49, "word": " 0,", "probability": 0.7578125}, {"start": 1390.55, "end": 1390.79, "word": " 0.", "probability": 0.98388671875}, {"start": 1391.75, "end": 1392.07, "word": " But", "probability": 0.92041015625}, {"start": 1392.07, "end": 1392.19, "word": " if", "probability": 0.83984375}, {"start": 1392.19, "end": 1392.23, "word": " you", "probability": 0.90869140625}, {"start": 1392.23, "end": 1392.45, "word": " ask", "probability": 0.9287109375}, {"start": 1392.45, "end": 1392.87, "word": " about", "probability": 0.90673828125}, {"start": 1392.87, "end": 1394.59, "word": " what's", "probability": 0.84326171875}, {"start": 1394.59, "end": 1394.65, "word": " the", "probability": 0.560546875}, {"start": 1394.65, "end": 1394.89, "word": " probability", "probability": 0.97802734375}, {"start": 1394.89, "end": 1395.21, "word": " of", "probability": 0.93896484375}, {"start": 1395.21, "end": 1395.45, "word": " Z", "probability": 0.64404296875}, {"start": 1395.45, "end": 1396.33, "word": " less", "probability": 0.86474609375}, {"start": 1396.33, "end": 1396.55, "word": " than", "probability": 0.94580078125}, {"start": 1396.55, "end": 1396.79, "word": " 2", "probability": 0.98046875}, {"start": 1396.79, "end": 1397.47, "word": ".05?", "probability": 0.965576171875}, {"start": 1399.47, "end": 1399.95, "word": " So", "probability": 0.2352294921875}, {"start": 1399.95, "end": 1400.19, "word": " this", "probability": 0.80712890625}, {"start": 1400.19, "end": 1400.39, "word": " is", "probability": 0.94677734375}, {"start": 1400.39, "end": 1400.59, "word": " 2.", "probability": 0.77783203125}, {"start": 1403.81, "end": 1404.29, "word": " Now", "probability": 0.236083984375}, {"start": 1404.29, "end": 1404.65, "word": " under", "probability": 0.5791015625}, {"start": 1404.65, "end": 1405.07, "word": " 5,", "probability": 0.87890625}, {"start": 1407.43, "end": 1407.69, "word": " 9,", "probability": 0.3798828125}, {"start": 1407.81, "end": 1408.09, "word": " 7,", "probability": 0.8076171875}, {"start": 1408.27, "end": 1408.71, "word": " 9,", "probability": 0.994140625}, {"start": 1408.83, "end": 1408.99, "word": " 8.", "probability": 0.9873046875}, {"start": 1409.15, "end": 1409.27, "word": " So", "probability": 0.91845703125}, {"start": 1409.27, "end": 1409.43, "word": " the", "probability": 0.89111328125}, {"start": 1409.43, "end": 1409.65, "word": " answer", "probability": 0.95361328125}, {"start": 1409.65, "end": 1409.87, "word": " is", "probability": 0.94580078125}, {"start": 1409.87, "end": 1410.09, "word": " 9,", "probability": 0.9423828125}, {"start": 1410.17, "end": 1410.37, "word": " 7.", "probability": 0.89697265625}], "temperature": 1.0}, {"id": 55, "seek": 143688, "start": 1414.36, "end": 1436.88, "text": " Because this is two, and we need five decimal places. So all the way up to 9798. So this value is 2.05. Now it's about, it's more than 1.5, exactly 1.5.", "tokens": [1436, 341, 307, 732, 11, 293, 321, 643, 1732, 26601, 3190, 13, 407, 439, 264, 636, 493, 281, 23399, 22516, 13, 407, 341, 2158, 307, 568, 13, 13328, 13, 823, 309, 311, 466, 11, 309, 311, 544, 813, 502, 13, 20, 11, 2293, 502, 13, 20, 13], "avg_logprob": -0.23681640811264515, "compression_ratio": 1.1953125, "no_speech_prob": 0.0, "words": [{"start": 1414.3600000000001, "end": 1415.16, "word": " Because", "probability": 0.2059326171875}, {"start": 1415.16, "end": 1415.96, "word": " this", "probability": 0.9267578125}, {"start": 1415.96, "end": 1416.1, "word": " is", "probability": 0.9453125}, {"start": 1416.1, "end": 1416.3, "word": " two,", "probability": 0.51708984375}, {"start": 1417.2, "end": 1417.68, "word": " and", "probability": 0.9296875}, {"start": 1417.68, "end": 1417.86, "word": " we", "probability": 0.955078125}, {"start": 1417.86, "end": 1418.14, "word": " need", "probability": 0.92041015625}, {"start": 1418.14, "end": 1418.56, "word": " five", "probability": 0.8896484375}, {"start": 1418.56, "end": 1418.9, "word": " decimal", "probability": 0.55029296875}, {"start": 1418.9, "end": 1419.32, "word": " places.", "probability": 0.96337890625}, {"start": 1420.74, "end": 1421.54, "word": " So", "probability": 0.919921875}, {"start": 1421.54, "end": 1421.8, "word": " all", "probability": 0.744140625}, {"start": 1421.8, "end": 1421.9, "word": " the", "probability": 0.91748046875}, {"start": 1421.9, "end": 1422.04, "word": " way", "probability": 0.95849609375}, {"start": 1422.04, "end": 1422.2, "word": " up", "probability": 0.9541015625}, {"start": 1422.2, "end": 1422.36, "word": " to", "probability": 0.96826171875}, {"start": 1422.36, "end": 1423.26, "word": " 9798.", "probability": 0.47412109375}, {"start": 1423.92, "end": 1424.18, "word": " So", "probability": 0.95068359375}, {"start": 1424.18, "end": 1424.42, "word": " this", "probability": 0.90771484375}, {"start": 1424.42, "end": 1424.82, "word": " value", "probability": 0.97607421875}, {"start": 1424.82, "end": 1427.24, "word": " is", "probability": 0.873046875}, {"start": 1427.24, "end": 1427.48, "word": " 2", "probability": 0.984375}, {"start": 1427.48, "end": 1428.96, "word": ".05.", "probability": 0.993408203125}, {"start": 1429.62, "end": 1429.92, "word": " Now", "probability": 0.5556640625}, {"start": 1429.92, "end": 1430.1, "word": " it's", "probability": 0.7288818359375}, {"start": 1430.1, "end": 1430.4, "word": " about,", "probability": 0.8828125}, {"start": 1431.98, "end": 1433.08, "word": " it's", "probability": 0.8642578125}, {"start": 1433.08, "end": 1433.32, "word": " more", "probability": 0.91162109375}, {"start": 1433.32, "end": 1433.6, "word": " than", "probability": 0.9482421875}, {"start": 1433.6, "end": 1433.82, "word": " 1", "probability": 0.9912109375}, {"start": 1433.82, "end": 1434.38, "word": ".5,", "probability": 0.991455078125}, {"start": 1435.6, "end": 1436.14, "word": " exactly", "probability": 0.90380859375}, {"start": 1436.14, "end": 1436.38, "word": " 1", "probability": 0.990234375}, {"start": 1436.38, "end": 1436.88, "word": ".5.", "probability": 0.9990234375}], "temperature": 1.0}, {"id": 56, "seek": 146320, "start": 1442.14, "end": 1463.2, "text": " 1.5. This is 1.5. 9332. 1.5. Exactly 1.5. So 9332. What's about probability less than 1.35?", "tokens": [502, 13, 20, 13, 639, 307, 502, 13, 20, 13, 1722, 10191, 17, 13, 502, 13, 20, 13, 7587, 502, 13, 20, 13, 407, 1722, 10191, 17, 13, 708, 311, 466, 8482, 1570, 813, 502, 13, 8794, 30], "avg_logprob": -0.24699519383601654, "compression_ratio": 1.15, "no_speech_prob": 0.0, "words": [{"start": 1442.1399999999999, "end": 1443.06, "word": " 1", "probability": 0.245849609375}, {"start": 1443.06, "end": 1443.68, "word": ".5.", "probability": 0.953125}, {"start": 1443.82, "end": 1444.08, "word": " This", "probability": 0.274169921875}, {"start": 1444.08, "end": 1444.2, "word": " is", "probability": 0.93408203125}, {"start": 1444.2, "end": 1444.38, "word": " 1", "probability": 0.978515625}, {"start": 1444.38, "end": 1444.88, "word": ".5.", "probability": 0.995849609375}, {"start": 1448.8, "end": 1449.72, "word": " 9332.", "probability": 0.90087890625}, {"start": 1452.44, "end": 1453.36, "word": " 1", "probability": 0.798828125}, {"start": 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with mean mu and standard deviation of sigma and let's see how can we compute the value of the probability mainly there are three steps to find the probability of x greater than a and less than b when x is distributed normally first step", "tokens": [498, 321, 362, 2031, 293, 2031, 300, 575, 2710, 7316, 365, 914, 2992, 293, 3832, 25163, 295, 12771, 293, 718, 311, 536, 577, 393, 321, 14722, 264, 2158, 295, 264, 8482, 8704, 456, 366, 1045, 4439, 281, 915, 264, 8482, 295, 2031, 5044, 813, 257, 293, 1570, 813, 272, 562, 2031, 307, 12631, 5646, 700, 1823], "avg_logprob": -0.14350328738229318, "compression_ratio": 1.672514619883041, "no_speech_prob": 0.0, "words": [{"start": 1575.45, "end": 1575.87, "word": " if", "probability": 0.357666015625}, {"start": 1575.87, "end": 1576.13, "word": " we", "probability": 0.9501953125}, {"start": 1576.13, "end": 1576.39, "word": " have", "probability": 0.94775390625}, {"start": 1576.39, "end": 1576.81, "word": " x", "probability": 0.55615234375}, 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So draw the normal curve first. Second, translate x values to z values by using the formula we have. z x minus mu divided by sigma. Then use the standardized normal table on page 570 and 571. For example,", "tokens": [20386, 2710, 7605, 337, 264, 1154, 294, 2115, 295, 2031, 13, 407, 2642, 264, 2710, 7605, 700, 13, 5736, 11, 13799, 2031, 4190, 281, 710, 4190, 538, 1228, 264, 8513, 321, 362, 13, 710, 2031, 3175, 2992, 6666, 538, 12771, 13, 1396, 764, 264, 31677, 2710, 3199, 322, 3028, 1025, 5867, 293, 21423, 16, 13, 1171, 1365, 11], "avg_logprob": -0.21729343775975501, "compression_ratio": 1.4350282485875707, "no_speech_prob": 0.0, "words": [{"start": 1605.64, "end": 1606.1, "word": " Draw", "probability": 0.5107421875}, {"start": 1606.1, "end": 1606.62, "word": " normal", "probability": 0.420166015625}, {"start": 1606.62, "end": 1607.0, "word": " curve", "probability": 0.9296875}, {"start": 1607.0, "end": 1609.94, "word": " for", "probability": 0.5029296875}, {"start": 1609.94, "end": 1610.1, "word": " the", "probability": 0.849609375}, {"start": 1610.1, "end": 1610.48, "word": " problem", "probability": 0.86767578125}, {"start": 1610.48, "end": 1611.06, "word": " in", "probability": 0.91796875}, {"start": 1611.06, "end": 1611.34, "word": " terms", "probability": 0.896484375}, {"start": 1611.34, "end": 1611.48, "word": " of", "probability": 0.96875}, {"start": 1611.48, "end": 1611.82, "word": " x.", "probability": 0.5185546875}, {"start": 1613.98, "end": 1614.22, "word": " So", "probability": 0.73828125}, {"start": 1614.22, "end": 1614.42, "word": " draw", "probability": 0.5537109375}, {"start": 1614.42, "end": 1614.64, "word": " the", "probability": 0.85400390625}, {"start": 1614.64, "end": 1614.88, "word": " normal", "probability": 0.87841796875}, {"start": 1614.88, "end": 1615.22, "word": " curve", "probability": 0.900390625}, {"start": 1615.22, "end": 1615.54, "word": " first.", "probability": 0.7763671875}, {"start": 1615.94, "end": 1616.34, "word": " Second,", "probability": 0.70947265625}, {"start": 1616.5, "end": 1617.06, "word": " translate", "probability": 0.8154296875}, {"start": 1617.06, "end": 1617.38, "word": " x", "probability": 0.94287109375}, {"start": 1617.38, "end": 1617.78, "word": " values", "probability": 0.642578125}, {"start": 1617.78, "end": 1617.96, "word": " to", "probability": 0.93017578125}, {"start": 1617.96, "end": 1618.14, "word": " z", "probability": 0.97509765625}, {"start": 1618.14, "end": 1618.56, "word": " values", "probability": 0.9619140625}, {"start": 1618.56, "end": 1619.2, "word": " by", "probability": 0.85302734375}, {"start": 1619.2, "end": 1619.54, "word": " using", "probability": 0.927734375}, {"start": 1619.54, "end": 1619.76, "word": " the", "probability": 0.91455078125}, {"start": 1619.76, "end": 1620.12, "word": " formula", "probability": 0.91064453125}, {"start": 1620.12, "end": 1620.32, "word": " we", "probability": 0.92041015625}, {"start": 1620.32, "end": 1620.62, "word": " have.", "probability": 0.9375}, {"start": 1621.94, "end": 1622.22, "word": " z", "probability": 0.64306640625}, {"start": 1622.22, "end": 1622.48, "word": " x", "probability": 0.55126953125}, {"start": 1622.48, "end": 1622.82, "word": " minus", "probability": 0.94970703125}, {"start": 1622.82, "end": 1623.04, "word": " mu", "probability": 0.85107421875}, {"start": 1623.04, "end": 1623.24, "word": " divided", "probability": 0.77001953125}, {"start": 1623.24, "end": 1623.44, "word": " by", "probability": 0.9755859375}, {"start": 1623.44, "end": 1623.68, "word": " sigma.", "probability": 0.8994140625}, {"start": 1624.54, "end": 1624.96, "word": " Then", "probability": 0.8603515625}, {"start": 1624.96, "end": 1625.28, "word": " use", "probability": 0.72900390625}, {"start": 1625.28, "end": 1625.5, "word": " the", "probability": 0.89892578125}, {"start": 1625.5, "end": 1626.08, "word": " standardized", "probability": 0.84033203125}, {"start": 1626.08, "end": 1626.44, "word": " normal", "probability": 0.87939453125}, {"start": 1626.44, "end": 1626.92, "word": " table", "probability": 0.84228515625}, {"start": 1626.92, "end": 1627.58, "word": " on", "probability": 0.92333984375}, {"start": 1627.58, "end": 1627.9, "word": " page", "probability": 0.8486328125}, {"start": 1627.9, "end": 1628.6, "word": " 570", "probability": 0.8955078125}, {"start": 1628.6, "end": 1628.9, "word": " and", "probability": 0.93115234375}, {"start": 1628.9, "end": 1630.58, "word": " 571.", "probability": 0.830810546875}, {"start": 1631.4, "end": 1631.76, "word": " For", "probability": 0.9482421875}, {"start": 1631.76, "end": 1632.1, "word": " example,", "probability": 0.96533203125}], "temperature": 1.0}, {"id": 63, "seek": 166310, "start": 1634.7, "end": 1663.1, "text": " Let's see how can we find normal probabilities. Let's assume that X represents the time it takes to download an image from the internet. So suppose X, time required to download an image file from the internet. And suppose we know that the time is normally distributed for with mean of eight minutes.", "tokens": [961, 311, 536, 577, 393, 321, 915, 2710, 33783, 13, 961, 311, 6552, 300, 1783, 8855, 264, 565, 309, 2516, 281, 5484, 364, 3256, 490, 264, 4705, 13, 407, 7297, 1783, 11, 565, 4739, 281, 5484, 364, 3256, 3991, 490, 264, 4705, 13, 400, 7297, 321, 458, 300, 264, 565, 307, 5646, 12631, 337, 365, 914, 295, 3180, 2077, 13], "avg_logprob": -0.17802253609797994, "compression_ratio": 1.675977653631285, "no_speech_prob": 0.0, "words": [{"start": 1634.7, "end": 1635.04, "word": " Let's", "probability": 0.821533203125}, {"start": 1635.04, "end": 1635.14, "word": " see", "probability": 0.9033203125}, {"start": 1635.14, "end": 1635.24, "word": " how", "probability": 0.90576171875}, {"start": 1635.24, "end": 1635.44, "word": " can", "probability": 0.6318359375}, {"start": 1635.44, "end": 1635.56, "word": " we", "probability": 0.94580078125}, {"start": 1635.56, "end": 1635.92, "word": " find", 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He asked about probability of X smaller than 8.6. So we are interested in the area below 8.6. So it matched the table we have. 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So just straightforward calculation, 8.6 is your value of x. 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Still, we have the same normal curve. 8, the mean. Now, the mean of z is 0, as we mentioned. 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probability of X smaller than 8.6, smaller than 1.12, so they are equivalent. So we transform here from normal distribution to standardized normal distribution in order to compute the probability we are looking for. Now, this is just a portion of the table we have.", "tokens": [407, 2602, 295, 5006, 8482, 295, 1783, 4356, 813, 1649, 13, 21, 11, 4356, 813, 502, 13, 4762, 11, 370, 436, 366, 10344, 13, 407, 321, 4088, 510, 490, 2710, 7316, 281, 31677, 2710, 7316, 294, 1668, 281, 14722, 264, 8482, 321, 366, 1237, 337, 13, 823, 11, 341, 307, 445, 257, 8044, 295, 264, 3199, 321, 362, 13], "avg_logprob": -0.16145832985639572, "compression_ratio": 1.5567567567567568, "no_speech_prob": 0.0, "words": [{"start": 1781.14, "end": 1781.46, "word": " So", "probability": 0.90380859375}, {"start": 1781.46, "end": 1781.88, "word": " instead", "probability": 0.7392578125}, {"start": 1781.88, "end": 1782.18, "word": " of", "probability": 0.96728515625}, {"start": 1782.18, "end": 1782.58, "word": " finding", "probability": 0.8642578125}, {"start": 1782.58, "end": 1783.0, "word": " probability", "probability": 0.61962890625}, {"start": 1783.0, "end": 1783.3, "word": " of", "probability": 0.91455078125}, {"start": 1783.3, "end": 1783.56, "word": " X", "probability": 0.775390625}, {"start": 1783.56, "end": 1784.1, "word": " smaller", "probability": 0.78369140625}, {"start": 1784.1, "end": 1784.38, "word": " than", "probability": 0.9423828125}, {"start": 1784.38, "end": 1784.58, "word": " 8", "probability": 0.923828125}, {"start": 1784.58, "end": 1785.2, "word": ".6,", "probability": 0.994140625}, {"start": 1785.72, "end": 1786.74, "word": " smaller", "probability": 0.52001953125}, {"start": 1786.74, "end": 1787.22, "word": " than", "probability": 0.92724609375}, {"start": 1787.22, "end": 1787.56, "word": " 1", "probability": 0.70849609375}, {"start": 1787.56, "end": 1788.06, "word": ".12,", "probability": 0.77197265625}, {"start": 1788.24, "end": 1788.44, "word": " so", "probability": 0.9140625}, {"start": 1788.44, "end": 1788.58, "word": " they", "probability": 0.880859375}, {"start": 1788.58, "end": 1788.72, "word": " are", "probability": 0.93310546875}, {"start": 1788.72, "end": 1789.06, "word": " equivalent.", "probability": 0.912109375}, {"start": 1789.88, "end": 1790.08, "word": " So", "probability": 0.95654296875}, {"start": 1790.08, "end": 1790.24, "word": " we", "probability": 0.73583984375}, {"start": 1790.24, "end": 1791.0, "word": " transform", "probability": 0.767578125}, {"start": 1791.0, "end": 1792.22, "word": " here", "probability": 0.80126953125}, {"start": 1792.22, "end": 1792.8, "word": " from", "probability": 0.8359375}, {"start": 1792.8, "end": 1793.76, "word": " normal", "probability": 0.84326171875}, {"start": 1793.76, "end": 1794.42, "word": " distribution", "probability": 0.845703125}, {"start": 1794.42, "end": 1795.5, "word": " to", "probability": 0.8876953125}, {"start": 1795.5, "end": 1796.02, "word": " standardized", "probability": 0.6337890625}, {"start": 1796.02, "end": 1796.38, "word": " normal", "probability": 0.87890625}, {"start": 1796.38, "end": 1796.98, "word": " distribution", "probability": 0.8671875}, {"start": 1796.98, "end": 1797.2, "word": " in", "probability": 0.7724609375}, {"start": 1797.2, "end": 1797.42, "word": " order", "probability": 0.91796875}, {"start": 1797.42, "end": 1797.64, "word": " to", "probability": 0.97021484375}, {"start": 1797.64, "end": 1798.04, "word": " compute", "probability": 0.9111328125}, {"start": 1798.04, "end": 1798.72, "word": " the", "probability": 0.9130859375}, {"start": 1798.72, "end": 1799.1, "word": " probability", "probability": 0.9638671875}, {"start": 1799.1, "end": 1799.58, "word": " we", "probability": 0.9482421875}, {"start": 1799.58, "end": 1799.74, "word": " are", "probability": 0.9345703125}, {"start": 1799.74, "end": 1799.98, "word": " looking", "probability": 0.916015625}, {"start": 1799.98, "end": 1800.34, "word": " for.", "probability": 0.95166015625}, {"start": 1801.28, "end": 1801.58, "word": " Now,", "probability": 0.95556640625}, {"start": 1801.76, "end": 1802.02, "word": " this", "probability": 0.943359375}, {"start": 1802.02, "end": 1802.22, "word": " is", "probability": 0.94873046875}, {"start": 1802.22, "end": 1802.54, "word": " just", "probability": 0.9189453125}, {"start": 1802.54, "end": 1802.76, "word": " a", "probability": 0.9560546875}, {"start": 1802.76, "end": 1803.12, "word": " portion", "probability": 0.85791015625}, {"start": 1803.12, "end": 1804.88, "word": " of", "probability": 0.9560546875}, {"start": 1804.88, "end": 1805.16, "word": " the", "probability": 0.91552734375}, {"start": 1805.16, "end": 1805.6, "word": " table", "probability": 0.8828125}, {"start": 1805.6, "end": 1805.82, "word": " we", "probability": 0.84619140625}, {"start": 1805.82, "end": 1806.1, "word": " have.", "probability": 0.94091796875}], "temperature": 1.0}, {"id": 69, "seek": 183819, "start": 1810.53, "end": 1838.19, "text": " So for positive z values. Now 0.1 is 0.1. Because here we are looking for z less than 0.1. So 0.1. Also, we have two. So move up to two decimal places, we get this value. So the answer is point.", "tokens": [407, 337, 3353, 710, 4190, 13, 823, 1958, 13, 16, 307, 1958, 13, 16, 13, 1436, 510, 321, 366, 1237, 337, 710, 1570, 813, 1958, 13, 16, 13, 407, 1958, 13, 16, 13, 2743, 11, 321, 362, 732, 13, 407, 1286, 493, 281, 732, 26601, 3190, 11, 321, 483, 341, 2158, 13, 407, 264, 1867, 307, 935, 13], "avg_logprob": -0.24218750151537233, "compression_ratio": 1.3928571428571428, "no_speech_prob": 0.0, "words": [{"start": 1810.53, "end": 1810.83, "word": " So", "probability": 0.63427734375}, {"start": 1810.83, "end": 1811.05, "word": " for", "probability": 0.69482421875}, {"start": 1811.05, "end": 1811.39, "word": " positive", "probability": 0.6318359375}, {"start": 1811.39, "end": 1811.59, "word": " z", "probability": 0.611328125}, {"start": 1811.59, "end": 1811.97, "word": " values.", "probability": 0.7041015625}, {"start": 1812.97, "end": 1813.27, "word": " Now", "probability": 0.87939453125}, {"start": 1813.27, "end": 1813.53, "word": " 0", "probability": 0.260009765625}, {"start": 1813.53, "end": 1813.93, "word": ".1", "probability": 0.99462890625}, {"start": 1813.93, "end": 1815.47, "word": " is", "probability": 0.32666015625}, {"start": 1815.47, "end": 1815.67, "word": " 0", "probability": 0.9052734375}, {"start": 1815.67, "end": 1816.03, "word": ".1.", "probability": 0.998046875}, {"start": 1817.85, "end": 1818.53, "word": " Because", "probability": 0.87890625}, {"start": 1818.53, "end": 1818.89, "word": " here", "probability": 0.841796875}, {"start": 1818.89, "end": 1819.03, "word": " we", "probability": 0.8095703125}, {"start": 1819.03, "end": 1819.17, "word": " are", "probability": 0.92919921875}, {"start": 1819.17, "end": 1819.45, "word": " looking", "probability": 0.91455078125}, {"start": 1819.45, "end": 1819.81, "word": " for", "probability": 0.95166015625}, {"start": 1819.81, "end": 1821.23, "word": " z", "probability": 0.935546875}, {"start": 1821.23, "end": 1821.51, "word": " less", "probability": 0.88623046875}, {"start": 1821.51, "end": 1821.69, "word": " than", "probability": 0.92578125}, {"start": 1821.69, "end": 1821.95, "word": " 0", "probability": 0.90869140625}, {"start": 1821.95, "end": 1822.19, "word": ".1.", "probability": 0.988525390625}, {"start": 1824.43, "end": 1825.07, "word": " So", "probability": 0.92041015625}, {"start": 1825.07, "end": 1825.33, "word": " 0", "probability": 0.88916015625}, {"start": 1825.33, "end": 1825.67, "word": ".1.", "probability": 0.9990234375}, {"start": 1827.21, "end": 1827.89, "word": " Also,", "probability": 0.90185546875}, {"start": 1827.99, "end": 1828.05, "word": " we", "probability": 0.8701171875}, {"start": 1828.05, "end": 1828.25, "word": " have", "probability": 0.94873046875}, {"start": 1828.25, "end": 1828.59, "word": " two.", "probability": 0.5576171875}, {"start": 1829.37, "end": 1829.61, "word": " So", "probability": 0.9052734375}, {"start": 1829.61, "end": 1831.15, "word": " move", "probability": 0.720703125}, {"start": 1831.15, "end": 1831.39, "word": " up", "probability": 0.96142578125}, {"start": 1831.39, "end": 1831.51, "word": " to", "probability": 0.9169921875}, {"start": 1831.51, "end": 1832.57, "word": " two", "probability": 0.73876953125}, {"start": 1832.57, "end": 1832.95, "word": " decimal", "probability": 0.80810546875}, {"start": 1832.95, "end": 1833.33, "word": " places,", "probability": 0.560546875}, {"start": 1833.93, "end": 1834.29, "word": " we", "probability": 0.93505859375}, {"start": 1834.29, "end": 1834.65, "word": " get", "probability": 0.9287109375}, {"start": 1834.65, "end": 1835.71, "word": " this", "probability": 0.9365234375}, {"start": 1835.71, "end": 1835.99, "word": " value.", "probability": 0.9423828125}, {"start": 1836.79, "end": 1837.09, "word": " So", "probability": 0.95654296875}, {"start": 1837.09, "end": 1837.25, "word": " the", "probability": 0.916015625}, {"start": 1837.25, "end": 1837.55, "word": " answer", "probability": 0.95703125}, {"start": 1837.55, "end": 1837.81, "word": " is", "probability": 0.94384765625}, {"start": 1837.81, "end": 1838.19, "word": " point.", "probability": 0.5634765625}], "temperature": 1.0}, {"id": 70, "seek": 186502, "start": 1842.12, "end": 1865.02, "text": " I think it's straightforward to compute the probability underneath the normal curve if X has normal distribution. So B of X is smaller than 8.6 is the same as B of Z less than 0.12, which is around 55%. Makes sense because the area to the left of 0 equals 1 half.", "tokens": [286, 519, 309, 311, 15325, 281, 14722, 264, 8482, 7223, 264, 2710, 7605, 498, 1783, 575, 2710, 7316, 13, 407, 363, 295, 1783, 307, 4356, 813, 1649, 13, 21, 307, 264, 912, 382, 363, 295, 1176, 1570, 813, 1958, 13, 4762, 11, 597, 307, 926, 12330, 6856, 25245, 2020, 570, 264, 1859, 281, 264, 1411, 295, 1958, 6915, 502, 1922, 13], "avg_logprob": -0.23538306980363785, "compression_ratio": 1.382198952879581, "no_speech_prob": 0.0, "words": [{"start": 1842.1200000000001, "end": 1842.72, "word": " I", "probability": 0.23779296875}, {"start": 1842.72, "end": 1843.32, "word": " think", "probability": 0.90478515625}, {"start": 1843.32, "end": 1843.5, "word": " it's", "probability": 0.632568359375}, {"start": 1843.5, "end": 1843.98, "word": " straightforward", "probability": 0.83154296875}, {"start": 1843.98, "end": 1845.2, "word": " to", "probability": 0.75537109375}, {"start": 1845.2, "end": 1845.56, "word": " compute", 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189254, "start": 1866.26, "end": 1892.54, "text": " But we are looking for the area below 0.12. So greater than zero. So this area actually is greater than 0.5. So it makes sense that your result is greater than 0.5. Questions? Next, suppose we are interested of probability of X greater than.", "tokens": [583, 321, 366, 1237, 337, 264, 1859, 2507, 1958, 13, 4762, 13, 407, 5044, 813, 4018, 13, 407, 341, 1859, 767, 307, 5044, 813, 1958, 13, 20, 13, 407, 309, 1669, 2020, 300, 428, 1874, 307, 5044, 813, 1958, 13, 20, 13, 27738, 30, 3087, 11, 7297, 321, 366, 3102, 295, 8482, 295, 1783, 5044, 813, 13], "avg_logprob": -0.1926185319135929, "compression_ratio": 1.5612903225806452, "no_speech_prob": 0.0, "words": [{"start": 1866.26, "end": 1866.42, "word": " But", "probability": 0.3984375}, {"start": 1866.42, "end": 1866.52, "word": " we", "probability": 0.80078125}, {"start": 1866.52, "end": 1866.62, "word": " are", "probability": 0.88525390625}, {"start": 1866.62, "end": 1866.84, "word": " looking", "probability": 0.91357421875}, {"start": 1866.84, "end": 1867.08, "word": " for", "probability": 0.94775390625}, {"start": 1867.08, "end": 1867.2, "word": " the", "probability": 0.904296875}, {"start": 1867.2, "end": 1867.58, "word": " area", "probability": 0.89501953125}, {"start": 1867.58, "end": 1868.76, "word": " below", "probability": 0.84521484375}, {"start": 1868.76, "end": 1869.26, "word": " 0", "probability": 0.2308349609375}, {"start": 1869.26, "end": 1869.84, "word": ".12.", "probability": 0.984130859375}, {"start": 1870.2, "end": 1870.84, "word": " So", "probability": 0.51171875}, {"start": 1870.84, "end": 1871.1, "word": " greater", "probability": 0.81005859375}, {"start": 1871.1, "end": 1871.42, "word": " than", "probability": 0.9423828125}, {"start": 1871.42, "end": 1871.72, "word": " zero.", "probability": 0.61865234375}, {"start": 1872.04, "end": 1872.18, "word": " So", "probability": 0.9404296875}, {"start": 1872.18, "end": 1872.44, "word": " this", "probability": 0.89453125}, {"start": 1872.44, "end": 1872.82, "word": " area", "probability": 0.890625}, {"start": 1872.82, "end": 1873.7, "word": " actually", "probability": 0.83154296875}, {"start": 1873.7, "end": 1873.94, "word": " is", "probability": 0.93994140625}, {"start": 1873.94, "end": 1874.26, "word": " greater", "probability": 0.83642578125}, {"start": 1874.26, "end": 1874.76, "word": " than", "probability": 0.93896484375}, {"start": 1874.76, "end": 1875.5, "word": " 0", "probability": 0.9306640625}, {"start": 1875.5, "end": 1875.84, "word": ".5.", "probability": 0.989990234375}, {"start": 1876.04, "end": 1876.24, "word": " So", "probability": 0.95263671875}, {"start": 1876.24, "end": 1876.36, "word": " it", "probability": 0.90966796875}, {"start": 1876.36, "end": 1876.6, "word": " makes", "probability": 0.81787109375}, {"start": 1876.6, "end": 1876.84, "word": " sense", "probability": 0.81494140625}, {"start": 1876.84, "end": 1877.1, "word": " that", "probability": 0.93017578125}, {"start": 1877.1, "end": 1877.3, "word": " your", "probability": 0.8154296875}, {"start": 1877.3, "end": 1877.7, "word": " result", "probability": 0.93408203125}, {"start": 1877.7, "end": 1879.54, "word": " is", "probability": 0.8896484375}, {"start": 1879.54, "end": 1879.8, "word": " greater", "probability": 0.8876953125}, {"start": 1879.8, "end": 1880.04, "word": " than", "probability": 0.93701171875}, {"start": 1880.04, "end": 1880.26, "word": " 0", "probability": 0.6845703125}, {"start": 1880.26, "end": 1880.44, "word": ".5.", "probability": 0.989990234375}, {"start": 1882.32, "end": 1882.96, "word": " Questions?", "probability": 0.79248046875}, {"start": 1885.48, "end": 1886.12, "word": " Next,", "probability": 0.88134765625}, {"start": 1887.16, "end": 1887.66, "word": " suppose", "probability": 0.9072265625}, {"start": 1887.66, "end": 1888.46, "word": " we", "probability": 0.94287109375}, {"start": 1888.46, "end": 1888.74, "word": " are", "probability": 0.94091796875}, {"start": 1888.74, "end": 1889.3, "word": " interested", "probability": 0.84814453125}, {"start": 1889.3, "end": 1890.12, "word": " of", "probability": 0.58154296875}, {"start": 1890.12, "end": 1890.5, "word": " probability", "probability": 0.6279296875}, {"start": 1890.5, "end": 1890.78, "word": " of", "probability": 0.92822265625}, {"start": 1890.78, "end": 1891.1, "word": " X", "probability": 0.88623046875}, {"start": 1891.1, "end": 1892.22, "word": " greater", "probability": 0.8798828125}, {"start": 1892.22, "end": 1892.54, "word": " than.", "probability": 0.947265625}], "temperature": 1.0}, {"id": 72, "seek": 191722, "start": 1893.14, "end": 1917.22, "text": " So that's how can we find normal upper tail probabilities. Again, the table we have gives the area to the left. In order to compute the area in the upper tail probabilities, I mean this area, since the normal distribution is symmetric and", "tokens": [407, 300, 311, 577, 393, 321, 915, 2710, 6597, 6838, 33783, 13, 3764, 11, 264, 3199, 321, 362, 2709, 264, 1859, 281, 264, 1411, 13, 682, 1668, 281, 14722, 264, 1859, 294, 264, 6597, 6838, 33783, 11, 286, 914, 341, 1859, 11, 1670, 264, 2710, 7316, 307, 32330, 293], "avg_logprob": -0.1706250062584877, "compression_ratio": 1.5620915032679739, "no_speech_prob": 0.0, "words": [{"start": 1893.14, "end": 1893.38, "word": " So", "probability": 0.9052734375}, {"start": 1893.38, "end": 1893.78, "word": " that's", "probability": 0.89697265625}, {"start": 1893.78, "end": 1893.96, "word": " how", "probability": 0.88720703125}, {"start": 1893.96, "end": 1894.18, "word": " can", "probability": 0.81884765625}, {"start": 1894.18, "end": 1894.34, "word": " we", "probability": 0.9326171875}, {"start": 1894.34, "end": 1894.68, "word": " find", "probability": 0.88134765625}, {"start": 1894.68, "end": 1895.38, "word": " normal", "probability": 0.82666015625}, {"start": 1895.38, "end": 1895.78, "word": " upper", "probability": 0.75048828125}, {"start": 1895.78, "end": 1896.12, "word": " tail", "probability": 0.475830078125}, {"start": 1896.12, "end": 1896.66, "word": " probabilities.", "probability": 0.8740234375}, {"start": 1898.24, "end": 1898.8, "word": " Again,", "probability": 0.9287109375}, {"start": 1900.88, "end": 1901.1, "word": " the", "probability": 0.8330078125}, {"start": 1901.1, "end": 1901.42, "word": " table", "probability": 0.86328125}, {"start": 1901.42, "end": 1901.64, "word": " we", "probability": 0.91845703125}, {"start": 1901.64, "end": 1901.98, "word": " have", "probability": 0.94580078125}, {"start": 1901.98, "end": 1902.52, "word": " gives", "probability": 0.7421875}, {"start": 1902.52, "end": 1902.66, "word": " the", "probability": 0.669921875}, {"start": 1902.66, "end": 1902.8, "word": " area", "probability": 0.85107421875}, {"start": 1902.8, "end": 1902.98, "word": " to", "probability": 0.962890625}, {"start": 1902.98, "end": 1903.1, "word": " the", "probability": 0.9140625}, {"start": 1903.1, "end": 1903.32, "word": " left.", "probability": 0.953125}, {"start": 1904.9, "end": 1905.42, "word": " In", "probability": 0.9677734375}, {"start": 1905.42, "end": 1905.62, "word": " order", "probability": 0.92138671875}, {"start": 1905.62, "end": 1905.96, "word": " to", "probability": 0.9716796875}, {"start": 1905.96, "end": 1906.58, "word": " compute", "probability": 0.91015625}, {"start": 1906.58, "end": 1906.78, "word": " the", "probability": 0.91357421875}, {"start": 1906.78, "end": 1907.1, "word": " area", "probability": 0.89697265625}, {"start": 1907.1, "end": 1907.8, "word": " in", "probability": 0.9130859375}, {"start": 1907.8, "end": 1907.96, "word": " the", "probability": 0.9267578125}, {"start": 1907.96, "end": 1908.24, "word": " upper", "probability": 0.80908203125}, {"start": 1908.24, "end": 1908.6, "word": " tail", "probability": 0.75146484375}, {"start": 1908.6, "end": 1909.4, "word": " probabilities,", "probability": 0.91357421875}, {"start": 1910.44, "end": 1910.78, "word": " I", "probability": 0.9833984375}, {"start": 1910.78, "end": 1910.88, "word": " mean", "probability": 0.70703125}, {"start": 1910.88, "end": 1911.14, "word": " this", "probability": 0.91162109375}, {"start": 1911.14, "end": 1911.48, "word": " area,", "probability": 0.88671875}, {"start": 1912.86, "end": 1913.28, "word": " since", "probability": 0.7138671875}, {"start": 1913.28, "end": 1913.48, "word": " the", "probability": 0.91064453125}, {"start": 1913.48, "end": 1913.76, "word": " normal", "probability": 0.87060546875}, {"start": 1913.76, "end": 1914.4, "word": " distribution", "probability": 0.8603515625}, {"start": 1914.4, "end": 1915.62, "word": " is", "probability": 0.92041015625}, {"start": 1915.62, "end": 1916.02, "word": " symmetric", "probability": 0.8173828125}, {"start": 1916.02, "end": 1917.22, "word": " and", "probability": 0.441162109375}], "temperature": 1.0}, {"id": 73, "seek": 194382, "start": 1917.74, "end": 1943.82, "text": " The total area underneath the curve is 1. So the probability of X greater than 8.6 is the same as 1 minus B of X less than 8.6. So first step, just find the probability we just have and subtract from 1. 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It's 1 minus the result we got from previous one. So this value 1 minus this value gives 0.452. So for the other tail probability, just subtract 1 from the lower tail probabilities. Now let's see how can we find", "tokens": [597, 307, 264, 912, 382, 502, 3175, 363, 295, 1176, 1570, 813, 1958, 13, 20, 13, 467, 311, 502, 3175, 264, 1874, 321, 658, 490, 3894, 472, 13, 407, 341, 2158, 502, 3175, 341, 2158, 2709, 1958, 13, 8465, 17, 13, 407, 337, 264, 661, 6838, 8482, 11, 445, 16390, 502, 490, 264, 3126, 6838, 33783, 13, 823, 718, 311, 536, 577, 393, 321, 915], "avg_logprob": -0.21969697127739587, "compression_ratio": 1.5202312138728324, "no_speech_prob": 0.0, "words": [{"start": 1945.23, "end": 1945.67, "word": " which", "probability": 0.51123046875}, {"start": 1945.67, "end": 1945.79, "word": " is", "probability": 0.94970703125}, {"start": 1945.79, "end": 1945.93, "word": " the", "probability": 0.87353515625}, {"start": 1945.93, "end": 1946.11, "word": " same", "probability": 0.91796875}, {"start": 1946.11, "end": 1946.35, "word": " as", "probability": 0.958984375}, {"start": 1946.35, "end": 1946.61, "word": " 1", "probability": 0.5029296875}, {"start": 1946.61, 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I mean if X, for example, for the same data we have, suppose X between 8 and 8.6. Now what's the area between these two?", "tokens": [21277, 8482, 1296, 732, 4190, 13, 286, 914, 498, 1783, 11, 337, 1365, 11, 337, 264, 912, 1412, 321, 362, 11, 7297, 1783, 1296, 1649, 293, 1649, 13, 21, 13, 823, 437, 311, 264, 1859, 1296, 613, 732, 30], "avg_logprob": -0.19511719085276127, "compression_ratio": 1.3114754098360655, "no_speech_prob": 0.0, "words": [{"start": 1974.83, "end": 1975.33, "word": " Normal", "probability": 0.41015625}, {"start": 1975.33, "end": 1975.75, "word": " probability", "probability": 0.939453125}, {"start": 1975.75, "end": 1976.73, "word": " between", "probability": 0.8818359375}, {"start": 1976.73, "end": 1976.99, "word": " two", "probability": 0.91845703125}, {"start": 1976.99, "end": 1977.39, "word": " values.", "probability": 0.97119140625}, {"start": 1979.11, "end": 1979.17, "word": " I", "probability": 0.88818359375}, {"start": 1979.17, "end": 1979.35, "word": " mean", "probability": 0.96923828125}, {"start": 1979.35, "end": 1979.55, "word": " if", "probability": 0.59033203125}, {"start": 1979.55, "end": 1980.07, "word": " X,", "probability": 0.60498046875}, {"start": 1980.77, "end": 1981.09, "word": " for", "probability": 0.95556640625}, {"start": 1981.09, "end": 1981.47, "word": " example,", "probability": 0.953125}, {"start": 1981.63, "end": 1981.75, "word": " for", "probability": 0.92333984375}, {"start": 1981.75, "end": 1981.93, "word": " the", "probability": 0.921875}, {"start": 1981.93, "end": 1982.25, "word": " same", "probability": 0.900390625}, {"start": 1982.25, "end": 1982.65, "word": " data", "probability": 0.95654296875}, {"start": 1982.65, "end": 1982.81, "word": " we", "probability": 0.9208984375}, {"start": 1982.81, "end": 1983.09, "word": " have,", "probability": 0.9453125}, {"start": 1983.85, "end": 1984.19, "word": " suppose", "probability": 0.884765625}, {"start": 1984.19, "end": 1984.65, "word": " X", "probability": 0.96533203125}, {"start": 1984.65, "end": 1985.77, "word": " between", "probability": 0.8212890625}, {"start": 1985.77, "end": 1986.11, "word": " 8", "probability": 0.59814453125}, {"start": 1986.11, "end": 1986.39, "word": " and", "probability": 0.93310546875}, {"start": 1986.39, "end": 1986.61, "word": " 8", "probability": 0.9951171875}, {"start": 1986.61, "end": 1987.21, "word": ".6.", "probability": 0.9892578125}, {"start": 1988.27, "end": 1988.61, "word": " Now", "probability": 0.9130859375}, {"start": 1988.61, "end": 1988.91, "word": " what's", "probability": 0.789794921875}, {"start": 1988.91, "end": 1989.05, "word": " the", "probability": 0.92041015625}, {"start": 1989.05, "end": 1989.31, "word": " area", "probability": 0.93212890625}, {"start": 1989.31, "end": 1989.61, "word": " between", "probability": 0.86865234375}, {"start": 1989.61, "end": 1989.83, "word": " these", "probability": 0.75927734375}, {"start": 1989.83, "end": 1989.97, "word": " two?", "probability": 0.6357421875}], "temperature": 1.0}, {"id": 76, "seek": 202098, "start": 1992.68, "end": 2020.98, "text": " Here we have two values of x, x is 8 and x is 8.6. Exactly, so below 8.6 minus below 8 and below 8 is 1 half. So the probability of x between 8 and", "tokens": [1692, 321, 362, 732, 4190, 295, 2031, 11, 2031, 307, 1649, 293, 2031, 307, 1649, 13, 21, 13, 7587, 11, 370, 2507, 1649, 13, 21, 3175, 2507, 1649, 293, 2507, 1649, 307, 502, 1922, 13, 407, 264, 8482, 295, 2031, 1296, 1649, 293], "avg_logprob": -0.3110795481638475, "compression_ratio": 1.2982456140350878, "no_speech_prob": 0.0, "words": [{"start": 1992.68, "end": 1993.22, "word": " Here", "probability": 0.434326171875}, {"start": 1993.22, "end": 1993.36, "word": " we", "probability": 0.767578125}, {"start": 1993.36, "end": 1993.48, "word": " have", "probability": 0.943359375}, {"start": 1993.48, "end": 1993.66, "word": " two", "probability": 0.82568359375}, {"start": 1993.66, "end": 1993.92, "word": " values", "probability": 0.94970703125}, {"start": 1993.92, "end": 1994.08, "word": " of", "probability": 0.94140625}, {"start": 1994.08, "end": 1994.3, "word": " x,", "probability": 0.48046875}, {"start": 1994.48, "end": 1994.64, "word": " x", "probability": 0.96435546875}, {"start": 1994.64, "end": 1994.76, "word": " is", "probability": 0.77880859375}, {"start": 1994.76, "end": 1995.08, "word": " 8", "probability": 0.47509765625}, {"start": 1995.08, "end": 1996.06, "word": " and", "probability": 0.72021484375}, {"start": 1996.06, "end": 1996.28, "word": " x", "probability": 0.9384765625}, {"start": 1996.28, "end": 1996.44, "word": " is", "probability": 0.94482421875}, {"start": 1996.44, "end": 1996.64, "word": " 8", "probability": 0.96240234375}, {"start": 1996.64, "end": 1997.22, "word": ".6.", "probability": 0.96875}, {"start": 2004.28, "end": 2004.88, "word": " Exactly,", "probability": 0.1279296875}, {"start": 2007.16, "end": 2007.26, "word": " so", "probability": 0.84423828125}, {"start": 2007.26, "end": 2007.66, "word": " below", "probability": 0.861328125}, {"start": 2007.66, "end": 2009.34, "word": " 8", "probability": 0.853515625}, {"start": 2009.34, "end": 2009.98, "word": ".6", "probability": 0.995361328125}, {"start": 2009.98, "end": 2011.26, "word": " minus", "probability": 0.71533203125}, {"start": 2011.26, "end": 2012.0, "word": " below", "probability": 0.873046875}, {"start": 2012.0, "end": 2012.42, "word": " 8", "probability": 0.86767578125}, {"start": 2012.42, "end": 2012.98, "word": " and", "probability": 0.544921875}, {"start": 2012.98, "end": 2013.24, "word": " below", "probability": 0.8818359375}, {"start": 2013.24, "end": 2013.54, "word": " 8", "probability": 0.90771484375}, {"start": 2013.54, "end": 2013.78, "word": " is", "probability": 0.93212890625}, {"start": 2013.78, "end": 2014.02, "word": " 1", "probability": 0.389404296875}, {"start": 2014.02, "end": 2014.32, "word": " half.", "probability": 0.047454833984375}, {"start": 2014.88, "end": 2015.22, "word": " So", "probability": 0.919921875}, {"start": 2015.22, "end": 2015.42, "word": " the", "probability": 0.77685546875}, {"start": 2015.42, "end": 2015.74, "word": " probability", "probability": 0.9345703125}, {"start": 2015.74, "end": 2016.2, "word": " of", "probability": 0.9619140625}, {"start": 2016.2, "end": 2017.06, "word": " x", "probability": 0.86328125}, {"start": 2017.06, "end": 2017.6, "word": " between", "probability": 0.8984375}, {"start": 2017.6, "end": 2020.84, "word": " 8", "probability": 0.8046875}, {"start": 2020.84, "end": 2020.98, "word": " and", "probability": 0.66796875}], "temperature": 1.0}, {"id": 77, "seek": 204967, "start": 2021.08, "end": 2049.68, "text": " And 8.2 and 8.6. You can find z-score for the first value, which is zero. Also compute the z-score for the other value, which as we computed before, 0.12. Now this problem becomes z between zero and 0.5. So B of x.", "tokens": [400, 1649, 13, 17, 293, 1649, 13, 21, 13, 509, 393, 915, 710, 12, 4417, 418, 337, 264, 700, 2158, 11, 597, 307, 4018, 13, 2743, 14722, 264, 710, 12, 4417, 418, 337, 264, 661, 2158, 11, 597, 382, 321, 40610, 949, 11, 1958, 13, 4762, 13, 823, 341, 1154, 3643, 710, 1296, 4018, 293, 1958, 13, 20, 13, 407, 363, 295, 2031, 13], "avg_logprob": -0.23569711538461538, "compression_ratio": 1.4429530201342282, "no_speech_prob": 0.0, "words": [{"start": 2021.08, "end": 2021.36, "word": " And", "probability": 0.431884765625}, {"start": 2021.36, "end": 2021.62, "word": " 8", "probability": 0.912109375}, {"start": 2021.62, "end": 2022.12, "word": ".2", "probability": 0.97412109375}, {"start": 2022.12, "end": 2022.52, "word": " and", "probability": 0.7685546875}, {"start": 2022.52, "end": 2022.72, "word": " 8", "probability": 0.99462890625}, {"start": 2022.72, "end": 2023.3, "word": ".6.", "probability": 0.998291015625}, {"start": 2024.64, "end": 2025.2, "word": " You", "probability": 0.953125}, {"start": 2025.2, "end": 2025.48, "word": " can", "probability": 0.94921875}, {"start": 2025.48, "end": 2025.92, "word": " find", "probability": 0.87841796875}, {"start": 2025.92, "end": 2026.38, "word": " z", "probability": 0.2890625}, {"start": 2026.38, "end": 2026.74, "word": "-score", "probability": 0.730224609375}, {"start": 2026.74, "end": 2027.16, "word": " for", "probability": 0.93994140625}, {"start": 2027.16, "end": 2027.34, "word": " the", "probability": 0.9150390625}, {"start": 2027.34, "end": 2027.58, "word": " first", "probability": 0.87548828125}, {"start": 2027.58, "end": 2027.94, "word": " value,", "probability": 0.974609375}, {"start": 2028.32, "end": 2028.5, "word": " which", "probability": 0.943359375}, {"start": 2028.5, "end": 2028.62, "word": " is", "probability": 0.94873046875}, {"start": 2028.62, "end": 2028.86, "word": " zero.", "probability": 0.59521484375}, {"start": 2030.92, "end": 2031.56, "word": " Also", "probability": 0.92724609375}, {"start": 2031.56, "end": 2032.06, "word": " compute", "probability": 0.8232421875}, {"start": 2032.06, "end": 2032.28, "word": " the", "probability": 0.88037109375}, {"start": 2032.28, "end": 2032.48, "word": " z", "probability": 0.98583984375}, {"start": 2032.48, "end": 2032.8, "word": "-score", "probability": 0.9187825520833334}, {"start": 2032.8, "end": 2033.04, "word": " for", "probability": 0.94482421875}, {"start": 2033.04, "end": 2033.18, "word": " the", "probability": 0.9091796875}, {"start": 2033.18, "end": 2033.42, "word": " other", "probability": 0.88134765625}, {"start": 2033.42, "end": 2033.74, "word": " value,", "probability": 0.962890625}, {"start": 2033.84, "end": 2033.94, "word": " which", "probability": 0.943359375}, {"start": 2033.94, "end": 2034.12, "word": " as", "probability": 0.486328125}, {"start": 2034.12, "end": 2034.38, "word": " we", "probability": 0.96337890625}, {"start": 2034.38, "end": 2035.54, "word": " computed", "probability": 0.90869140625}, {"start": 2035.54, "end": 2035.9, "word": " before,", "probability": 0.873046875}, {"start": 2036.02, "end": 2036.26, "word": " 0", "probability": 0.451416015625}, {"start": 2036.26, "end": 2036.58, "word": ".12.", "probability": 0.990234375}, {"start": 2037.28, "end": 2037.56, "word": " Now", "probability": 0.93603515625}, {"start": 2037.56, "end": 2037.86, "word": " this", "probability": 0.7021484375}, {"start": 2037.86, "end": 2038.26, "word": " problem", "probability": 0.8544921875}, {"start": 2038.26, "end": 2038.98, "word": " becomes", "probability": 0.85546875}, {"start": 2038.98, "end": 2041.3, "word": " z", "probability": 0.87060546875}, {"start": 2041.3, "end": 2041.58, "word": " between", "probability": 0.8828125}, {"start": 2041.58, "end": 2042.1, "word": " zero", "probability": 0.6533203125}, {"start": 2042.1, "end": 2043.98, "word": " and", "probability": 0.89111328125}, {"start": 2043.98, "end": 2044.24, "word": " 0", "probability": 0.625}, {"start": 2044.24, "end": 2044.54, "word": ".5.", "probability": 0.826171875}, {"start": 2047.48, "end": 2047.9, "word": " So", "probability": 0.953125}, {"start": 2047.9, "end": 2049.18, "word": " B", "probability": 0.09246826171875}, {"start": 2049.18, "end": 2049.32, "word": " of", "probability": 0.6484375}, {"start": 2049.32, "end": 2049.68, "word": " x.", "probability": 0.56005859375}], "temperature": 1.0}, {"id": 78, "seek": 207900, "start": 2050.56, "end": 2079.0, "text": " Greater than 8 and smaller than 8.6 is the same as z between 0 and 0.12. Now this area equals b of z smaller than 0.12 minus the area below z which is 1.5. So again, b of z between 0 and 1.5 equal b of z small.", "tokens": [38410, 813, 1649, 293, 4356, 813, 1649, 13, 21, 307, 264, 912, 382, 710, 1296, 1958, 293, 1958, 13, 4762, 13, 823, 341, 1859, 6915, 272, 295, 710, 4356, 813, 1958, 13, 4762, 3175, 264, 1859, 2507, 710, 597, 307, 502, 13, 20, 13, 407, 797, 11, 272, 295, 710, 1296, 1958, 293, 502, 13, 20, 2681, 272, 295, 710, 1359, 13], "avg_logprob": -0.22743055129808093, "compression_ratio": 1.5746268656716418, "no_speech_prob": 0.0, "words": [{"start": 2050.56, "end": 2051.0, "word": " Greater", "probability": 0.269287109375}, {"start": 2051.0, "end": 2051.38, "word": " than", "probability": 0.9443359375}, {"start": 2051.38, "end": 2051.72, "word": " 8", "probability": 0.556640625}, {"start": 2051.72, "end": 2052.74, "word": " and", "probability": 0.5439453125}, {"start": 2052.74, "end": 2053.12, "word": " smaller", "probability": 0.85888671875}, {"start": 2053.12, "end": 2053.36, "word": " than", "probability": 0.943359375}, {"start": 2053.36, "end": 2053.52, "word": " 8", "probability": 0.98388671875}, {"start": 2053.52, "end": 2054.14, "word": ".6", "probability": 0.986572265625}, {"start": 2054.14, "end": 2055.12, "word": " is", "probability": 0.767578125}, {"start": 2055.12, "end": 2055.28, "word": " the", "probability": 0.91650390625}, {"start": 2055.28, "end": 2055.52, "word": " same", "probability": 0.91650390625}, {"start": 2055.52, "end": 2055.9, "word": " as", "probability": 0.94921875}, {"start": 2055.9, "end": 2056.24, "word": " z", "probability": 0.65283203125}, {"start": 2056.24, "end": 2056.58, "word": " between", "probability": 0.89990234375}, {"start": 2056.58, "end": 2057.04, "word": " 0", "probability": 0.79541015625}, {"start": 2057.04, "end": 2057.74, "word": " and", "probability": 0.92626953125}, {"start": 2057.74, "end": 2058.0, "word": " 0", "probability": 0.52099609375}, {"start": 2058.0, "end": 2058.26, "word": ".12.", "probability": 0.830322265625}, {"start": 2059.26, "end": 2059.94, "word": " Now", "probability": 0.943359375}, {"start": 2059.94, "end": 2060.34, "word": " this", "probability": 0.64013671875}, {"start": 2060.34, "end": 2060.8, "word": " area", "probability": 0.900390625}, {"start": 2060.8, "end": 2061.6, "word": " equals", "probability": 0.90234375}, {"start": 2061.6, "end": 2062.08, "word": " b", "probability": 0.427734375}, {"start": 2062.08, "end": 2062.24, "word": " of", "probability": 0.951171875}, {"start": 2062.24, "end": 2062.38, "word": " z", "probability": 0.9755859375}, {"start": 2062.38, "end": 2062.78, "word": " smaller", "probability": 0.748046875}, {"start": 2062.78, "end": 2063.06, "word": " than", "probability": 0.9365234375}, {"start": 2063.06, "end": 2063.32, "word": " 0", "probability": 0.880859375}, {"start": 2063.32, "end": 2063.66, "word": ".12", "probability": 0.98388671875}, {"start": 2063.66, "end": 2064.72, "word": " minus", "probability": 0.931640625}, {"start": 2064.72, "end": 2065.1, "word": " the", "probability": 0.9091796875}, {"start": 2065.1, "end": 2065.32, "word": " area", "probability": 0.91455078125}, {"start": 2065.32, "end": 2065.56, "word": " below", "probability": 0.86767578125}, {"start": 2065.56, "end": 2065.8, "word": " z", "probability": 0.9306640625}, {"start": 2065.8, "end": 2065.98, "word": " which", "probability": 0.64306640625}, {"start": 2065.98, "end": 2066.08, "word": " is", "probability": 0.9501953125}, {"start": 2066.08, "end": 2066.28, "word": " 1", "probability": 0.58740234375}, {"start": 2066.28, "end": 2066.52, "word": ".5.", "probability": 0.336212158203125}, {"start": 2071.1, "end": 2071.78, "word": " So", "probability": 0.955078125}, {"start": 2071.78, "end": 2072.12, "word": " again,", "probability": 0.90869140625}, {"start": 2073.28, "end": 2074.24, "word": " b", "probability": 0.81396484375}, {"start": 2074.24, "end": 2074.4, "word": " of", "probability": 0.9677734375}, {"start": 2074.4, "end": 2074.76, "word": " z", "probability": 0.994140625}, {"start": 2074.76, "end": 2075.16, "word": " between", "probability": 0.87060546875}, {"start": 2075.16, "end": 2075.46, "word": " 0", "probability": 0.91162109375}, {"start": 2075.46, "end": 2075.58, "word": " and", "probability": 0.93359375}, {"start": 2075.58, "end": 2075.74, "word": " 1", "probability": 0.7900390625}, {"start": 2075.74, "end": 2075.96, "word": ".5", "probability": 0.8779296875}, {"start": 2075.96, "end": 2076.38, "word": " equal", "probability": 0.455078125}, {"start": 2076.38, "end": 2076.92, "word": " b", "probability": 0.91845703125}, {"start": 2076.92, "end": 2077.1, "word": " of", "probability": 0.97607421875}, {"start": 2077.1, "end": 2077.38, "word": " z", "probability": 0.99560546875}, {"start": 2077.38, "end": 2079.0, "word": " small.", "probability": 0.55029296875}], "temperature": 1.0}, {"id": 79, "seek": 210838, "start": 2079.74, "end": 2108.38, "text": " larger than 0.12 minus b of z less than zero. Now, b of z less than 0.12 gives this result, 0.5478. The probability below zero is one-half because we know that the area to the left is zero, same as to the right is one-half. So the answer is going to be 0.478. So that's how can we compute the probabilities for lower 10 directly from the table.", "tokens": [4833, 813, 1958, 13, 4762, 3175, 272, 295, 710, 1570, 813, 4018, 13, 823, 11, 272, 295, 710, 1570, 813, 1958, 13, 4762, 2709, 341, 1874, 11, 1958, 13, 20, 14060, 23, 13, 440, 8482, 2507, 4018, 307, 472, 12, 25461, 570, 321, 458, 300, 264, 1859, 281, 264, 1411, 307, 4018, 11, 912, 382, 281, 264, 558, 307, 472, 12, 25461, 13, 407, 264, 1867, 307, 516, 281, 312, 1958, 13, 14060, 23, 13, 407, 300, 311, 577, 393, 321, 14722, 264, 33783, 337, 3126, 1266, 3838, 490, 264, 3199, 13], "avg_logprob": -0.24596773488547213, "compression_ratio": 1.6121495327102804, "no_speech_prob": 0.0, "words": [{"start": 2079.74, "end": 2080.12, "word": " larger", "probability": 0.31640625}, {"start": 2080.12, "end": 2080.38, "word": " than", "probability": 0.91064453125}, {"start": 2080.38, "end": 2080.54, "word": " 0", "probability": 0.587890625}, {"start": 2080.54, "end": 2080.94, "word": ".12", "probability": 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Is it clear? Now here we converted or we transformed from normal distribution to standardized. So instead of X between 7.4 and 8, we have now Z between minus 0.12 and 0. 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So B of z less than zero minus negative 0.12. That will give the area between minus 0.12 and zero. This is one-half. 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B of Z less than negative 0.12 equals minus 0.4522. That will give 0 forcibility. 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Because of symmetric, since this area equals the same for the other part. So from 0 up to 0.12 is the same as minus 0.12 up to 0. So equal, so the normal distribution is symmetric. 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Again, the equal sign does not matter. Because here we have the complement. 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So we should have just permutation, the equality. But it doesn't matter. If in the problem we don't have equal sign in the complement, we should have equal sign. But it doesn't matter actually if we have equal sign or not. 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Now what's the complement of that? 1 minus less than or equal to A. But if X is greater than or equal to A, the complement is without equal sign. But in continuous distribution, the equal sign does not matter. Any question?", "tokens": [363, 295, 1783, 5044, 813, 316, 13, 823, 437, 311, 264, 17103, 295, 300, 30, 502, 3175, 1570, 813, 420, 2681, 281, 316, 13, 583, 498, 1783, 307, 5044, 813, 420, 2681, 281, 316, 11, 264, 17103, 307, 1553, 2681, 1465, 13, 583, 294, 10957, 7316, 11, 264, 2681, 1465, 775, 406, 1871, 13, 2639, 1168, 30], "avg_logprob": -0.23679957359001555, "compression_ratio": 1.6466666666666667, "no_speech_prob": 0.0, "words": [{"start": 2416.23, "end": 2416.45, "word": " B", "probability": 0.07550048828125}, {"start": 2416.45, "end": 2416.61, "word": " of", "probability": 0.9140625}, {"start": 2416.61, "end": 2416.81, "word": " X", "probability": 0.70947265625}, {"start": 2416.81, "end": 2417.15, "word": " greater", "probability": 0.7138671875}, {"start": 2417.15, "end": 2417.45, "word": " than", "probability": 0.9541015625}, {"start": 2417.45, "end": 2417.65, "word": " A.", "probability": 0.681640625}, {"start": 2418.57, "end": 2419.05, "word": " Now", "probability": 0.861328125}, {"start": 2419.05, "end": 2419.31, "word": " what's", "probability": 0.701416015625}, {"start": 2419.31, "end": 2419.43, "word": " the", "probability": 0.91259765625}, {"start": 2419.43, "end": 2419.81, "word": " complement", "probability": 0.79638671875}, {"start": 2419.81, "end": 2420.07, "word": " of", "probability": 0.93603515625}, {"start": 2420.07, "end": 2420.33, "word": " that?", "probability": 0.92578125}, {"start": 2421.03, "end": 2421.61, "word": " 1", "probability": 0.45263671875}, {"start": 2421.61, "end": 2422.25, "word": " minus", "probability": 0.9638671875}, {"start": 2422.25, "end": 2425.95, "word": " less", "probability": 0.82275390625}, {"start": 2425.95, "end": 2426.21, "word": " than", "probability": 0.947265625}, {"start": 2426.21, "end": 2426.35, "word": " or", "probability": 0.95263671875}, {"start": 2426.35, "end": 2426.55, "word": " equal", "probability": 0.9052734375}, {"start": 2426.55, "end": 2426.65, "word": " to", "probability": 0.5380859375}, {"start": 2426.65, "end": 2426.81, "word": " A.", "probability": 0.94091796875}, {"start": 2428.51, "end": 2429.09, "word": " But", "probability": 0.90576171875}, {"start": 2429.09, "end": 2429.77, "word": " if", "probability": 0.8212890625}, {"start": 2429.77, "end": 2431.37, "word": " X", "probability": 0.888671875}, {"start": 2431.37, "end": 2431.71, "word": " is", "probability": 0.473876953125}, {"start": 2431.71, "end": 2431.99, "word": " greater", "probability": 0.9091796875}, {"start": 2431.99, "end": 2432.27, "word": " than", "probability": 0.9306640625}, {"start": 2432.27, "end": 2432.45, "word": " or", "probability": 0.955078125}, {"start": 2432.45, "end": 2432.63, "word": " equal", "probability": 0.90673828125}, {"start": 2432.63, "end": 2432.79, "word": " to", "probability": 0.91064453125}, {"start": 2432.79, "end": 2432.87, "word": " A,", "probability": 0.970703125}, {"start": 2432.93, "end": 2433.03, "word": " the", "probability": 0.81591796875}, {"start": 2433.03, "end": 2433.35, "word": " complement", "probability": 0.9072265625}, {"start": 2433.35, "end": 2433.93, "word": " is", "probability": 0.91357421875}, {"start": 2433.93, "end": 2436.43, "word": " without", "probability": 0.1600341796875}, {"start": 2436.43, "end": 2437.51, "word": " equal", "probability": 0.79443359375}, {"start": 2437.51, "end": 2437.87, "word": " sign.", "probability": 0.88916015625}, {"start": 2438.31, "end": 2438.75, "word": " But", "probability": 0.9404296875}, {"start": 2438.75, "end": 2439.19, "word": " in", "probability": 0.90283203125}, {"start": 2439.19, "end": 2439.65, "word": " continuous", "probability": 0.828125}, {"start": 2439.65, "end": 2440.15, "word": " distribution,", "probability": 0.83056640625}, {"start": 2440.39, "end": 2440.49, "word": " the", "probability": 0.9169921875}, {"start": 2440.49, "end": 2440.71, "word": " equal", "probability": 0.8798828125}, {"start": 2440.71, "end": 2440.97, "word": " sign", "probability": 0.896484375}, {"start": 2440.97, "end": 2441.15, "word": " does", "probability": 0.9169921875}, {"start": 2441.15, "end": 2441.33, "word": " not", "probability": 0.94482421875}, {"start": 2441.33, "end": 2441.59, "word": " matter.", "probability": 0.85791015625}, {"start": 2444.05, "end": 2444.63, "word": " Any", "probability": 0.90869140625}, {"start": 2444.63, "end": 2444.99, "word": " question?", "probability": 0.61865234375}], "temperature": 1.0}, {"id": 92, "seek": 247675, "start": 2452.19, "end": 2476.75, "text": " comments. Let's move to the next topic which talks about the empirical rule. If you remember before we said there is an empirical rule for 68, 95, 95,", "tokens": [3053, 13, 961, 311, 1286, 281, 264, 958, 4829, 597, 6686, 466, 264, 31886, 4978, 13, 759, 291, 1604, 949, 321, 848, 456, 307, 364, 31886, 4978, 337, 23317, 11, 13420, 11, 13420, 11], "avg_logprob": -0.23325893027441844, "compression_ratio": 1.2796610169491525, "no_speech_prob": 0.0, "words": [{"start": 2452.1900000000005, "end": 2453.4300000000003, "word": " comments.", "probability": 0.46435546875}, {"start": 2453.4300000000003, "end": 2454.67, "word": " Let's", "probability": 0.91796875}, {"start": 2454.67, "end": 2454.99, "word": " move", "probability": 0.94482421875}, {"start": 2454.99, "end": 2455.33, "word": " to", "probability": 0.96337890625}, {"start": 2455.33, "end": 2455.61, "word": " the", "probability": 0.91650390625}, {"start": 2455.61, "end": 2455.83, "word": " next", "probability": 0.9404296875}, {"start": 2455.83, "end": 2456.29, "word": " topic", "probability": 0.9658203125}, {"start": 2456.29, "end": 2457.81, "word": " which", "probability": 0.546875}, {"start": 2457.81, "end": 2458.13, "word": " talks", "probability": 0.87158203125}, {"start": 2458.13, "end": 2458.75, "word": " about", "probability": 0.91259765625}, {"start": 2458.75, "end": 2459.95, "word": " the", "probability": 0.85888671875}, {"start": 2459.95, "end": 2460.51, "word": " empirical", "probability": 0.80078125}, {"start": 2460.51, "end": 2460.89, "word": " rule.", "probability": 0.79638671875}, {"start": 2463.21, "end": 2464.45, "word": " If", "probability": 0.94677734375}, {"start": 2464.45, "end": 2464.57, "word": " you", "probability": 0.9619140625}, {"start": 2464.57, "end": 2464.87, "word": " remember", "probability": 0.875}, {"start": 2464.87, "end": 2465.51, "word": " before", "probability": 0.76513671875}, {"start": 2465.51, "end": 2466.51, "word": " we", "probability": 0.662109375}, {"start": 2466.51, "end": 2466.87, "word": " said", "probability": 0.94970703125}, {"start": 2466.87, "end": 2467.21, "word": " there", "probability": 0.86669921875}, {"start": 2467.21, "end": 2467.87, "word": " is", "probability": 0.82177734375}, {"start": 2467.87, "end": 2468.49, "word": " an", "probability": 0.9189453125}, {"start": 2468.49, "end": 2468.79, "word": " empirical", "probability": 0.9033203125}, {"start": 2468.79, "end": 2469.29, "word": " rule", "probability": 0.91748046875}, {"start": 2469.29, "end": 2470.53, "word": " for", "probability": 0.9306640625}, {"start": 2470.53, "end": 2471.19, "word": " 68,", "probability": 0.7666015625}, {"start": 2472.49, "end": 2473.71, "word": " 95,", "probability": 0.95361328125}, {"start": 2474.99, "end": 2476.75, "word": " 95,", "probability": 0.638671875}], "temperature": 1.0}, {"id": 93, "seek": 250562, "start": 2477.42, "end": 2505.62, "text": " 99.71. Now let's see the exact meaning of this rule. Now we have to apply the empirical rule not to Chebyshev's inequality because the distribution is normal. 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For symmetric, we have to apply the empirical rule. Here, we assume the distribution is normal. And today, we are talking about normal distribution. So we have to use the empirical rules. Now, the mean is the value in the middle. Suppose we are far away.", "tokens": [337, 8756, 26896, 37870, 13, 1171, 32330, 11, 321, 362, 281, 3079, 264, 31886, 4978, 13, 1692, 11, 321, 6552, 264, 7316, 307, 2710, 13, 400, 965, 11, 321, 366, 1417, 466, 2710, 7316, 13, 407, 321, 362, 281, 764, 264, 31886, 4474, 13, 823, 11, 264, 914, 307, 264, 2158, 294, 264, 2808, 13, 21360, 321, 366, 1400, 1314, 13], "avg_logprob": -0.1499495902849782, "compression_ratio": 1.6726190476190477, "no_speech_prob": 0.0, "words": [{"start": 2507.4300000000003, "end": 2508.05, "word": " for", "probability": 0.3681640625}, {"start": 2508.05, "end": 2508.67, "word": " skewed", "probability": 0.953125}, {"start": 2508.67, "end": 2509.33, "word": " distributions.", "probability": 0.84375}, {"start": 2510.49, "end": 2511.11, "word": " For", "probability": 0.82080078125}, {"start": 2511.11, "end": 2511.51, "word": " symmetric,", "probability": 0.5546875}, {"start": 2511.69, "end": 2511.77, "word": " we", "probability": 0.6083984375}, {"start": 2511.77, 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Now let's see how can we compute the exact area, area not just say 68%. Now X has normal distribution with mean mu and standard deviation sigma. 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So this is the first value here. Now the z-score, the general formula is x minus the mean divided by sigma. Now the first quantity is mu minus sigma. So instead of x here, so first z is, now this x should be replaced by mu minus sigma. 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Mu minus sigma minus mu mu cancels, so plus one. And let's see how can we compute that area. I mean between minus one and plus one. 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The area below 1 is 0.3413. Okay, now the area below minus 1. This is minus 1.", "tokens": [823, 264, 1859, 2507, 11, 341, 307, 502, 13, 440, 1859, 2507, 502, 307, 1958, 13, 12249, 7668, 13, 1033, 11, 586, 264, 1859, 2507, 3175, 502, 13, 639, 307, 3175, 502, 13], "avg_logprob": -0.3177849343594383, "compression_ratio": 1.5277777777777777, "no_speech_prob": 1.1324882507324219e-06, "words": [{"start": 2723.4300000000003, "end": 2724.11, "word": " Now", "probability": 0.463134765625}, {"start": 2724.11, "end": 2724.29, "word": " the", "probability": 0.607421875}, {"start": 2724.29, "end": 2724.63, "word": " area", "probability": 0.8828125}, {"start": 2724.63, "end": 2725.41, "word": " below,", "probability": 0.869140625}, {"start": 2725.55, "end": 2725.71, "word": " this", "probability": 0.91259765625}, {"start": 2725.71, "end": 2725.85, "word": " is", "probability": 0.9365234375}, {"start": 2725.85, "end": 2726.05, "word": " 1.", "probability": 0.4130859375}, {"start": 2727.31, "end": 2727.99, "word": " The", "probability": 0.78466796875}, {"start": 2727.99, "end": 2728.23, "word": " area", "probability": 0.8837890625}, {"start": 2728.23, "end": 2728.45, "word": " below", "probability": 0.9072265625}, {"start": 2728.45, "end": 2728.77, "word": " 1", "probability": 0.91455078125}, {"start": 2728.77, "end": 2729.87, "word": " is", "probability": 0.796875}, {"start": 2729.87, "end": 2730.13, "word": " 0", "probability": 0.486572265625}, {"start": 2730.13, "end": 2731.31, "word": ".3413.", "probability": 0.73486328125}, {"start": 2734.43, "end": 2734.77, "word": " Okay,", "probability": 0.4677734375}, {"start": 2735.23, "end": 2735.61, "word": " now", "probability": 0.9384765625}, {"start": 2735.61, "end": 2736.25, "word": " the", "probability": 0.716796875}, {"start": 2736.25, "end": 2736.57, "word": " area", "probability": 0.8876953125}, {"start": 2736.57, "end": 2736.89, "word": " below", "probability": 0.919921875}, {"start": 2736.89, "end": 2737.21, "word": " minus", "probability": 0.8076171875}, {"start": 2737.21, "end": 2737.59, "word": " 1.", "probability": 0.91552734375}, {"start": 2740.77, "end": 2741.45, "word": " This", "probability": 0.6904296875}, {"start": 2741.45, "end": 2741.53, "word": " is", "probability": 0.9072265625}, {"start": 2741.53, "end": 2741.77, "word": " minus", "probability": 0.9736328125}, {"start": 2741.77, "end": 2742.05, "word": " 1.", "probability": 0.94482421875}], "temperature": 1.0}, {"id": 102, "seek": 277609, "start": 2746.81, "end": 2776.09, "text": " Now, the area below minus 1 is the same as above 1. These are the two areas here are equal. So the area below minus 1, I mean b of z less than minus 1 is the same as b of z greater than 1. And b of z greater than 1 is the same as 1 minus b of z smaller than 1. 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Just subtract 1 from this value. Make sense? Here we compute the value of B of Z less than 1, which is 0.8413. We are looking for B of Z less than minus 1, which is the same as B of Z greater than 1. Now, greater than means our tail. It's 1 minus the lower tail probability.", "tokens": [509, 4659, 380, 643, 281, 574, 797, 281, 264, 3199, 13, 1449, 16390, 502, 490, 341, 2158, 13, 4387, 2020, 30, 1692, 321, 14722, 264, 2158, 295, 363, 295, 1176, 1570, 813, 502, 11, 597, 307, 1958, 13, 23, 17344, 18, 13, 492, 366, 1237, 337, 363, 295, 1176, 1570, 813, 3175, 502, 11, 597, 307, 264, 912, 382, 363, 295, 1176, 5044, 813, 502, 13, 823, 11, 5044, 813, 1355, 527, 6838, 13, 467, 311, 502, 3175, 264, 3126, 6838, 8482, 13], "avg_logprob": -0.17559524075615973, "compression_ratio": 1.5631067961165048, "no_speech_prob": 0.0, "words": [{"start": 2777.07, "end": 2777.31, "word": " You", "probability": 0.73681640625}, {"start": 2777.31, "end": 2777.71, "word": " shouldn't", "probability": 0.964599609375}, {"start": 2777.71, "end": 2778.11, "word": " need", "probability": 0.8857421875}, {"start": 2778.11, "end": 2778.25, "word": " to", "probability": 0.96435546875}, {"start": 2778.25, "end": 2778.41, "word": " look", "probability": 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So the answer again is 1 minus 0.8413. So 8413 minus 0.1587. So 8413.", "tokens": [407, 341, 307, 502, 3175, 13, 407, 264, 1867, 797, 307, 502, 3175, 1958, 13, 25494, 7668, 13, 407, 29018, 7668, 3175, 1958, 13, 5211, 23853, 13, 407, 29018, 7668, 13], "avg_logprob": -0.2462158128619194, "compression_ratio": 1.3846153846153846, "no_speech_prob": 0.0, "words": [{"start": 2806.26, "end": 2806.62, "word": " So", "probability": 0.70751953125}, {"start": 2806.62, "end": 2806.88, "word": " this", "probability": 0.57763671875}, {"start": 2806.88, "end": 2806.98, "word": " is", "probability": 0.943359375}, {"start": 2806.98, "end": 2807.2, "word": " 1", "probability": 0.43896484375}, {"start": 2807.2, "end": 2807.62, "word": " minus.", "probability": 0.85791015625}, {"start": 2808.0, "end": 2808.36, "word": " So", "probability": 0.87109375}, {"start": 2808.36, "end": 2808.7, "word": " the", "probability": 0.7646484375}, {"start": 2808.7, "end": 2809.0, "word": " answer", "probability": 0.96044921875}, {"start": 2809.0, "end": 2809.62, "word": " again", "probability": 0.6689453125}, {"start": 2809.62, "end": 2809.84, "word": " is", "probability": 0.90283203125}, {"start": 2809.84, "end": 2810.06, "word": " 1", "probability": 0.88525390625}, {"start": 2810.06, "end": 2810.56, "word": " minus", "probability": 0.96630859375}, {"start": 2810.56, "end": 2810.98, "word": " 0", "probability": 0.478271484375}, {"start": 2810.98, "end": 2812.24, "word": ".8413.", "probability": 0.869140625}, {"start": 2814.28, "end": 2815.16, "word": " So", "probability": 0.317626953125}, {"start": 2815.16, "end": 2816.76, "word": " 8413", "probability": 0.775634765625}, {"start": 2816.76, "end": 2817.76, "word": " minus", "probability": 0.982421875}, {"start": 2817.76, "end": 2818.62, "word": " 0", "probability": 0.97705078125}, {"start": 2818.62, "end": 2820.04, "word": ".1587.", "probability": 0.9879557291666666}, {"start": 2831.38, "end": 2832.26, "word": " So", "probability": 0.91259765625}, {"start": 2832.26, "end": 2832.98, "word": " 8413.", "probability": 0.862060546875}], "temperature": 1.0}, {"id": 105, "seek": 286059, "start": 2833.61, "end": 2860.59, "text": " minus 1.1587. Okay, so that gives 0.6826. Multiply this one by 100, we get 68.1826. So roughly 60-80%", "tokens": [3175, 502, 13, 5211, 23853, 13, 1033, 11, 370, 300, 2709, 1958, 13, 27102, 10880, 13, 31150, 356, 341, 472, 538, 2319, 11, 321, 483, 23317, 13, 6494, 10880, 13, 407, 9810, 4060, 12, 4702, 4], "avg_logprob": -0.3236908687127603, "compression_ratio": 1.0515463917525774, "no_speech_prob": 0.0, "words": [{"start": 2833.61, "end": 2834.05, "word": " minus", "probability": 0.1898193359375}, {"start": 2834.05, "end": 2834.45, "word": " 1", "probability": 0.67919921875}, {"start": 2834.45, "end": 2837.03, "word": ".1587.", "probability": 0.91015625}, {"start": 2841.63, "end": 2841.93, "word": " Okay,", "probability": 0.5888671875}, {"start": 2842.15, "end": 2842.35, "word": " so", "probability": 0.85693359375}, {"start": 2842.35, "end": 2842.67, "word": " that", "probability": 0.91357421875}, {"start": 2842.67, "end": 2843.33, "word": " gives", "probability": 0.64013671875}, {"start": 2843.33, "end": 2843.67, "word": " 0", "probability": 0.83740234375}, {"start": 2843.67, "end": 2847.57, "word": ".6826.", "probability": 0.9326171875}, {"start": 2849.09, "end": 2849.83, "word": " Multiply", "probability": 0.91455078125}, {"start": 2849.83, "end": 2850.05, "word": " this", "probability": 0.93603515625}, {"start": 2850.05, "end": 2850.25, "word": " one", "probability": 0.8779296875}, {"start": 2850.25, "end": 2850.41, "word": " by", "probability": 0.9736328125}, {"start": 2850.41, "end": 2850.93, "word": " 100,", "probability": 0.75634765625}, {"start": 2852.61, "end": 2852.95, "word": " we", "probability": 0.5341796875}, {"start": 2852.95, "end": 2853.25, "word": " get", "probability": 0.5078125}, {"start": 2853.25, "end": 2854.15, "word": " 68", "probability": 0.97021484375}, {"start": 2854.15, "end": 2857.55, "word": ".1826.", "probability": 0.78662109375}, {"start": 2858.75, "end": 2859.01, "word": " So", "probability": 0.9111328125}, {"start": 2859.01, "end": 2859.31, "word": " roughly", "probability": 0.73681640625}, {"start": 2859.31, "end": 2860.03, "word": " 60", "probability": 0.5390625}, {"start": 2860.03, "end": 2860.27, "word": "-80", "probability": 0.68212890625}, {"start": 2860.27, "end": 2860.59, "word": "%", "probability": 0.548828125}], "temperature": 1.0}, {"id": 106, "seek": 288963, "start": 2861.59, "end": 2889.63, "text": " of the observations lie between one standard deviation around the mean. So this is the way how can we compute the area below one standard deviation or above one standard deviation of the mean. Do the same for not mu minus sigma, mu plus minus two sigma and mu plus two sigma.", "tokens": [295, 264, 18163, 4544, 1296, 472, 3832, 25163, 926, 264, 914, 13, 407, 341, 307, 264, 636, 577, 393, 321, 14722, 264, 1859, 2507, 472, 3832, 25163, 420, 3673, 472, 3832, 25163, 295, 264, 914, 13, 1144, 264, 912, 337, 406, 2992, 3175, 12771, 11, 2992, 1804, 3175, 732, 12771, 293, 2992, 1804, 732, 12771, 13], "avg_logprob": -0.1414473705124437, "compression_ratio": 1.84, "no_speech_prob": 0.0, "words": [{"start": 2861.59, "end": 2861.91, "word": " of", "probability": 0.42138671875}, {"start": 2861.91, "end": 2862.05, "word": " the", "probability": 0.91552734375}, {"start": 2862.05, "end": 2862.63, "word": " observations", "probability": 0.76123046875}, {"start": 2862.63, "end": 2863.43, "word": " lie", "probability": 0.82275390625}, {"start": 2863.43, "end": 2864.01, "word": " between", "probability": 0.86083984375}, {"start": 2864.01, "end": 2865.29, "word": " one", "probability": 0.771484375}, {"start": 2865.29, "end": 2865.57, "word": " standard", "probability": 0.728515625}, {"start": 2865.57, "end": 2865.97, "word": " deviation", "probability": 0.9208984375}, {"start": 2865.97, "end": 2867.15, "word": " around", "probability": 0.873046875}, {"start": 2867.15, "end": 2867.41, "word": " the", "probability": 0.4990234375}, {"start": 2867.41, "end": 2867.53, "word": " mean.", "probability": 0.94921875}, {"start": 2869.45, "end": 2870.09, "word": " So", "probability": 0.9072265625}, {"start": 2870.09, "end": 2870.33, "word": " this", "probability": 0.841796875}, {"start": 2870.33, "end": 2870.47, "word": " is", "probability": 0.947265625}, {"start": 2870.47, "end": 2870.59, "word": " the", "probability": 0.8759765625}, {"start": 2870.59, "end": 2870.75, "word": " way", "probability": 0.96337890625}, {"start": 2870.75, "end": 2870.91, "word": " how", "probability": 0.82958984375}, {"start": 2870.91, "end": 2871.15, "word": " can", "probability": 0.83642578125}, {"start": 2871.15, "end": 2871.45, "word": " we", "probability": 0.95556640625}, {"start": 2871.45, "end": 2872.05, "word": " compute", "probability": 0.896484375}, {"start": 2872.05, "end": 2872.51, "word": " the", "probability": 0.90673828125}, {"start": 2872.51, "end": 2872.87, "word": " area", "probability": 0.88818359375}, {"start": 2872.87, "end": 2873.59, "word": " below", "probability": 0.888671875}, {"start": 2873.59, "end": 2873.85, "word": " one", "probability": 0.92236328125}, {"start": 2873.85, "end": 2874.07, "word": " standard", "probability": 0.96875}, {"start": 2874.07, "end": 2874.43, "word": " deviation", "probability": 0.92431640625}, {"start": 2874.43, "end": 2875.19, "word": " or", "probability": 0.80419921875}, {"start": 2875.19, "end": 2875.55, "word": " above", "probability": 0.96435546875}, {"start": 2875.55, "end": 2876.67, "word": " one", "probability": 0.9169921875}, {"start": 2876.67, "end": 2876.89, "word": " standard", "probability": 0.96728515625}, {"start": 2876.89, "end": 2877.25, "word": " deviation", "probability": 0.9365234375}, {"start": 2877.25, "end": 2877.49, "word": " of", "probability": 0.96337890625}, {"start": 2877.49, "end": 2877.61, "word": " the", "probability": 0.92724609375}, {"start": 2877.61, "end": 2877.77, "word": " mean.", "probability": 0.9716796875}, {"start": 2879.53, "end": 2879.71, "word": " Do", "probability": 0.9482421875}, {"start": 2879.71, "end": 2879.93, "word": " the", "probability": 0.91650390625}, {"start": 2879.93, "end": 2880.27, "word": " same", "probability": 0.90478515625}, {"start": 2880.27, "end": 2882.15, "word": " for", "probability": 0.93310546875}, {"start": 2882.15, "end": 2882.87, "word": " not", "probability": 0.85205078125}, {"start": 2882.87, "end": 2883.05, "word": " mu", "probability": 0.6884765625}, {"start": 2883.05, "end": 2883.33, "word": " minus", "probability": 0.97509765625}, {"start": 2883.33, "end": 2883.79, "word": " sigma,", "probability": 0.908203125}, {"start": 2885.23, "end": 2885.47, "word": " mu", "probability": 0.91552734375}, {"start": 2885.47, "end": 2885.83, "word": " plus", "probability": 0.9169921875}, {"start": 2885.83, "end": 2887.37, "word": " minus", "probability": 0.9609375}, {"start": 2887.37, "end": 2887.91, "word": " two", "probability": 0.615234375}, {"start": 2887.91, "end": 2888.55, "word": " sigma", "probability": 0.9140625}, {"start": 2888.55, "end": 2888.71, "word": " and", "probability": 0.79150390625}, {"start": 2888.71, "end": 2888.87, "word": " mu", "probability": 0.94287109375}, {"start": 2888.87, "end": 2889.11, "word": " plus", "probability": 0.955078125}, {"start": 2889.11, "end": 2889.31, "word": " two", "probability": 0.92822265625}, {"start": 2889.31, "end": 2889.63, "word": " sigma.", "probability": 0.9228515625}], "temperature": 1.0}, {"id": 107, "seek": 291690, "start": 2890.94, "end": 2916.9, "text": " The only difference is that this one is going to be minus 2 and do the same. That's the empirical rule we discussed in chapter 3. So here we can find any probability, not just 95 or 68 or 99.7. We can use the normal table to give or to find or to compute any probability.", "tokens": [440, 787, 2649, 307, 300, 341, 472, 307, 516, 281, 312, 3175, 568, 293, 360, 264, 912, 13, 663, 311, 264, 31886, 4978, 321, 7152, 294, 7187, 805, 13, 407, 510, 321, 393, 915, 604, 8482, 11, 406, 445, 13420, 420, 23317, 420, 11803, 13, 22, 13, 492, 393, 764, 264, 2710, 3199, 281, 976, 420, 281, 915, 420, 281, 14722, 604, 8482, 13], "avg_logprob": -0.20120192307692308, "compression_ratio": 1.4702702702702704, "no_speech_prob": 0.0, "words": [{"start": 2890.94, "end": 2891.54, "word": " The", "probability": 0.6552734375}, {"start": 2891.54, "end": 2892.14, "word": " only", "probability": 0.9248046875}, {"start": 2892.14, "end": 2892.68, "word": " difference", "probability": 0.8671875}, {"start": 2892.68, "end": 2893.04, "word": " is", "probability": 0.93017578125}, {"start": 2893.04, "end": 2893.38, "word": " that", "probability": 0.8837890625}, {"start": 2893.38, "end": 2894.02, "word": " this", 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For mu plus or minus three sigma, it covers around all the data, 99.7. So just do it at home, you will see that the exact area is 95.44 instead of 95.", "tokens": [407, 797, 11, 337, 264, 661, 472, 11, 2992, 1804, 420, 3175, 732, 12771, 11, 309, 10538, 466, 13420, 4, 295, 264, 10298, 13, 1171, 2992, 1804, 420, 3175, 1045, 12771, 11, 309, 10538, 926, 439, 264, 1412, 11, 11803, 13, 22, 13, 407, 445, 360, 309, 412, 1280, 11, 291, 486, 536, 300, 264, 1900, 1859, 307, 13420, 13, 13912, 2602, 295, 13420, 13], "avg_logprob": -0.1667850430716168, "compression_ratio": 1.50625, "no_speech_prob": 0.0, "words": [{"start": 2928.27, "end": 2928.47, "word": " So", "probability": 0.921875}, {"start": 2928.47, "end": 2928.75, "word": " again,", "probability": 0.8046875}, {"start": 2928.87, "end": 2929.05, "word": " for", "probability": 0.95556640625}, {"start": 2929.05, "end": 2929.35, "word": " the", "probability": 0.9267578125}, {"start": 2929.35, "end": 2929.97, "word": " other", "probability": 0.8994140625}, {"start": 2929.97, "end": 2930.29, "word": " one,", 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So that's the empirical rule we discussed in chapter three. I'm going to stop at this point, which is the x value for the normal probability. 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