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Normal distribution excel exercise | Probability and Statistics | Khan Academy

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(Long-26 minutes) Presentation on spreadsheet to show that the normal distribution approximates the binomial distribution for a large number of trials.
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{'title': 'Normal distribution excel exercise | Probability and Statistics | Khan Academy', 'heatmap': [{'end': 205.354, 'start': 183.452, 'weight': 0.72}, {'end': 474.781, 'start': 405.134, 'weight': 0.806}], 'summary': 'Covers the significance and applications of normal distribution in statistics, the formula for normal distribution, simulating games with coin flips, probability of reaching a final position after a series of moves, and the convergence of binomial distribution to normal distribution as the number of moves increases.', 'chapters': [{'end': 103.787, 'segs': [{'end': 86.543, 'src': 'embed', 'start': 21.651, 'weight': 0, 'content': [{'end': 25.772, 'text': 'Everyone should know about this, because it touches on every single aspect of our lives.', 'start': 21.651, 'duration': 4.121}, {'end': 30.173, 'text': "And that's the normal distribution, or the Gaussian distribution, or the bell curve.", 'start': 25.812, 'duration': 4.361}, {'end': 40.89, 'text': 'And just to kind of give you a preview of what it is, and my preview will actually make it seem pretty strange, but as we go through this video,', 'start': 31.147, 'duration': 9.743}, {'end': 43.31, 'text': "hopefully you'll get a little bit more of an intuition of what it's all about.", 'start': 40.89, 'duration': 2.42}, {'end': 49.992, 'text': "But the Gaussian distribution, or the normal distribution, they're just two words for the same thing.", 'start': 43.771, 'duration': 6.221}, {'end': 52.053, 'text': 'It was actually Gauss who came up with it.', 'start': 50.032, 'duration': 2.021}, {'end': 56.728, 'text': 'I think he was studying astronomical phenomenon when he did.', 'start': 52.985, 'duration': 3.743}, {'end': 61.011, 'text': "But it's a probability density function, just like we studied the Poisson distribution.", 'start': 57.329, 'duration': 3.682}, {'end': 61.732, 'text': "It's just like that.", 'start': 61.051, 'duration': 0.681}, {'end': 64.294, 'text': 'And just to give you the preview, it looks like this.', 'start': 62.232, 'duration': 2.062}, {'end': 68.557, 'text': 'of getting any x.', 'start': 67.697, 'duration': 0.86}, {'end': 74.019, 'text': "And it's a class of probability distribution functions, just like the binomial distribution is,", 'start': 68.557, 'duration': 5.462}, {'end': 76.54, 'text': 'and the Poisson distribution is based on a bunch of parameters.', 'start': 74.019, 'duration': 2.521}, {'end': 81.781, 'text': "But it's equal to, and this is how you would traditionally see it written in a lot of textbooks.", 'start': 77.08, 'duration': 4.701}, {'end': 86.543, 'text': "And if we have time, I'd like to rearrange the algebra, just so you get a little bit more intuition of how it all works out.", 'start': 81.841, 'duration': 4.702}], 'summary': 'The gaussian distribution, also known as the normal distribution, is a probability density function, first introduced by gauss, and has wide-ranging implications in various aspects of our lives.', 'duration': 64.892, 'max_score': 21.651, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ21651.jpg'}], 'start': 2.188, 'title': 'The normal distribution', 'summary': 'Covers the significance and applications of the normal distribution in statistics, emphasizing its importance in various scientific fields and its relation to probability density functions as pioneered by gauss.', 'chapters': [{'end': 103.787, 'start': 2.188, 'title': 'Understanding the normal distribution', 'summary': 'Covers the significance of the normal distribution in statistics and its widespread application in scientific fields, emphasizing its importance in various aspects of life and its relation to probability density functions, as pioneered by gauss.', 'duration': 101.599, 'highlights': ['The normal distribution, or the Gaussian distribution, is considered the single most important concept in statistics and widely applicable across scientific fields, touching every aspect of our lives.', 'It is a probability density function, similar to the Poisson distribution, and is based on a class of probability distribution functions with a set of parameters.', 'The normal distribution, also known as the bell curve, was pioneered by Gauss and is a fundamental concept that should be included in core curriculums.', 'Gauss developed the normal distribution while studying astronomical phenomena, and it is represented by a specific probability density function.', "The chapter provides a preview of the normal distribution and its significance, aiming to enhance viewers' intuition about its nature and applications."]}], 'duration': 101.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ2188.jpg', 'highlights': ['The normal distribution, or the Gaussian distribution, is considered the single most important concept in statistics and widely applicable across scientific fields, touching every aspect of our lives.', 'The normal distribution, also known as the bell curve, was pioneered by Gauss and is a fundamental concept that should be included in core curriculums.', 'Gauss developed the normal distribution while studying astronomical phenomena, and it is represented by a specific probability density function.', 'It is a probability density function, similar to the Poisson distribution, and is based on a class of probability distribution functions with a set of parameters.', "The chapter provides a preview of the normal distribution and its significance, aiming to enhance viewers' intuition about its nature and applications."]}, {'end': 323.865, 'segs': [{'end': 153.261, 'src': 'embed', 'start': 103.827, 'weight': 0, 'content': [{'end': 108.909, 'text': 'It approximates factorials with essentially a continuous function.', 'start': 103.827, 'duration': 5.082}, {'end': 110.149, 'text': "But I won't go into that right now.", 'start': 108.949, 'duration': 1.2}, {'end': 112.629, 'text': 'But the binomial.', 'start': 110.189, 'duration': 2.44}, {'end': 116.517, 'text': "I'm sorry, the normal distribution is 1 over.", 'start': 112.629, 'duration': 3.888}, {'end': 118.038, 'text': "this is how it's normally written.", 'start': 116.517, 'duration': 1.521}, {'end': 127.624, 'text': 'the standard deviation times the square root of 2 pi times e to the minus 1 half.', 'start': 118.038, 'duration': 9.586}, {'end': 130.726, 'text': 'Well, I like to write it this way.', 'start': 128.324, 'duration': 2.402}, {'end': 131.527, 'text': "It's easier to remember.", 'start': 130.765, 'duration': 0.762}, {'end': 141.834, 'text': "Times whatever value we're trying to get minus the mean of our distribution divided by the standard deviation of our distribution.", 'start': 131.967, 'duration': 9.867}, {'end': 147.937, 'text': 'squared And so if you think about it, actually, this is a good thing to just notice right now.', 'start': 143.334, 'duration': 4.603}, {'end': 153.261, 'text': "This is how far I'm from the mean, and we're dividing that by the standard deviation of whatever our distribution is.", 'start': 148.358, 'duration': 4.903}], 'summary': 'Discusses approximating factorials and normal distribution calculations using a simpler formula.', 'duration': 49.434, 'max_score': 103.827, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ103827.jpg'}, {'end': 214.105, 'src': 'heatmap', 'start': 183.452, 'weight': 4, 'content': [{'end': 189.497, 'text': 'So this whole term right here is how many standard deviations we are away from the mean.', 'start': 183.452, 'duration': 6.045}, {'end': 191.498, 'text': 'And this is actually called the standard z-score.', 'start': 189.517, 'duration': 1.981}, {'end': 197.743, 'text': "One thing I've found in statistics is there's a lot of words and a lot of definitions, and they all sound very fancy.", 'start': 192.339, 'duration': 5.404}, {'end': 199.344, 'text': 'The standard z-score.', 'start': 197.943, 'duration': 1.401}, {'end': 205.354, 'text': 'But the underlying concept is pretty straightforward.', 'start': 199.925, 'duration': 5.429}, {'end': 214.105, 'text': "Let's say I had a probability distribution, and I get some x value that's out here, and it's 3 and 1 half the standard deviations away from the mean.", 'start': 205.374, 'duration': 8.731}], 'summary': 'Standard z-score measures distance from mean in units of standard deviations', 'duration': 24.588, 'max_score': 183.452, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ183452.jpg'}, {'end': 283.687, 'src': 'embed', 'start': 256.61, 'weight': 3, 'content': [{'end': 261.512, 'text': 'But the distribution of the sum of all of those trials approaches the normal distribution.', 'start': 256.61, 'duration': 4.902}, {'end': 262.914, 'text': "And I'll talk more about that later.", 'start': 261.533, 'duration': 1.381}, {'end': 272.36, 'text': "But that's why it's such a good distribution to kind of assume for a lot of underlying phenomenon if you're kind of modeling weather patterns or drug interactions.", 'start': 262.954, 'duration': 9.406}, {'end': 276.503, 'text': "And we'll talk about where it might work well and where it might not work so well.", 'start': 272.48, 'duration': 4.023}, {'end': 280.325, 'text': 'Like sometimes people might assume things like a normal distribution in finance.', 'start': 276.563, 'duration': 3.762}, {'end': 283.687, 'text': "And we see in the financial crisis that's led to a lot of things blowing up.", 'start': 280.365, 'duration': 3.322}], 'summary': 'Normal distribution is useful for modeling weather patterns and drug interactions, but may lead to issues in finance.', 'duration': 27.077, 'max_score': 256.61, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ256610.jpg'}], 'start': 103.827, 'title': 'Normal distribution', 'summary': 'Explains the normal distribution and its formula, highlighting the method to calculate how far a value is from the mean using the standard deviation. it also introduces the normal distribution as a good approximation for the binomial distribution and emphasizes its use in modeling various phenomena and the central limit theorem.', 'chapters': [{'end': 153.261, 'start': 103.827, 'title': 'Normal distribution explanation', 'summary': 'Explains the normal distribution and its formula, highlighting the method to calculate how far a value is from the mean using the standard deviation.', 'duration': 49.434, 'highlights': ['The formula of normal distribution is 1/(standard deviation * √(2π)) * e^(-1/2)', 'The expression for calculating how far a value is from the mean is (value - mean) / standard deviation', 'The explanation of approximating factorials with a continuous function']}, {'end': 323.865, 'start': 153.281, 'title': 'Normal distribution and its applications', 'summary': 'Introduces the normal distribution as a good approximation for the binomial distribution, emphasizing its use in modeling various phenomena and the central limit theorem.', 'duration': 170.584, 'highlights': ['The normal distribution is a good approximation for the binomial distribution and vice versa, especially when taking enough trials. The normal distribution is introduced as a good approximation for the binomial distribution, highlighting the importance of taking enough trials for accuracy.', 'The standard z-score is explained as a measure of how many standard deviations a value is away from the mean in a probability distribution. The concept of the standard z-score is explained as a measure of deviation from the mean in a probability distribution.', 'The normal distribution is emphasized as a suitable model for phenomena like weather patterns and drug interactions, but cautioned against its blind application in finance. The normal distribution is recommended for modeling weather patterns and drug interactions, but its potential pitfalls in finance are highlighted.']}], 'duration': 220.038, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ103827.jpg', 'highlights': ['The formula of normal distribution is 1/(standard deviation * √(2π)) * e^(-1/2)', 'The expression for calculating how far a value is from the mean is (value - mean) / standard deviation', 'The explanation of approximating factorials with a continuous function', 'The normal distribution is a good approximation for the binomial distribution and vice versa, especially when taking enough trials', 'The standard z-score is explained as a measure of how many standard deviations a value is away from the mean in a probability distribution', 'The normal distribution is emphasized as a suitable model for phenomena like weather patterns and drug interactions, but cautioned against its blind application in finance']}, {'end': 933.404, 'segs': [{'end': 359.275, 'src': 'embed', 'start': 323.885, 'weight': 0, 'content': [{'end': 328.688, 'text': 'And I encourage you to play with it and maybe do other spreadsheets where you experiment with it.', 'start': 323.885, 'duration': 4.803}, {'end': 335.353, 'text': "So this spreadsheet, what we do is we're doing a game where let's say I'm sitting, I'm on a street, and I flip a coin.", 'start': 328.728, 'duration': 6.625}, {'end': 336.875, 'text': 'I flip a completely fair coin.', 'start': 335.393, 'duration': 1.482}, {'end': 345.745, 'text': "If I get heads, let's say this is heads, I take a step backwards, or let's say a step to the left.", 'start': 337.395, 'duration': 8.35}, {'end': 348.473, 'text': 'And if I get a tails, I take a step to the right.', 'start': 346.106, 'duration': 2.367}, {'end': 359.275, 'text': 'So in general, I always have a 50% chance of taking a step to the left, and I have a 50% chance of taking a step to the right.', 'start': 351.109, 'duration': 8.166}], 'summary': 'Experiment with a coin flip game in a spreadsheet for 50% chance to move left or right.', 'duration': 35.39, 'max_score': 323.885, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ323885.jpg'}, {'end': 453.704, 'src': 'embed', 'start': 422.567, 'weight': 6, 'content': [{'end': 430.893, 'text': 'The mean of the binomial distribution is essentially the probability of taking a left step times the total number of trials.', 'start': 422.567, 'duration': 8.326}, {'end': 432.234, 'text': "And so that's equal to 5.", 'start': 431.193, 'duration': 1.041}, {'end': 433.335, 'text': "That's where that number comes from.", 'start': 432.234, 'duration': 1.101}, {'end': 437.315, 'text': "And then the variance, and I'm not sure if I went over this.", 'start': 434.053, 'duration': 3.262}, {'end': 438.796, 'text': 'I need to prove this to you if I have.', 'start': 437.355, 'duration': 1.441}, {'end': 443.138, 'text': "And I'll make a whole other video on the variance of the binomial distribution.", 'start': 438.816, 'duration': 4.322}, {'end': 453.704, 'text': 'But the variance is essentially equal to the number of trials, 10, times the probability of taking the left step, or kind of a successful trial.', 'start': 443.178, 'duration': 10.526}], 'summary': 'The mean of the binomial distribution is 5 and the variance is 10.', 'duration': 31.137, 'max_score': 422.567, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ422567.jpg'}, {'end': 494.806, 'src': 'heatmap', 'start': 405.134, 'weight': 5, 'content': [{'end': 412.039, 'text': "So what I've done right here is I have this little assumption here, and I encourage you to fill that out and change it as you like.", 'start': 405.134, 'duration': 6.905}, {'end': 413.9, 'text': 'This is the number of steps I take.', 'start': 412.379, 'duration': 1.521}, {'end': 417.643, 'text': 'This is the mean number of left steps.', 'start': 415.362, 'duration': 2.281}, {'end': 422.507, 'text': 'And all I did is I got the probability, and we figured out the mean of the binomial distribution.', 'start': 418.104, 'duration': 4.403}, {'end': 430.893, 'text': 'The mean of the binomial distribution is essentially the probability of taking a left step times the total number of trials.', 'start': 422.567, 'duration': 8.326}, {'end': 432.234, 'text': "And so that's equal to 5.", 'start': 431.193, 'duration': 1.041}, {'end': 433.335, 'text': "That's where that number comes from.", 'start': 432.234, 'duration': 1.101}, {'end': 437.315, 'text': "And then the variance, and I'm not sure if I went over this.", 'start': 434.053, 'duration': 3.262}, {'end': 438.796, 'text': 'I need to prove this to you if I have.', 'start': 437.355, 'duration': 1.441}, {'end': 443.138, 'text': "And I'll make a whole other video on the variance of the binomial distribution.", 'start': 438.816, 'duration': 4.322}, {'end': 453.704, 'text': 'But the variance is essentially equal to the number of trials, 10, times the probability of taking the left step, or kind of a successful trial.', 'start': 443.178, 'duration': 10.526}, {'end': 464.498, 'text': "I'm defining left as a successful trial, but it could be right as well, times the probability of 1 minus a successful trial or a non-successful trial.", 'start': 454.364, 'duration': 10.134}, {'end': 468.199, 'text': "In this case, they're equally probable, and that's where I got the 2.5 from.", 'start': 464.538, 'duration': 3.661}, {'end': 469.539, 'text': "And that's all in the spreadsheet.", 'start': 468.619, 'duration': 0.92}, {'end': 474.781, 'text': 'If you actually click on the cell and look at the actual form that I did, that, although sometimes, when you see it in Excel,', 'start': 469.579, 'duration': 5.202}, {'end': 475.661, 'text': "it's a little bit confusing.", 'start': 474.781, 'duration': 0.88}, {'end': 479.922, 'text': 'And this is just the square root of that number, right? The standard deviation is just the square root of the variance.', 'start': 475.721, 'duration': 4.201}, {'end': 484.683, 'text': "So that's just the square root of 2.5.", 'start': 480.462, 'duration': 4.221}, {'end': 486.984, 'text': 'And so if you look here, this says, OK.', 'start': 484.683, 'duration': 2.301}, {'end': 494.806, 'text': 'What is the probability that I take 0 steps? So I take a total of 10 steps, just to understand the spreadsheet.', 'start': 488.422, 'duration': 6.384}], 'summary': 'Using binomial distribution, the mean of left steps is 5, with a variance of 2.5.', 'duration': 89.672, 'max_score': 405.134, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ405134.jpg'}, {'end': 676.133, 'src': 'embed', 'start': 651.01, 'weight': 1, 'content': [{'end': 656.614, 'text': 'And then, once again, the whole point of this is to see that the normal distribution is a good approximation,', 'start': 651.01, 'duration': 5.604}, {'end': 659.076, 'text': "but they're so close that you can't even see the difference.", 'start': 656.614, 'duration': 2.462}, {'end': 666.441, 'text': 'If you only took four steps, OK, I think you can see here, the blue here is definitely, let me get my screen drawer on.', 'start': 659.616, 'duration': 6.825}, {'end': 670.024, 'text': 'So let me draw this.', 'start': 669.283, 'duration': 0.741}, {'end': 672.186, 'text': 'The blue curve is right around there.', 'start': 670.424, 'duration': 1.762}, {'end': 673.907, 'text': 'So this is the binomial.', 'start': 672.786, 'duration': 1.121}, {'end': 676.133, 'text': "There's only a few points here.", 'start': 674.952, 'duration': 1.181}], 'summary': 'Normal distribution closely approximates binomial distribution with few points.', 'duration': 25.123, 'max_score': 651.01, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ651010.jpg'}, {'end': 843.551, 'src': 'embed', 'start': 818.026, 'weight': 2, 'content': [{'end': 825.53, 'text': 'And then this is a little bit tricky because the binomial distribution is a discrete probability distribution.', 'start': 818.026, 'duration': 7.504}, {'end': 836.657, 'text': 'You could just look at this chart, or look here and you say what is the probability of having exactly one left step and three right steps,', 'start': 825.55, 'duration': 11.107}, {'end': 837.858, 'text': 'which puts me right here?', 'start': 836.657, 'duration': 1.201}, {'end': 841.32, 'text': 'Well, you just look at this chart and you say oh, that puts me right there.', 'start': 838.338, 'duration': 2.982}, {'end': 843.551, 'text': 'I just read that probability.', 'start': 842.429, 'duration': 1.122}], 'summary': 'Explains binomial distribution with 1 left step, 3 right steps.', 'duration': 25.525, 'max_score': 818.026, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ818026.jpg'}], 'start': 323.885, 'title': 'Binomial distribution in spreadsheets', 'summary': 'Discusses simulating a game with coin flips, calculating mean and variance, and understanding standard deviation in spreadsheets. it also explains the convergence of binomial distribution to normal distribution and challenges of handling discrete and continuous probability distributions.', 'chapters': [{'end': 486.984, 'start': 323.885, 'title': 'Binomial distribution and variance in spreadsheets', 'summary': 'Discusses using spreadsheets to simulate a game involving coin flips, calculating the mean and variance of the binomial distribution, and understanding the concept of standard deviation.', 'duration': 163.099, 'highlights': ['The chapter discusses using spreadsheets to simulate a game involving coin flips. The speaker encourages experimentation with spreadsheets to simulate a game involving coin flips.', 'Calculating the mean and variance of the binomial distribution. The speaker explains how to calculate the mean and variance of the binomial distribution using the probability of taking a left step and the total number of trials.', 'Understanding the concept of standard deviation. The speaker explains the concept of standard deviation as the square root of the variance and demonstrates its calculation using the spreadsheet.']}, {'end': 933.404, 'start': 488.422, 'title': 'Binomial distribution convergence', 'summary': 'Explains the binomial distribution by calculating the probability of taking 0 left steps out of 10 total steps, highlighting the convergence to the normal distribution and the challenge of handling discrete and continuous probability distributions.', 'duration': 444.982, 'highlights': ['The binomial distribution calculation for the probability of taking 0 left steps out of 10 total steps yields a value of 0.001, demonstrating the application of the binomial distribution in a specific scenario.', 'The visualization of the binomial distribution using a scatter plot chart illustrates the probabilities of taking different numbers of left steps and the corresponding final positions, emphasizing the discrete nature of the binomial distribution.', 'The comparison between the binomial distribution and the normal distribution underlines the challenge of dealing with the discrete and continuous nature of probability distributions, necessitating the consideration of ranges and approximation methods for continuous distributions.']}], 'duration': 609.519, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ323885.jpg', 'highlights': ['The speaker encourages experimentation with spreadsheets to simulate a game involving coin flips.', 'The comparison between the binomial distribution and the normal distribution underlines the challenge of dealing with the discrete and continuous nature of probability distributions, necessitating the consideration of ranges and approximation methods for continuous distributions.', 'The visualization of the binomial distribution using a scatter plot chart illustrates the probabilities of taking different numbers of left steps and the corresponding final positions, emphasizing the discrete nature of the binomial distribution.', 'The chapter discusses using spreadsheets to simulate a game involving coin flips.', 'The binomial distribution calculation for the probability of taking 0 left steps out of 10 total steps yields a value of 0.001, demonstrating the application of the binomial distribution in a specific scenario.', 'Understanding the concept of standard deviation. The speaker explains the concept of standard deviation as the square root of the variance and demonstrates its calculation using the spreadsheet.', 'Calculating the mean and variance of the binomial distribution. The speaker explains how to calculate the mean and variance of the binomial distribution using the probability of taking a left step and the total number of trials.']}, {'end': 1201.388, 'segs': [{'end': 978.502, 'src': 'embed', 'start': 933.444, 'weight': 3, 'content': [{'end': 940.437, 'text': 'So what is this number right here? Well, I said, what is the probability? Well, let me do something like this.', 'start': 933.444, 'duration': 6.993}, {'end': 943.078, 'text': "Let's say this one right here, because I don't want to use the 0.", 'start': 940.457, 'duration': 2.621}, {'end': 948.722, 'text': 'So what is the probability that I take one left step? I kind of used left steps as a success.', 'start': 943.078, 'duration': 5.644}, {'end': 962.594, 'text': 'So what is the probability of 1, and that equaled 1 over The standard deviation, when I only took four steps, the standard deviation was 1.', 'start': 948.742, 'duration': 13.852}, {'end': 963.855, 'text': 'So 1 over 1.', 'start': 962.594, 'duration': 1.261}, {'end': 969.978, 'text': 'Actually, let me change this, because I think it might be, let me change it to a higher number.', 'start': 963.855, 'duration': 6.123}, {'end': 973.619, 'text': "Let's see, if I make this, I don't know.", 'start': 970.658, 'duration': 2.961}, {'end': 978.102, 'text': "Let me go back to the example where I'm at 10.", 'start': 973.84, 'duration': 4.262}, {'end': 978.502, 'text': 'All right.', 'start': 978.102, 'duration': 0.4}], 'summary': 'Calculating the probability of left steps, with standard deviation of 1 over 4 steps.', 'duration': 45.058, 'max_score': 933.444, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ933444.jpg'}, {'end': 1050.307, 'src': 'embed', 'start': 1002.854, 'weight': 1, 'content': [{'end': 1004.595, 'text': "So that's this point right here.", 'start': 1002.854, 'duration': 1.741}, {'end': 1007.237, 'text': "So what's that probability?", 'start': 1006.257, 'duration': 0.98}, {'end': 1008.338, 'text': 'How do I figure this out?', 'start': 1007.277, 'duration': 1.061}, {'end': 1009.879, 'text': 'using the probability density function?', 'start': 1008.338, 'duration': 1.541}, {'end': 1011.14, 'text': 'How do I figure this out?', 'start': 1010.36, 'duration': 0.78}, {'end': 1015.978, 'text': 'Well, I say the probability of taking two left steps.', 'start': 1011.18, 'duration': 4.798}, {'end': 1016.938, 'text': "That's how I calculated it.", 'start': 1016.018, 'duration': 0.92}, {'end': 1018.719, 'text': "If you actually click on the cell, you'll see that.", 'start': 1016.958, 'duration': 1.761}, {'end': 1026.784, 'text': 'Is equal to 1 over the standard deviation, 1.581.', 'start': 1019.26, 'duration': 7.524}, {'end': 1028.665, 'text': 'And I just directly referenced the cell there.', 'start': 1026.784, 'duration': 1.881}, {'end': 1032.327, 'text': 'Divided times the square root of 2 pi.', 'start': 1030.286, 'duration': 2.041}, {'end': 1039.94, 'text': 'And I always go out in awe of the whole notion of e to the i pi is equal to negative 1 and all of that.', 'start': 1034.717, 'duration': 5.223}, {'end': 1045.184, 'text': 'But this is another amazing thing that all of a sudden we have this as we take many, many, many trials.', 'start': 1039.98, 'duration': 5.204}, {'end': 1048.326, 'text': 'we have this formula that has e and pi in it and square roots.', 'start': 1045.184, 'duration': 3.142}, {'end': 1050.307, 'text': 'But once again, these two numbers just keep showing up.', 'start': 1048.346, 'duration': 1.961}], 'summary': 'Calculating probability using density function, result: 1/1.581', 'duration': 47.453, 'max_score': 1002.854, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ1002854.jpg'}, {'end': 1190.901, 'src': 'embed', 'start': 1163.627, 'weight': 0, 'content': [{'end': 1169.19, 'text': 'The curves for the normal distribution here is in purple and the binomial distribution is in blue.', 'start': 1163.627, 'duration': 5.563}, {'end': 1171.751, 'text': "So they're almost right on top of each other.", 'start': 1169.23, 'duration': 2.521}, {'end': 1178.595, 'text': 'When the number of steps I took was a little bit smaller and as you take many, many, many, many more steps,', 'start': 1172.412, 'duration': 6.183}, {'end': 1180.476, 'text': 'they almost converge right on top of each other.', 'start': 1178.595, 'duration': 1.881}, {'end': 1182.216, 'text': 'And I encourage you to play with this spreadsheet.', 'start': 1180.496, 'duration': 1.72}, {'end': 1185.418, 'text': 'And actually, let me show you that they converge.', 'start': 1182.637, 'duration': 2.781}, {'end': 1190.901, 'text': "There's a convergence worksheet on this spreadsheet as well, if you click on the bottom tab on convergence.", 'start': 1185.638, 'duration': 5.263}], 'summary': 'Normal and binomial distributions converge with more steps.', 'duration': 27.274, 'max_score': 1163.627, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ1163627.jpg'}], 'start': 933.444, 'title': 'Probability and occurrence', 'summary': 'Discusses the probability of left steps and occurrence of e, pi, and square roots, using probability density function, standard deviation, and demonstrating convergence of distributions as the number of steps increases.', 'chapters': [{'end': 1028.665, 'start': 933.444, 'title': 'Probability of left steps', 'summary': 'Discusses the probability of taking a specific number of left steps out of a total number of steps, utilizing the probability density function and standard deviation calculations.', 'duration': 95.221, 'highlights': ['The probability of taking two left steps out of a total of 10 steps is calculated as 1 divided by the standard deviation, which equals 1.581. The probability of taking two left steps out of a total of 10 steps is calculated as 1 divided by the standard deviation, which equals 1.581.', 'The probability of taking one left step out of a total of four steps is calculated as 1 divided by the standard deviation, which equals 1. The probability of taking one left step out of a total of four steps is calculated as 1 divided by the standard deviation, which equals 1.', 'The chapter discusses the concept of using left steps as a measure of success in the context of probability calculations. The chapter discusses the concept of using left steps as a measure of success in the context of probability calculations.']}, {'end': 1201.388, 'start': 1030.286, 'title': 'Occurrence of e, pi, and square roots', 'summary': 'Discusses the occurrence of e, pi, and square roots in a formula derived from many trials, illustrating their significance in the order of the universe. it also demonstrates the convergence of normal and binomial distributions as the number of steps increases.', 'duration': 171.102, 'highlights': ['The chapter discusses the occurrence of e, pi, and square roots in a formula derived from many trials, illustrating their significance in the order of the universe. The formula derived from many trials contains e, pi, and square roots, demonstrating their recurring presence, suggesting their significance in the order of the universe.', 'It also demonstrates the convergence of normal and binomial distributions as the number of steps increases. The convergence of normal and binomial distributions is illustrated as the number of steps increases, showing their almost identical alignment with a smaller number of steps and convergence with many more steps.']}], 'duration': 267.944, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ933444.jpg', 'highlights': ['The convergence of normal and binomial distributions is illustrated as the number of steps increases, showing their almost identical alignment with a smaller number of steps and convergence with many more steps.', 'The formula derived from many trials contains e, pi, and square roots, demonstrating their recurring presence, suggesting their significance in the order of the universe.', 'The probability of taking two left steps out of a total of 10 steps is calculated as 1 divided by the standard deviation, which equals 1.581.', 'The probability of taking one left step out of a total of four steps is calculated as 1 divided by the standard deviation, which equals 1.', 'The chapter discusses the concept of using left steps as a measure of success in the context of probability calculations.']}, {'end': 1562.877, 'segs': [{'end': 1234.208, 'src': 'embed', 'start': 1202.709, 'weight': 0, 'content': [{'end': 1208.914, 'text': "So this is, you know, what's the probability of moving left, right? So this is just saying, I'm just fixing a point.", 'start': 1202.709, 'duration': 6.205}, {'end': 1213.318, 'text': "What's the probability and you could change this of my final position being 10?", 'start': 1208.954, 'duration': 4.364}, {'end': 1220.333, 'text': 'And then this essentially tells you that if I take 10 moves, Then for my final position to be 10 to the right,', 'start': 1213.318, 'duration': 7.015}, {'end': 1222.837, 'text': 'I have to take 10 right moves and 0 left moves.', 'start': 1220.333, 'duration': 2.504}, {'end': 1223.738, 'text': "That's a typo right there.", 'start': 1222.977, 'duration': 0.761}, {'end': 1225.06, 'text': 'It should be moves, not movest.', 'start': 1223.758, 'duration': 1.302}, {'end': 1234.208, 'text': 'If I take 20 moves to end up 10 moves to the right, then I have to make 15 right moves and 5 left moves.', 'start': 1226.382, 'duration': 7.826}], 'summary': 'Analyzing probability of moving left or right with specific moves and positions.', 'duration': 31.499, 'max_score': 1202.709, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ1202709.jpg'}, {'end': 1476.281, 'src': 'embed', 'start': 1458.988, 'weight': 3, 'content': [{'end': 1472.118, 'text': 'The difference between what the normal probability distribution tells us and the binomial probability distribution tells us gets smaller and smaller in terms of the probability of you ending up 10 moves to the right.', 'start': 1458.988, 'duration': 13.13}, {'end': 1474.479, 'text': 'And you can change this number here.', 'start': 1472.158, 'duration': 2.321}, {'end': 1476.281, 'text': 'Let me change it just to show you.', 'start': 1475.12, 'duration': 1.161}], 'summary': 'Normal vs binomial distribution difference decreases with probability of moving 10 steps right.', 'duration': 17.293, 'max_score': 1458.988, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ1458988.jpg'}, {'end': 1562.877, 'src': 'embed', 'start': 1540.678, 'weight': 2, 'content': [{'end': 1551.747, 'text': 'As you take more and more flips of your coin, the normal distribution becomes a much better approximation for the actual binomial distribution.', 'start': 1540.678, 'duration': 11.069}, {'end': 1554.81, 'text': 'And as you approach infinity, they actually converge to each other.', 'start': 1551.807, 'duration': 3.003}, {'end': 1556.812, 'text': "Anyway, that's all for this video.", 'start': 1555.23, 'duration': 1.582}, {'end': 1562.416, 'text': "I'll actually do several more videos on the normal distribution just because it is such an important concept.", 'start': 1557.152, 'duration': 5.264}, {'end': 1562.877, 'text': 'See you soon.', 'start': 1562.496, 'duration': 0.381}], 'summary': 'With more coin flips, normal distribution approximates binomial distribution; as they approach infinity, they converge.', 'duration': 22.199, 'max_score': 1540.678, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ1540678.jpg'}], 'start': 1202.709, 'title': 'Probability and normal distribution', 'summary': 'Discusses the probability of reaching a final position 10 moves to the right after a series of moves, showcasing how the probability changes with an increasing number of moves and demonstrating that as the number of moves increases, the probability of ending up 10 moves to the right converges to a specific value. it also explains how the normal distribution becomes a better approximation for the binomial distribution as the total number of moves increases, and as it converges to infinity, showing the decreasing difference between the two distributions in terms of probability.', 'chapters': [{'end': 1268.29, 'start': 1202.709, 'title': 'Probability of final position', 'summary': 'Discusses the probability of reaching a final position 10 moves to the right after a series of moves, showcasing how the probability changes with an increasing number of moves and demonstrating that as the number of moves increases, the probability of ending up 10 moves to the right converges to a specific value.', 'duration': 65.581, 'highlights': ['The chapter discusses the probability of reaching a final position 10 moves to the right after a series of moves. The speaker introduces the concept of determining the probability of ending up 10 moves to the right after a series of moves.', 'The probability changes with an increasing number of moves. The speaker demonstrates that the probability of ending up 10 moves to the right changes based on the total number of moves taken.', 'As the number of moves increases, the probability of ending up 10 moves to the right converges to a specific value. It is explained that as the number of moves increases, the probability of ending up 10 moves to the right converges to a specific value, showcasing the convergence behavior of the probability.']}, {'end': 1562.877, 'start': 1268.982, 'title': 'Normal distribution approximation', 'summary': 'Explains how the normal distribution becomes a better approximation for the binomial distribution as the total number of moves increases, and as it converges to infinity, showing the decreasing difference between the two distributions in terms of probability.', 'duration': 293.895, 'highlights': ['The normal distribution becomes a better approximation for the binomial distribution as the total number of moves increases, and as it converges to infinity. Convergence to infinity, increasing total number of moves, improving approximation', 'The decreasing difference between the normal and binomial probability distributions in terms of the probability of ending up a certain number of moves to the right, as the total number of moves increases. Decreasing difference, increasing total number of moves, probability of ending up a certain number of moves to the right', 'Explanation of how the normal probability distribution becomes a much better approximation for the actual binomial distribution as more flips of the coin are taken, leading to convergence to each other as the total number of moves approaches infinity. Improving approximation, more flips of the coin, convergence to each other, approaching infinity']}], 'duration': 360.168, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yTGEMoaWDCQ/pics/yTGEMoaWDCQ1202709.jpg', 'highlights': ['The probability of ending up 10 moves to the right changes based on the total number of moves taken.', 'As the number of moves increases, the probability of ending up 10 moves to the right converges to a specific value.', 'The normal distribution becomes a better approximation for the binomial distribution as the total number of moves increases, and as it converges to infinity.', 'The decreasing difference between the normal and binomial probability distributions in terms of the probability of ending up a certain number of moves to the right, as the total number of moves increases.']}], 'highlights': ['The normal distribution, or the Gaussian distribution, is considered the single most important concept in statistics and widely applicable across scientific fields, touching every aspect of our lives.', 'The normal distribution, also known as the bell curve, was pioneered by Gauss and is a fundamental concept that should be included in core curriculums.', 'The normal distribution becomes a better approximation for the binomial distribution as the total number of moves increases, and as it converges to infinity.', 'The formula of normal distribution is 1/(standard deviation * √(2π)) * e^(-1/2)', 'The convergence of normal and binomial distributions is illustrated as the number of steps increases, showing their almost identical alignment with a smaller number of steps and convergence with many more steps.', 'The comparison between the binomial distribution and the normal distribution underlines the challenge of dealing with the discrete and continuous nature of probability distributions, necessitating the consideration of ranges and approximation methods for continuous distributions.']}