title

Lecture 29: Law of Large Numbers and Central Limit Theorem | Statistics 110

description

We introduce and prove versions of the Law of Large Numbers and Central Limit Theorem, which are two of the most famous and important theorems in all of statistics.

detail

{'title': 'Lecture 29: Law of Large Numbers and Central Limit Theorem | Statistics 110', 'heatmap': [{'end': 917.198, 'start': 835.087, 'weight': 0.792}], 'summary': 'Covers probability theorems, discussing the law of large numbers and central limit theorem, convergence of random variables, and significance in estimating probability. it explores the proof of the weak law of large numbers and provides informative convergence as the sample size increases.', 'chapters': [{'end': 238.351, 'segs': [{'end': 32.796, 'src': 'embed', 'start': 0.447, 'weight': 0, 'content': [{'end': 2.109, 'text': "All right, so let's get started.", 'start': 0.447, 'duration': 1.662}, {'end': 11.64, 'text': "So today we're gonna talk about what are probably the two most famous theorems in the entire history of probability.", 'start': 2.469, 'duration': 9.171}, {'end': 15.524, 'text': "They're called the Law of Large Numbers and the Central Limit Theorem.", 'start': 13.082, 'duration': 2.442}, {'end': 20.991, 'text': "They're closely related, so it makes sense to do them together, kind of compare and contrast them.", 'start': 15.865, 'duration': 5.126}, {'end': 27.093, 'text': "I can't think of a more famous probability theorem than these two.", 'start': 22.271, 'duration': 4.822}, {'end': 32.796, 'text': 'So the setup for today is that we have iid random variables.', 'start': 27.694, 'duration': 5.102}], 'summary': 'Discussion on the law of large numbers and central limit theorem in probability theory.', 'duration': 32.349, 'max_score': 0.447, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA447.jpg'}, {'end': 86.444, 'src': 'embed', 'start': 56.471, 'weight': 1, 'content': [{'end': 60.312, 'text': "So we're assuming that these are finite for now, that the mean and variance exist.", 'start': 56.471, 'duration': 3.841}, {'end': 69.232, 'text': 'And both of these theorems, Tell us what happens to the sample mean as n gets large.', 'start': 60.852, 'duration': 8.38}, {'end': 72.414, 'text': 'So the sample mean is just defined as Xn bar.', 'start': 69.532, 'duration': 2.882}, {'end': 81.14, 'text': "Standard notation in statistics, we put a bar to mean averages, and that's just the average of the first n.", 'start': 74.115, 'duration': 7.025}, {'end': 86.444, 'text': "So to take the first n random variables and average them, so that's just called the sample mean.", 'start': 81.14, 'duration': 5.304}], 'summary': 'Theorems demonstrate sample mean behavior as n increases.', 'duration': 29.973, 'max_score': 56.471, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA56471.jpg'}, {'end': 198.69, 'src': 'embed', 'start': 127.191, 'weight': 2, 'content': [{'end': 131.974, 'text': "All right, so first here's what the law of large numbers says.", 'start': 127.191, 'duration': 4.783}, {'end': 142.199, 'text': "It's a very simple statement, and hopefully pretty intuitive too.", 'start': 131.994, 'duration': 10.205}, {'end': 159.681, 'text': 'Law of large numbers says that Xn bar converges to mu as n goes to infinity with probability 1.', 'start': 143.159, 'duration': 16.522}, {'end': 164.142, 'text': "That's the fine print, probability 1.", 'start': 159.681, 'duration': 4.461}, {'end': 170.424, 'text': "With probability 0, so something really crazy could happen, but we don't worry too much about it because it has probability 0.", 'start': 164.142, 'duration': 6.282}, {'end': 178.98, 'text': 'With probability 1, this is the sample mean, and it says that the sample mean, converges to the true mean.', 'start': 170.424, 'duration': 8.556}, {'end': 192.607, 'text': 'Okay, so that is a pretty nice, intuitive, easy to remember result, right?', 'start': 182.901, 'duration': 9.706}, {'end': 195.008, 'text': 'By true, I mean the theoretical mean.', 'start': 193.047, 'duration': 1.961}, {'end': 198.69, 'text': 'that is the expected value of Xj, for any j is mu.', 'start': 195.008, 'duration': 3.682}], 'summary': 'Law of large numbers states xn bar converges to mu as n goes to infinity with probability 1.', 'duration': 71.499, 'max_score': 127.191, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA127191.jpg'}], 'start': 0.447, 'title': 'Probability theorems', 'summary': 'Discusses the law of large numbers and the central limit theorem, highlighting the behavior of sample mean for iid random variables, and explains the convergence of sample mean to true mean with probability 1.', 'chapters': [{'end': 127.171, 'start': 0.447, 'title': 'Probability theorems: law of large numbers and central limit theorem', 'summary': 'Discusses the law of large numbers and the central limit theorem, focusing on the behavior of the sample mean as n gets large for iid random variables with the same mean and variance, aiming to understand their implications for data analysis.', 'duration': 126.724, 'highlights': ['The Law of Large Numbers and the Central Limit Theorem are the two most famous theorems in the history of probability. These theorems hold significant prominence in the realm of probability.', 'The theorems focus on the behavior of the sample mean as n gets large for iid random variables with the same mean and variance. The theorems specifically address the behavior of the sample mean as the sample size increases for independent and identically distributed random variables with identical mean and variance.', 'The sample mean, denoted as Xn bar, represents the average of the first n random variables. The sample mean, represented as Xn bar, is the average of the initial n random variables, serving as a fundamental concept in statistics.']}, {'end': 238.351, 'start': 127.191, 'title': 'Law of large numbers', 'summary': 'Explains the law of large numbers, stating that the sample mean converges to the true mean with probability 1, providing an intuitive and easy-to-remember result for understanding the convergence of random variables.', 'duration': 111.16, 'highlights': ['The law of large numbers states that Xn bar converges to mu as n goes to infinity with probability 1, offering a simple and intuitive understanding of random variable convergence.', "The sample mean converges to the true mean, indicated by the fact that the expected value of Xj for any j is mu, providing a clear understanding of the theoretical mean's relationship to the sample mean.", 'The convergence statement of the law of large numbers is defined point-wise, emphasizing the careful consideration required when defining limits of random variables.']}], 'duration': 237.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA447.jpg', 'highlights': ['The Law of Large Numbers and the Central Limit Theorem are the two most famous theorems in the history of probability.', 'The theorems focus on the behavior of the sample mean as n gets large for iid random variables with the same mean and variance.', 'The law of large numbers states that Xn bar converges to mu as n goes to infinity with probability 1.', 'The sample mean converges to the true mean, indicated by the fact that the expected value of Xj for any j is mu.']}, {'end': 492.897, 'segs': [{'end': 333.173, 'src': 'embed', 'start': 260.719, 'weight': 0, 'content': [{'end': 271.544, 'text': "In other words, this is an event, okay? Either these random variables converge or they don't, and we say that event has probability 1.", 'start': 260.719, 'duration': 10.825}, {'end': 285.309, 'text': "That's what the statement of the theorem is, okay? So to just give a simple example, let's think about what happens if we have Bernoulli p.", 'start': 271.544, 'duration': 13.765}, {'end': 296.251, 'text': "So if Xj is Bernoulli p, Then intuitively we're just imagining an infinite sequence of coin tosses, right?", 'start': 285.309, 'duration': 10.942}, {'end': 299.513, 'text': 'Where the probability of heads is p, okay?', 'start': 296.911, 'duration': 2.602}, {'end': 312.763, 'text': 'And then this says that if we add up all of these Bernoullis up to n, that is just in the first coin flips.', 'start': 300.093, 'duration': 12.67}, {'end': 314.584, 'text': 'how many times did the coin land heads?', 'start': 312.763, 'duration': 1.821}, {'end': 318.267, 'text': 'Divided by the number of flips, should converge to p.', 'start': 314.944, 'duration': 3.323}, {'end': 320.347, 'text': 'probability one.', 'start': 319.566, 'duration': 0.781}, {'end': 326.91, 'text': 'So, for example.', 'start': 325.309, 'duration': 1.601}, {'end': 328.551, 'text': 'so this is a very intuitive statement, right?', 'start': 326.91, 'duration': 1.641}, {'end': 333.173, 'text': "Like if it's a fair coin and you flip the coin a million times.", 'start': 328.591, 'duration': 4.582}], 'summary': 'In a sequence of bernoulli trials, the probability of heads converging to p with a fair coin and a million flips.', 'duration': 72.454, 'max_score': 260.719, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA260719.jpg'}, {'end': 492.897, 'src': 'embed', 'start': 441.862, 'weight': 3, 'content': [{'end': 446.609, 'text': "and then it's the feeling that you're due to win right?", 'start': 441.862, 'duration': 4.747}, {'end': 457.203, 'text': "You lost all these times and then And you might try to justify that using the law of large numbers and say well, the coin landed, let's say, heads,", 'start': 446.969, 'duration': 10.234}, {'end': 458.944, 'text': 'you win money, tails, you lose money.', 'start': 457.203, 'duration': 1.741}, {'end': 466.689, 'text': "You just lost money ten times in a row, but the law of large numbers says in the long run it's gonna go back to one half if it's fair.", 'start': 459.265, 'duration': 7.424}, {'end': 472.612, 'text': "So somehow you need to start winning a lot to compensate, right? That's not the way it works.", 'start': 467.049, 'duration': 5.563}, {'end': 475.993, 'text': 'Okay, the coin is memoryless.', 'start': 474.432, 'duration': 1.561}, {'end': 479.654, 'text': 'The coin does not care how many failures or how many losses you had before.', 'start': 476.013, 'duration': 3.641}, {'end': 489.276, 'text': "So the way it works is not through like, if you're unlucky at the beginning that somehow it gets offset later by an increase in heads.", 'start': 480.334, 'duration': 8.942}, {'end': 492.897, 'text': 'The way it works is through what we might call swamping.', 'start': 489.856, 'duration': 3.041}], 'summary': 'The law of large numbers does not guarantee compensation for previous losses in gambling.', 'duration': 51.035, 'max_score': 441.862, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA441862.jpg'}], 'start': 238.411, 'title': 'Random variables convergence and law of large numbers', 'summary': "Discusses the convergence of random variables and the significance of the law of large numbers in estimating probability. it uses the example of bernoulli p and emphasizes the necessity and relevance to the gambler's fallacy, while debunking the idea of compensating for previous losses through subsequent wins.", 'chapters': [{'end': 398.639, 'start': 238.411, 'title': 'Convergence of random variables', 'summary': 'Discusses the convergence of random variables, stating that the numbers obtained from the evaluation of specific outcomes of an experiment converge to a certain value with probability one, using the example of bernoulli p to illustrate the concept.', 'duration': 160.228, 'highlights': ['The chapter discusses the convergence of random variables, stating that the numbers obtained from the evaluation of specific outcomes of an experiment converge to a certain value with probability one. The convergence of random variables to a certain value with probability one. Example: Bernoulli p.', 'The statement of the theorem is that the event of random variables converging has a probability of one. The theorem states that the event of random variables converging has a probability of one.', 'Using the example of Bernoulli p, it is illustrated that the sum of Bernoulli trials should converge to p with probability one. The sum of Bernoulli trials should converge to p with probability one.', "The qualification 'with probability one' is necessary due to mathematical possibilities, such as a fair coin landing heads forever, although this is practically impossible. The qualification 'with probability one' accounts for mathematical possibilities, such as a fair coin landing heads forever, despite being practically impossible."]}, {'end': 492.897, 'start': 399.059, 'title': 'Law of large numbers in estimating probability', 'summary': "Discusses the significance of the law of large numbers in estimating probability, emphasizing its necessity and relevance to the gambler's fallacy, highlighting the concept of memoryless coin and debunking the idea of compensating for previous losses through subsequent wins.", 'duration': 93.838, 'highlights': ['The law of large numbers is crucial for estimating probability, as it provides a justification for using the proportion of heads obtained from flipping a coin many times as an approximation for p.', 'The chapter emphasizes that the law of large numbers is a necessary result for justifying the approximation of probability through coin flipping.', 'The concept of the memoryless coin is highlighted, debunking the idea that previous losses can be compensated for by subsequent wins, emphasizing that the coin does not care about past outcomes but instead operates through swamping.']}], 'duration': 254.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA238411.jpg', 'highlights': ['The convergence of random variables to a certain value with probability one is discussed, using the example of Bernoulli p.', 'The theorem states that the event of random variables converging has a probability of one, accounting for mathematical possibilities.', 'The sum of Bernoulli trials should converge to p with probability one, despite mathematical possibilities like a fair coin landing heads forever.', 'The law of large numbers is crucial for estimating probability, justifying the approximation of probability through coin flipping.', 'The concept of the memoryless coin is highlighted, debunking the idea that previous losses can be compensated for by subsequent wins.']}, {'end': 1060.719, 'segs': [{'end': 552.72, 'src': 'embed', 'start': 517.755, 'weight': 0, 'content': [{'end': 522.938, 'text': 'So those first million just get swamped out by the entire infinite future.', 'start': 517.755, 'duration': 5.183}, {'end': 527.883, 'text': "So that's what's going on here.", 'start': 523.96, 'duration': 3.923}, {'end': 541.014, 'text': 'Yeah, so to tell you one little story about the law of large numbers, a colleague of mine told me this story.', 'start': 530.826, 'duration': 10.188}, {'end': 545.178, 'text': 'He had a student once who said he hated statistics.', 'start': 541.094, 'duration': 4.084}, {'end': 550.639, 'text': 'And of course, my colleague was very shocked, like how can anyone hate statistics?', 'start': 546.497, 'duration': 4.142}, {'end': 552.72, 'text': 'And so he asked why?', 'start': 551.48, 'duration': 1.24}], 'summary': 'Law of large numbers: first million swamped by infinite future.', 'duration': 34.965, 'max_score': 517.755, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA517755.jpg'}, {'end': 634.956, 'src': 'embed', 'start': 605.613, 'weight': 2, 'content': [{'end': 614.219, 'text': 'Because if you kind of imagine kind of hypothetical, counterfactual world where this theorem was actually false,', 'start': 605.613, 'duration': 8.606}, {'end': 618.122, 'text': 'that would be really depressing to try to ever learn about the world, right?', 'start': 614.219, 'duration': 3.903}, {'end': 626.668, 'text': "Cuz, this is saying you're collecting more and more data, you're letting your sample size go to infinity, and this says you converge to the truth.", 'start': 618.142, 'duration': 8.526}, {'end': 634.956, 'text': "It would be like some weird setting where you get more and more data and more and more data, and yet you're not able to converge to the truth,", 'start': 627.769, 'duration': 7.187}], 'summary': 'In a hypothetical world, the theorem being false would hinder learning and data convergence.', 'duration': 29.343, 'max_score': 605.613, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA605613.jpg'}, {'end': 715.506, 'src': 'embed', 'start': 683.596, 'weight': 1, 'content': [{'end': 702.293, 'text': 'The weak law of large numbers says that for any C greater than 0, the probability that Xn bar minus the mean is greater than C goes to 0.', 'start': 683.596, 'duration': 18.697}, {'end': 706.197, 'text': "It's a very similar looking statement.", 'start': 702.293, 'duration': 3.904}, {'end': 709.298, 'text': "It's not exactly equivalent.", 'start': 707.836, 'duration': 1.462}, {'end': 715.506, 'text': "It's possible to show if you have to go through some real analysis for this, that is not necessary for our purposes.", 'start': 709.418, 'duration': 6.088}], 'summary': 'Weak law of large numbers: p(xn bar - mean > c) -> 0 for c > 0.', 'duration': 31.91, 'max_score': 683.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA683596.jpg'}, {'end': 917.198, 'src': 'heatmap', 'start': 835.087, 'weight': 0.792, 'content': [{'end': 844.794, 'text': "So by Chebyshev's inequality, this is less than or equal to the variance of Xn bar divided by C squared.", 'start': 835.087, 'duration': 9.707}, {'end': 847.837, 'text': "That's just exactly Chebyshev from last time.", 'start': 844.814, 'duration': 3.023}, {'end': 851.459, 'text': 'Now we just need the variance of Xn bar.', 'start': 849.658, 'duration': 1.801}, {'end': 858.504, 'text': "Variance of Xn bar, we'll just stare at the definition of Xn bar for a second.", 'start': 853.261, 'duration': 5.243}, {'end': 863.148, 'text': "There's a 1 over n in front, that comes out as 1 over n squared.", 'start': 858.865, 'duration': 4.283}, {'end': 871.982, 'text': "And then since I'm assuming they're iid and independent, the variance of the sum is just n times the variance of one term.", 'start': 865.017, 'duration': 6.965}, {'end': 880.147, 'text': "So that's n sigma squared divided by c squared, which is sigma squared over nc squared.", 'start': 872.022, 'duration': 8.125}, {'end': 889.714, 'text': 'Sigma is a constant, c is a constant, n goes to infinity, so this goes to 0.', 'start': 881.848, 'duration': 7.866}, {'end': 894.317, 'text': 'So that proves the weak law of large numbers, just only a one line thing.', 'start': 889.714, 'duration': 4.603}, {'end': 910.554, 'text': 'Okay, so that tells us what happens point wise when we average a bunch of IID random variables and it converges to the mean.', 'start': 899.767, 'duration': 10.787}, {'end': 917.198, 'text': "So let me just rewrite that statement and then we'll write the central limit theorem and kind of compare them.", 'start': 911.835, 'duration': 5.363}], 'summary': "Using chebyshev's inequality and variance, the weak law of large numbers is proven with the pointwise convergence to the mean for averaged iid random variables.", 'duration': 82.111, 'max_score': 835.087, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA835087.jpg'}, {'end': 1014.767, 'src': 'embed', 'start': 988.028, 'weight': 5, 'content': [{'end': 992.913, 'text': 'then one way to not just in here, but just as a general approach to that kind of problem.', 'start': 988.028, 'duration': 4.885}, {'end': 996.797, 'text': "We know this goes to 0, but we don't know how fast.", 'start': 993.334, 'duration': 3.463}, {'end': 1001.963, 'text': 'One way to study that would be multiply by something that goes to infinity, right?', 'start': 997.378, 'duration': 4.585}, {'end': 1011.085, 'text': 'Now, if we multiply by something that goes to infinity, such that this times this goes to infinity,', 'start': 1002.504, 'duration': 8.581}, {'end': 1014.767, 'text': 'then we know that this part that blows up is dominating over this part.', 'start': 1011.085, 'duration': 3.682}], 'summary': 'Multiplying by something going to infinity can show domination over another part.', 'duration': 26.739, 'max_score': 988.028, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA988028.jpg'}], 'start': 494.244, 'title': 'Law of large numbers', 'summary': "Explains the law of large numbers, emphasizing that as the sample size approaches infinity, the initial trials become insignificant, and it explores the proof of the weak law of large numbers using chebyshev's inequality.", 'chapters': [{'end': 773.165, 'start': 494.244, 'title': 'Law of large numbers', 'summary': 'Explains the law of large numbers, showing that as the sample size goes to infinity, the first million trials get swamped out by the entire infinite future, and how this theorem is crucial for science to be possible.', 'duration': 278.921, 'highlights': ['The law of large numbers states that as the sample size goes to infinity, the first million trials get swamped out by the entire infinite future, becoming insignificant.', 'The theorem is crucial for science to be possible, as it allows for the convergence to the truth as more data is collected.', "The weak law of large numbers states that for any C greater than 0, the probability that Xn bar minus the mean is greater than C goes to 0, indicating that as n goes to infinity, it's extremely likely that the sample mean is very close to the true mean."]}, {'end': 1060.719, 'start': 774.471, 'title': 'Proving weak law of large numbers', 'summary': "Discusses the proof of the weak law of large numbers using chebyshev's inequality, demonstrating that xn bar minus mu goes to 0 as n goes to infinity, and introduces the concept of studying the rate at which it converges to 0 by multiplying it by n to some power.", 'duration': 286.248, 'highlights': ["Chebyshev's inequality is used to prove the weak law of large numbers, showing that Xn bar minus mu goes to 0 as n goes to infinity.", 'The variance of Xn bar is derived as sigma squared over nc squared, which goes to 0 as n goes to infinity, proving the weak law of large numbers.', 'The concept of studying the rate at which a value goes to 0 is introduced, suggesting the approach of multiplying by n to some power and comparing it to the rate of increase.']}], 'duration': 566.475, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA494244.jpg', 'highlights': ['The law of large numbers states that as the sample size goes to infinity, the first million trials get swamped out by the entire infinite future, becoming insignificant.', "The weak law of large numbers states that for any C greater than 0, the probability that Xn bar minus the mean is greater than C goes to 0, indicating that as n goes to infinity, it's extremely likely that the sample mean is very close to the true mean.", 'The theorem is crucial for science to be possible, as it allows for the convergence to the truth as more data is collected.', "Chebyshev's inequality is used to prove the weak law of large numbers, showing that Xn bar minus mu goes to 0 as n goes to infinity.", 'The variance of Xn bar is derived as sigma squared over nc squared, which goes to 0 as n goes to infinity, proving the weak law of large numbers.', 'The concept of studying the rate at which a value goes to 0 is introduced, suggesting the approach of multiplying by n to some power and comparing it to the rate of increase.']}, {'end': 1429.583, 'segs': [{'end': 1149.419, 'src': 'embed', 'start': 1084.402, 'weight': 0, 'content': [{'end': 1089.945, 'text': "This is kind of the happy medium where we're gonna get a non-degenerate distribution.", 'start': 1084.402, 'duration': 5.543}, {'end': 1095.047, 'text': 'This is gonna converge in distribution to an actual distribution.', 'start': 1090.485, 'duration': 4.562}, {'end': 1099.009, 'text': "It's not gonna just get killed to zero or blow up to infinity.", 'start': 1095.107, 'duration': 3.902}, {'end': 1101.371, 'text': "It's actually gonna give us a nice distribution.", 'start': 1099.089, 'duration': 2.282}, {'end': 1106.652, 'text': "And I'm also gonna divide by the sigma here, makes it a little bit cleaner.", 'start': 1102.711, 'duration': 3.941}, {'end': 1110.333, 'text': 'So this is the central limit theorem now.', 'start': 1108.652, 'duration': 1.681}, {'end': 1113.113, 'text': "I'm stating it, then we'll prove it.", 'start': 1111.613, 'duration': 1.5}, {'end': 1129.237, 'text': 'Central limit theorem says if you take this and look at what happens as n goes to infinity, Converges to standard normal.', 'start': 1117.474, 'duration': 11.763}, {'end': 1134.009, 'text': 'in distribution.', 'start': 1133.213, 'duration': 0.796}, {'end': 1146.577, 'text': 'By convergence and distribution, what we mean is that the distribution of this converges to the standard normal distribution.', 'start': 1140.413, 'duration': 6.164}, {'end': 1149.419, 'text': 'In other words, you could take the CDF.', 'start': 1146.978, 'duration': 2.441}], 'summary': 'The central limit theorem states that as n goes to infinity, the distribution converges to the standard normal distribution.', 'duration': 65.017, 'max_score': 1084.402, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1084402.jpg'}, {'end': 1275.852, 'src': 'embed', 'start': 1224.881, 'weight': 2, 'content': [{'end': 1229.463, 'text': "But the way it's used in practice is then people use normal approximations all the time.", 'start': 1224.881, 'duration': 4.582}, {'end': 1233.846, 'text': 'And a lot of the justification for normal approximations is coming from this.', 'start': 1229.704, 'duration': 4.142}, {'end': 1241.068, 'text': 'If n is large, then the sample mean will approximately have a normal distribution.', 'start': 1236.425, 'duration': 4.643}, {'end': 1252.474, 'text': 'Even if the original data did not look like they came from a normal distribution, when you average lots and lots of them, it looks normal.', 'start': 1244.389, 'duration': 8.085}, {'end': 1259.404, 'text': 'Okay, so this is, in a sense, is a better theorem than the law of large numbers.', 'start': 1254.562, 'duration': 4.842}, {'end': 1264.687, 'text': "But because it's kind of more informative to know the distribution and know something about the rate.", 'start': 1259.524, 'duration': 5.163}, {'end': 1272.93, 'text': "And it's interesting that its square root of n is kind of the power of n that's just right, right? Larger power, it's gonna blow up.", 'start': 1265.227, 'duration': 7.703}, {'end': 1275.852, 'text': "Smaller power, it's gonna go to 0.", 'start': 1272.99, 'duration': 2.862}], 'summary': 'Normal approximations are justified when n is large, as sample mean will approximately have a normal distribution, even if original data did not look normal.', 'duration': 50.971, 'max_score': 1224.881, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1224881.jpg'}], 'start': 1060.939, 'title': 'Central limit theorem', 'summary': 'Discusses the convergence to standard normal distribution as sample size increases, highlighting the magic threshold value of 1 half and the normal approximation of sample means as n increases, providing informative convergence.', 'chapters': [{'end': 1198.01, 'start': 1060.939, 'title': 'Central limit theorem: convergence to standard normal', 'summary': 'Discusses the central limit theorem, which states that as the sample size increases, the distribution of the sample mean converges to a standard normal distribution, providing a fundamental result applicable to various types of distributions and highlighting the magic threshold value of 1 half.', 'duration': 137.071, 'highlights': ['The central limit theorem states that as the sample size goes to infinity, the distribution of the sample mean converges to a standard normal distribution, providing a fundamental result applicable to various types of distributions.', 'The magic threshold value for the central limit theorem is 1 half, representing the point where the distribution becomes non-degenerate and converges to an actual distribution.', 'The chapter emphasizes that the normal distribution is just one particular distribution among many possible variations, including discrete, continuous, and complex distributions, thus highlighting the generality of the central limit theorem.', 'The central limit theorem provides a significant insight into the behavior of sample means, showcasing the convergence of their distribution to a standard normal distribution as the sample size increases.']}, {'end': 1429.583, 'start': 1199.051, 'title': 'Central limit theorem and normal approximation', 'summary': 'Discusses the importance of the standard normal distribution, the central limit theorem as n goes to infinity, and the normal approximation of sample means, emphasizing that as n increases, the sample mean will approximately have a normal distribution, providing more informative convergence than the law of large numbers.', 'duration': 230.532, 'highlights': ['As n increases, the sample mean will approximately have a normal distribution. The theorem states that as n goes to infinity, the sample mean will approximately have a normal distribution, even if the original data did not look like they came from a normal distribution, emphasizing the importance and wide usage of the standard normal distribution.', 'The importance of the standard normal distribution and its wide usage. The standard normal distribution is crucial and widely used, especially as a theorem as n goes to infinity, signifying its significance in practical applications.', "The square root of n is the power of n that's just right for convergence to a normal distribution. The square root of n is identified as the power of n that leads to the convergence of the sample mean to a normal distribution, providing more informative convergence and a compromise for ensuring a normal distribution.", 'The difference between convergence of random variables and convergence in distribution. The chapter distinguishes between convergence of random variables and convergence in distribution, indicating that both are informing about the behavior of Xn bar when n is large, but in different senses of convergence.', "The standardization process for the sum of x's to match a normal distribution. The process involves centering the sum of x's by subtracting n mu to make the mean 0 and then dividing by the standard deviation to standardize it, ensuring that it matches a normal distribution."]}], 'duration': 368.644, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1060939.jpg', 'highlights': ['The central limit theorem states that as the sample size goes to infinity, the distribution of the sample mean converges to a standard normal distribution, providing a fundamental result applicable to various types of distributions.', 'The magic threshold value for the central limit theorem is 1 half, representing the point where the distribution becomes non-degenerate and converges to an actual distribution.', 'As n increases, the sample mean will approximately have a normal distribution. The theorem states that as n goes to infinity, the sample mean will approximately have a normal distribution, even if the original data did not look like they came from a normal distribution, emphasizing the importance and wide usage of the standard normal distribution.', "The square root of n is the power of n that's just right for convergence to a normal distribution. The square root of n is identified as the power of n that leads to the convergence of the sample mean to a normal distribution, providing more informative convergence and a compromise for ensuring a normal distribution."]}, {'end': 2202.362, 'segs': [{'end': 1462.854, 'src': 'embed', 'start': 1430.244, 'weight': 0, 'content': [{'end': 1437.661, 'text': 'So over there we showed that the variance, of Xn bar is sigma squared over n.', 'start': 1430.244, 'duration': 7.417}, {'end': 1441.644, 'text': 'And the variance of this sum is just n sigma squared.', 'start': 1437.661, 'duration': 3.983}, {'end': 1454.135, 'text': "So it's just divide by the standard deviation, right? Which is square root of n times sigma, okay? Cuz the variance is n sigma squared.", 'start': 1442.025, 'duration': 12.11}, {'end': 1462.854, 'text': "So that's just the standardized version, and the statement is again that this converges to standard normal in distribution.", 'start': 1455.132, 'duration': 7.722}], 'summary': "Variance of xn bar is sigma squared over n, and the sum's variance is n sigma squared, converging to standard normal in distribution.", 'duration': 32.61, 'max_score': 1430.244, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1430244.jpg'}, {'end': 1584.226, 'src': 'embed', 'start': 1524.118, 'weight': 1, 'content': [{'end': 1528.882, 'text': 'But for our purposes, we may as well just assume MGF exists.', 'start': 1524.118, 'duration': 4.764}, {'end': 1540.749, 'text': "So assuming MGF, let's call it M they're IIDs.", 'start': 1530.062, 'duration': 10.687}, {'end': 1543.831, 'text': 'So if one of them has an MGF, they all have the same MGF.', 'start': 1540.949, 'duration': 2.882}, {'end': 1546.273, 'text': "We'll just assume that that exists.", 'start': 1544.772, 'duration': 1.501}, {'end': 1561.079, 'text': 'Once we have MGFs, then our strategy is to show that the MGFs converge.', 'start': 1554.835, 'duration': 6.244}, {'end': 1572.68, 'text': "So that's a theorem about MGFs that if the MGFs converge some other MGF, then the random variables converge in distribution right?", 'start': 1561.279, 'duration': 11.401}, {'end': 1575.562, 'text': 'We had a homework problem related to that,', 'start': 1572.921, 'duration': 2.641}, {'end': 1582.025, 'text': 'where you found that the MGFs converged to some MGF and that implies convergence of the distributions right?', 'start': 1575.562, 'duration': 6.463}, {'end': 1584.226, 'text': "Okay, so that's the whole strategy.", 'start': 1582.745, 'duration': 1.481}], 'summary': 'Assuming mgf exists, we aim to show mgf convergence, implying distribution convergence.', 'duration': 60.108, 'max_score': 1524.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1524118.jpg'}, {'end': 1711.002, 'src': 'embed', 'start': 1659.587, 'weight': 3, 'content': [{'end': 1663.19, 'text': 'But then you could just, I mean, if you want, just call this thing yj.', 'start': 1659.587, 'duration': 3.603}, {'end': 1667.953, 'text': "And once you have the central limit theorem for yj, then you know that that's true.", 'start': 1663.41, 'duration': 4.543}, {'end': 1670.355, 'text': "So you may as well just assume that they've already been standardized.", 'start': 1667.973, 'duration': 2.382}, {'end': 1679.922, 'text': "And so just to have some notation, let's just let Sn equal the sum, S for sum of the first n terms.", 'start': 1671.756, 'duration': 8.166}, {'end': 1690.695, 'text': "And what we wanna show is that the MGF, of Sn over root n, that's what we're looking at, right? Cuz I let mu equals 0, sigma equals 1.", 'start': 1680.322, 'duration': 10.373}, {'end': 1697.82, 'text': "So we're looking at Sn over root n, and we wanna show that that goes to the standard normal MGF.", 'start': 1690.695, 'duration': 7.125}, {'end': 1705.386, 'text': 'Right, so we just need to find this MGF, take a limit.', 'start': 1702.284, 'duration': 3.102}, {'end': 1711.002, 'text': "Okay, so let's just find the MGF.", 'start': 1707.034, 'duration': 3.968}], 'summary': 'Analyzing mgf of sn to show convergence to standard normal mgf', 'duration': 51.415, 'max_score': 1659.587, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1659587.jpg'}, {'end': 1920.749, 'src': 'embed', 'start': 1878.573, 'weight': 5, 'content': [{'end': 1881.617, 'text': "After taking the log, and we're trying to do a limit.", 'start': 1878.573, 'duration': 3.044}, {'end': 1885.703, 'text': "So we're doing the limit as n goes to infinity, and we take the log.", 'start': 1882.098, 'duration': 3.605}, {'end': 1893.335, 'text': "It's n log m of t over root n.", 'start': 1886.424, 'duration': 6.911}, {'end': 1901.296, 'text': "And so that's of the form infinity times 0.", 'start': 1895.732, 'duration': 5.564}, {'end': 1908.04, 'text': 'If we want 0 over 0 or infinity over infinity, we can just write it as 1 over n in the denominator.', 'start': 1901.296, 'duration': 6.744}, {'end': 1917.086, 'text': "Okay, now it's of the form 0 over 0.", 'start': 1908.06, 'duration': 9.026}, {'end': 1920.749, 'text': "So we can almost use L'Hopital's rule, but not quite.", 'start': 1917.086, 'duration': 3.663}], 'summary': "Analyzing a limit as n goes to infinity, using logarithms and l'hopital's rule.", 'duration': 42.176, 'max_score': 1878.573, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1878573.jpg'}], 'start': 1430.244, 'title': 'Convergence of mgfs', 'summary': "Discusses the convergence of standardized sum of independent random variables to a standard normal distribution, assuming the existence of mgfs. it also emphasizes the theorem that states the convergence of mgfs implies the convergence of the random variables in distribution. additionally, it covers finding mgf, taking limit, and utilizing standardization, independence, l'hopital's rule, and properties of mgfs.", 'chapters': [{'end': 1584.226, 'start': 1430.244, 'title': 'Convergence of mgfs to standard normal', 'summary': 'Discusses the convergence of the standardized sum of independent random variables to a standard normal distribution, assuming the existence of moment generating functions (mgfs). it also emphasizes the theorem that states the convergence of mgfs implies the convergence of the random variables in distribution.', 'duration': 153.982, 'highlights': ['The standardized sum of independent random variables converges to standard normal distribution. The chapter emphasizes the convergence of the standardized sum of independent random variables to a standard normal distribution, stating that this sum, when standardized, converges to standard normal.', 'The theorem that states the convergence of MGFs implies the convergence of the random variables in distribution. The chapter discusses a theorem regarding moment generating functions (MGFs), stating that if the MGFs converge to some other MGF, then the random variables also converge in distribution.', 'The assumption of the existence of MGFs to work with MGFs in the proof. The chapter explains the assumption of the existence of moment generating functions (MGFs) to work with them in the proof, allowing for the use of MGFs when dealing with the sum of independent random variables.']}, {'end': 2202.362, 'start': 1584.426, 'title': 'Mgf limit and standardization', 'summary': "Discusses the process of finding the moment generating function (mgf) and taking the limit to show that the mgf of sn over root n converges to the standard normal mgf, using standardization, independence, l'hopital's rule, and the properties of mgfs.", 'duration': 617.936, 'highlights': ['The chapter discusses finding the moment generating function (MGF) of Sn over root n and taking the limit to show its convergence to the standard normal MGF, using standardization and the properties of MGFs.', 'It explains the process of standardizing each x separately, considering independence, and using the central limit theorem for yj to assume that variables have already been standardized.', "It elaborates on using L'Hopital's rule and a change of variables to deal with the indeterminate form 1 to the infinity, and simplifying the limit using L'Hopital's rule to obtain the desired result of t squared over 2."]}], 'duration': 772.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA1430244.jpg', 'highlights': ['The standardized sum of independent random variables converges to standard normal distribution.', 'The theorem that states the convergence of MGFs implies the convergence of the random variables in distribution.', 'The assumption of the existence of MGFs to work with MGFs in the proof.', 'The chapter discusses finding the moment generating function (MGF) of Sn over root n and taking the limit to show its convergence to the standard normal MGF, using standardization and the properties of MGFs.', 'It explains the process of standardizing each x separately, considering independence, and using the central limit theorem for yj to assume that variables have already been standardized.', "It elaborates on using L'Hopital's rule and a change of variables to deal with the indeterminate form 1 to the infinity, and simplifying the limit using L'Hopital's rule to obtain the desired result of t squared over 2."]}, {'end': 2987.271, 'segs': [{'end': 2244.324, 'src': 'embed', 'start': 2205.149, 'weight': 0, 'content': [{'end': 2215.239, 'text': 'of e to the t squared over 2, but e to the t squared over 2 is exactly the normal 0, 1 MGF.', 'start': 2205.149, 'duration': 10.09}, {'end': 2223.147, 'text': 'Okay, so.', 'start': 2222.486, 'duration': 0.661}, {'end': 2228.526, 'text': "To prove that theorem, and that's the end of the proof of the central limit theorem.", 'start': 2223.9, 'duration': 4.626}, {'end': 2232.692, 'text': 'all we had to do was just basic facts about MGF.', 'start': 2228.526, 'duration': 4.166}, {'end': 2234.234, 'text': "use L'Hopital's rule twice.", 'start': 2232.692, 'duration': 1.542}, {'end': 2239.04, 'text': 'And there we have one of the most famous important theorems in statistics.', 'start': 2234.694, 'duration': 4.346}, {'end': 2244.324, 'text': 'So there are more general versions of this.', 'start': 2241.52, 'duration': 2.804}], 'summary': "Proof of central limit theorem using mgf and l'hopital's rule. important theorem in statistics.", 'duration': 39.175, 'max_score': 2205.149, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA2205149.jpg'}, {'end': 2322.618, 'src': 'embed', 'start': 2291.374, 'weight': 1, 'content': [{'end': 2303.187, 'text': 'So, historically, the first version of the central limit theorem that was ever proven, I think, was four binomials.', 'start': 2291.374, 'duration': 11.813}, {'end': 2312.976, 'text': "Okay, so what we're saying is that the binomial NP, under some conditions, will be approximately normal.", 'start': 2304.954, 'duration': 8.022}, {'end': 2320.318, 'text': "And well, in the old days that was an incredibly important fact because they didn't have computers to.", 'start': 2313.476, 'duration': 6.842}, {'end': 2322.618, 'text': 'binomials are hard to deal with right?', 'start': 2320.318, 'duration': 2.3}], 'summary': 'The central limit theorem was first proven with four binomials, showing that binomial np can be approximately normal under certain conditions.', 'duration': 31.244, 'max_score': 2291.374, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA2291374.jpg'}, {'end': 2548.443, 'src': 'embed', 'start': 2522.771, 'weight': 3, 'content': [{'end': 2529.295, 'text': "And then, now that we've standardized it, we can apply the central limit theorem if n is large enough.", 'start': 2522.771, 'duration': 6.524}, {'end': 2532.897, 'text': 'The central limit theorem said n goes to infinity.', 'start': 2530.075, 'duration': 2.822}, {'end': 2536.079, 'text': "That doesn't answer the question of how large does n have to be.", 'start': 2533.157, 'duration': 2.922}, {'end': 2540.121, 'text': "And for that, there's various theorems and various rules of thumb.", 'start': 2536.399, 'duration': 3.722}, {'end': 2548.443, 'text': 'A lot of books will say, how large does n have to be? And some books, at least, will say 30.', 'start': 2540.941, 'duration': 7.502}], 'summary': 'Central limit theorem applies if n is large enough, some books suggest n should be at least 30', 'duration': 25.672, 'max_score': 2522.771, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA2522771.jpg'}, {'end': 2727.491, 'src': 'embed', 'start': 2695.717, 'weight': 4, 'content': [{'end': 2702.3, 'text': "So Poisson is relevant when we're dealing with a large number of very rare, unlikely things.", 'start': 2695.717, 'duration': 6.583}, {'end': 2708.966, 'text': "That's really in contrast to this, in this case for the normal approximation.", 'start': 2702.844, 'duration': 6.122}, {'end': 2712.747, 'text': 'Then, well, we still want n to be large.', 'start': 2710.786, 'duration': 1.961}, {'end': 2722.729, 'text': 'But if we kind of think intuitively about when is this gonna work well, we actually want p to be close to 1 half.', 'start': 2713.487, 'duration': 9.242}, {'end': 2727.491, 'text': 'Because think about the symmetry.', 'start': 2725.15, 'duration': 2.341}], 'summary': 'Poisson applies to rare events, while normal approximation needs large n and p close to 0.5 for symmetry.', 'duration': 31.774, 'max_score': 2695.717, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA2695717.jpg'}, {'end': 2809.495, 'src': 'embed', 'start': 2782.434, 'weight': 5, 'content': [{'end': 2788.157, 'text': 'But as a practical matter, as an approximation, if p is close to 1 half, this is gonna work.', 'start': 2782.434, 'duration': 5.723}, {'end': 2793.621, 'text': "quite well if n is like 30 or 50 or 100, it'll work fine.", 'start': 2789.577, 'duration': 4.044}, {'end': 2798.145, 'text': 'But if p is 0.001, the central limit theorem is still true.', 'start': 2793.661, 'duration': 4.484}, {'end': 2801.908, 'text': "that as n goes to infinity, it's gonna work okay?", 'start': 2798.145, 'duration': 3.763}, {'end': 2809.495, 'text': "But if n is just kind of not that enormous of a number, then it's gonna be a pretty bad approximation.", 'start': 2802.129, 'duration': 7.366}], 'summary': 'For p close to 0.5, works well with n = 30, 50, 100; for p = 0.001, not accurate for small n.', 'duration': 27.061, 'max_score': 2782.434, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA2782434.jpg'}], 'start': 2205.149, 'title': 'Approximations and conditions', 'summary': 'Covers central limit theorem, binomial approximation, normal approximation, and poisson approximation, highlighting the historical significance and conditions for their validity.', 'chapters': [{'end': 2491.865, 'start': 2205.149, 'title': 'Central limit theorem and binomial approximation', 'summary': "Discusses the central limit theorem, its proof using basic facts about mgf and l'hopital's rule, the historical significance of binomial approximation to normal distribution, and the conditions for valid binomial approximation to normal.", 'duration': 286.716, 'highlights': ["The chapter proves the central limit theorem using basic facts about MGF and L'Hopital's rule. Proof of central limit theorem, use of basic facts about MGF, application of L'Hopital's rule", 'Historically, the binomial NP, under certain conditions, will be approximately normal, which was significant due to the complexity of binomial distributions in the absence of computers. Historical significance of binomial approximation to normal distribution, complexity of binomial distributions before computers', 'The chapter explores the conditions for valid binomial approximation to normal and discusses the framework of the central limit theorem for the approximation. Conditions for valid binomial approximation, framework of central limit theorem for approximation']}, {'end': 2987.271, 'start': 2492.583, 'title': 'Normal approximation and central limit theorem', 'summary': 'Discusses the process of standardizing a variable, applying the central limit theorem, and the relevance of normal and poisson approximations, emphasizing the conditions under which they work effectively.', 'duration': 494.688, 'highlights': ["The central limit theorem is applicable when the sample size is large enough, and various theorems and rules of thumb determine the minimum 'n', with some suggesting a minimum of 30.", "The normal approximation works well when 'p' is close to 1 half, as it results in a symmetric distribution, while the Poisson approximation is relevant in scenarios with a large number of very rare events where 'p' is very small.", "The central limit theorem holds true for 'p' close to 1 half and as 'n' approaches infinity, though for practical approximation, a large 'n' is necessary, making it a bad approximation for smaller 'n' values."]}], 'duration': 782.122, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/OprNqnHsVIA/pics/OprNqnHsVIA2205149.jpg', 'highlights': ["The chapter proves the central limit theorem using basic facts about MGF and L'Hopital's rule.", 'The chapter explores the conditions for valid binomial approximation to normal and discusses the framework of the central limit theorem for the approximation.', 'Historically, the binomial NP, under certain conditions, will be approximately normal, which was significant due to the complexity of binomial distributions in the absence of computers.', "The central limit theorem is applicable when the sample size is large enough, and various theorems and rules of thumb determine the minimum 'n', with some suggesting a minimum of 30.", "The normal approximation works well when 'p' is close to 1 half, as it results in a symmetric distribution, while the Poisson approximation is relevant in scenarios with a large number of very rare events where 'p' is very small.", "The central limit theorem holds true for 'p' close to 1 half and as 'n' approaches infinity, though for practical approximation, a large 'n' is necessary, making it a bad approximation for smaller 'n' values."]}], 'highlights': ['The central limit theorem states that as the sample size goes to infinity, the distribution of the sample mean converges to a standard normal distribution, providing a fundamental result applicable to various types of distributions.', 'The law of large numbers states that Xn bar converges to mu as n goes to infinity with probability 1.', "The weak law of large numbers states that for any C greater than 0, the probability that Xn bar minus the mean is greater than C goes to 0, indicating that as n goes to infinity, it's extremely likely that the sample mean is very close to the true mean.", 'The convergence of random variables to a certain value with probability one is discussed, using the example of Bernoulli p.', 'The theorem that states the convergence of MGFs implies the convergence of the random variables in distribution.', "The chapter proves the central limit theorem using basic facts about MGF and L'Hopital's rule.", 'The concept of studying the rate at which a value goes to 0 is introduced, suggesting the approach of multiplying by n to some power and comparing it to the rate of increase.']}