title

Lecture 17: Moment Generating Functions | Statistics 110

description

We introduce moment generating functions (MGFs), which have many uses in probability. We also discuss Laplace's rule of succession and the "hybrid" version of Bayes' rule.

detail

{'title': 'Lecture 17: Moment Generating Functions | Statistics 110', 'heatmap': [{'end': 1006.568, 'start': 942.028, 'weight': 0.824}, {'end': 1126.777, 'start': 1099.598, 'weight': 0.716}, {'end': 1900.624, 'start': 1852.49, 'weight': 0.725}, {'end': 3037.251, 'start': 2919.314, 'weight': 0.731}], 'summary': "The lecture covers topics including memoryless property of exponential distribution, life expectancy statistics, moment generating function (mgf) and its importance in probability and statistics, mgf for binomial and normal distributions, and probability calculation using laplace's rule of succession and bayes' rule, addressing philosophical debates between bayesians and frequentists.", 'chapters': [{'end': 88.192, 'segs': [{'end': 69.633, 'src': 'embed', 'start': 21.423, 'weight': 0, 'content': [{'end': 27.226, 'text': 'So what I want to talk about now is just the kind of fact that I find pretty amazing,', 'start': 21.423, 'duration': 5.803}, {'end': 31.787, 'text': 'which is that the exponential is the only memoryless distribution in continuous time.', 'start': 27.226, 'duration': 4.561}, {'end': 33.648, 'text': 'In discrete time, we have the geometric.', 'start': 32.107, 'duration': 1.541}, {'end': 38.91, 'text': 'So in a very deep sense, the geometric is the discrete analog of the exponential.', 'start': 34.068, 'duration': 4.842}, {'end': 42.092, 'text': 'The exponential is the continuous analog of the geometric.', 'start': 38.97, 'duration': 3.122}, {'end': 45.153, 'text': 'So those two distributions are very closely related.', 'start': 42.392, 'duration': 2.761}, {'end': 51.741, 'text': 'So, just to remind you what the memoryless properties said, and also just cuz,', 'start': 46.215, 'duration': 5.526}, {'end': 61.411, 'text': "I saw some news article recently that completely misunderstood the concept of life expectancy, and that's not the first time that that's happened.", 'start': 51.741, 'duration': 9.67}, {'end': 69.633, 'text': "Basically, it's a mistake of not understanding the difference between expectation and conditional expectation.", 'start': 64.827, 'duration': 4.806}], 'summary': 'Exponential is the only memoryless distribution in continuous time, with geometric as its discrete analog.', 'duration': 48.21, 'max_score': 21.423, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ821423.jpg'}], 'start': 0.989, 'title': 'Memoryless property of exponential distribution', 'summary': 'Discusses the memoryless property of the exponential distribution in continuous time and its relationship with the geometric distribution in discrete time, emphasizing the distinction between expectation and conditional expectation.', 'chapters': [{'end': 88.192, 'start': 0.989, 'title': 'Memoryless property of exponential distribution', 'summary': 'Discusses the memoryless property of the exponential distribution, highlighting its uniqueness in continuous time and its relationship with the geometric distribution in discrete time, and emphasizes the distinction between expectation and conditional expectation.', 'duration': 87.203, 'highlights': ['The exponential distribution is the only memoryless distribution in continuous time, while in discrete time, we have the geometric distribution, which is its discrete analog.', 'The relationship between the exponential and geometric distributions is very close, with the exponential being the continuous analog of the geometric.', 'The chapter emphasizes the distinction between expectation and conditional expectation, pointing out the common misunderstanding of the concept of life expectancy as a mistake of not understanding the difference between expectation and conditional expectation.']}], 'duration': 87.203, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8989.jpg', 'highlights': ['The exponential distribution is the only memoryless distribution in continuous time, while in discrete time, we have the geometric distribution, which is its discrete analog.', 'The relationship between the exponential and geometric distributions is very close, with the exponential being the continuous analog of the geometric.', 'The chapter emphasizes the distinction between expectation and conditional expectation, pointing out the common misunderstanding of the concept of life expectancy as a mistake of not understanding the difference between expectation and conditional expectation.']}, {'end': 348.207, 'segs': [{'end': 116.173, 'src': 'embed', 'start': 88.553, 'weight': 0, 'content': [{'end': 92.815, 'text': 'So we will spend a lot of time on conditional expectation later as a topic in its own right.', 'start': 88.553, 'duration': 4.262}, {'end': 96.397, 'text': "But it's already something that's familiar, right? Just use conditional probability.", 'start': 93.195, 'duration': 3.202}, {'end': 104.003, 'text': "Okay, so for this life expectancy thing, here's the common misconception that I've seen in various news articles.", 'start': 97.077, 'duration': 6.926}, {'end': 113.43, 'text': 'Last time I looked, the life expectancy in the US was 76 years for men, 81 years for women.', 'start': 105.464, 'duration': 7.966}, {'end': 116.173, 'text': "It's different in different countries and whatever.", 'start': 114.131, 'duration': 2.042}], 'summary': 'Life expectancy: us men 76 years, women 81 years.', 'duration': 27.62, 'max_score': 88.553, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ888553.jpg'}, {'end': 200.535, 'src': 'embed', 'start': 154.025, 'weight': 1, 'content': [{'end': 157.387, 'text': 'But secondly, at some point in time you want an answer right?', 'start': 154.025, 'duration': 3.362}, {'end': 164.392, 'text': "But if you only look at the ones who've died up to that point and average those, that's gonna be a very biased answer, right?", 'start': 157.807, 'duration': 6.585}, {'end': 168.674, 'text': "Because you're ignoring all the ones who had longer lifetimes, okay?", 'start': 164.512, 'duration': 4.162}, {'end': 171.396, 'text': "So that's an example of what's called censored data.", 'start': 168.695, 'duration': 2.701}, {'end': 172.937, 'text': "That's a good kind of censoring.", 'start': 171.716, 'duration': 1.221}, {'end': 175.619, 'text': "It's censored because they're still alive.", 'start': 172.977, 'duration': 2.642}, {'end': 180.142, 'text': "So anyway, that's a hard statistical problem and an interesting one.", 'start': 176.82, 'duration': 3.322}, {'end': 186.906, 'text': "The reason I'm mentioning it now is kind of like, Good news and bad news about life expectancy.", 'start': 180.562, 'duration': 6.344}, {'end': 191.549, 'text': "So let's just assume it's 80 years for simplicity.", 'start': 187.387, 'duration': 4.162}, {'end': 200.535, 'text': "The mistake that I saw in this news article was basically assuming that it's like 80 years for everyone, including people.", 'start': 192.87, 'duration': 7.665}], 'summary': 'Statistical problem: censored data affects life expectancy calculation.', 'duration': 46.51, 'max_score': 154.025, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8154025.jpg'}, {'end': 254.006, 'src': 'embed', 'start': 229.884, 'weight': 5, 'content': [{'end': 237.948, 'text': "given that that person lives to be at least 20, that's gonna be greater than just the expected value of t.", 'start': 229.884, 'duration': 8.064}, {'end': 244.618, 'text': "It's kind of intuitively clear, so that's a conditional expectation.", 'start': 240.194, 'duration': 4.424}, {'end': 252.585, 'text': 'It just means, given this information and we compute our expectation based on conditional probabilities rather than unconditional probabilities.', 'start': 244.658, 'duration': 7.927}, {'end': 254.006, 'text': 'this should be pretty intuitive, right?', 'start': 252.585, 'duration': 1.421}], 'summary': 'Conditional expectation is computed based on given information, using conditional probabilities.', 'duration': 24.122, 'max_score': 229.884, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8229884.jpg'}, {'end': 301.182, 'src': 'embed', 'start': 274.643, 'weight': 4, 'content': [{'end': 279.167, 'text': 'And so this is just to illustrate what the memoryless property would say.', 'start': 274.643, 'duration': 4.524}, {'end': 287.911, 'text': 'If human lifetimes were memoryless, it would say and if the average is 80 years, then it would say if you live to be 20,', 'start': 280.745, 'duration': 7.166}, {'end': 290.834, 'text': 'then your new expectation is 100, right?', 'start': 287.911, 'duration': 2.923}, {'end': 293.896, 'text': "Because memoryless says you're good as new, right?", 'start': 291.094, 'duration': 2.802}, {'end': 298.56, 'text': "So as you get an extra 80, no matter how long you've lived, you get an extra 80 years on average.", 'start': 293.916, 'duration': 4.644}, {'end': 301.182, 'text': "and that's not true empirically, okay?", 'start': 298.56, 'duration': 2.622}], 'summary': 'The memoryless property implies an average of 80 more years, not empirically true.', 'duration': 26.539, 'max_score': 274.643, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8274643.jpg'}], 'start': 88.553, 'title': 'Life expectancy statistics and censored data', 'summary': "Delves into life expectancy statistics, revealing 76 years for men and 81 years for women in the us, and explores censored data's impact on life expectancy predictions, addressing biases and the memoryless property of human lifetimes.", 'chapters': [{'end': 129.842, 'start': 88.553, 'title': 'Life expectancy statistics', 'summary': 'Discusses the common misconception in news articles about life expectancy, citing statistics of 76 years for men and 81 years for women in the us, and raises the question of how these numbers are derived.', 'duration': 41.289, 'highlights': ['The latest reported life expectancy in the US is 76 years for men and 81 years for women, with variations in different countries.', 'The chapter raises the question of how these life expectancy numbers are derived and highlights it as an interesting statistical problem.', 'Conditional expectation and probability are discussed as familiar concepts that will be explored further in the topic.']}, {'end': 348.207, 'start': 129.942, 'title': 'Life expectancy and censored data', 'summary': 'Discusses the concept of censored data in life expectancy, highlighting the bias in averaging lifetimes and the impact of conditional expectations on human lifetime, while also addressing the memoryless property of human lifetimes.', 'duration': 218.265, 'highlights': ['The concept of censored data in life expectancy is explained, highlighting the bias in averaging lifetimes and the impact of ignoring individuals with longer lifespans. Censored data in life expectancy is discussed, illustrating the bias in averaging lifetimes and the impact of ignoring individuals with longer lifespans.', 'The impact of conditional expectations on human lifetime is discussed, emphasizing the relationship between longevity and expected lifespan based on conditional probabilities. The discussion on conditional expectations highlights the relationship between longevity and expected lifespan based on conditional probabilities.', 'The memoryless property of human lifetimes is addressed, showcasing the unrealistic nature of this property in the context of human lifespans and its implications on expected lifespan. The discussion on the memoryless property of human lifetimes emphasizes its unrealistic nature in the context of human lifespans and its implications on expected lifespan.']}], 'duration': 259.654, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ888553.jpg', 'highlights': ['The latest reported life expectancy in the US is 76 years for men and 81 years for women, with variations in different countries.', 'The concept of censored data in life expectancy is explained, highlighting the bias in averaging lifetimes and the impact of ignoring individuals with longer lifespans.', 'The impact of conditional expectations on human lifetime is discussed, emphasizing the relationship between longevity and expected lifespan based on conditional probabilities.', 'The chapter raises the question of how these life expectancy numbers are derived and highlights it as an interesting statistical problem.', 'The memoryless property of human lifetimes is addressed, showcasing the unrealistic nature of this property in the context of human lifespans and its implications on expected lifespan.', 'Conditional expectation and probability are discussed as familiar concepts that will be explored further in the topic.']}, {'end': 1006.568, 'segs': [{'end': 458.096, 'src': 'embed', 'start': 429.716, 'weight': 0, 'content': [{'end': 438.862, 'text': 'So if you go in and study what distribution people actually use in practice for something like this capital T survival time,', 'start': 429.716, 'duration': 9.146}, {'end': 443.105, 'text': "the most popular distribution that's used in practice is what's called the Weibull.", 'start': 438.862, 'duration': 4.243}, {'end': 450.55, 'text': "And you don't need to know Weibulls right now, but just to mention it right now, a Weibull is just obtained by taking an exponential to a power.", 'start': 443.706, 'duration': 6.844}, {'end': 458.096, 'text': "And if you take an exponential random variable and then cube it, that's not gonna be exponential anymore, it's not gonna be memoryless anymore.", 'start': 451.611, 'duration': 6.485}], 'summary': 'The weibull distribution is popular for survival time analysis.', 'duration': 28.38, 'max_score': 429.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8429716.jpg'}, {'end': 486.237, 'src': 'embed', 'start': 464.501, 'weight': 1, 'content': [{'end': 473.908, 'text': 'So exponentials are a crucial building block, but in some cases memoryless is not an unreasonable assumption or it may be a reasonable approximation,', 'start': 464.501, 'duration': 9.407}, {'end': 476.41, 'text': "even if it's not exactly true, okay?", 'start': 473.908, 'duration': 2.502}, {'end': 478.833, 'text': "So that's just the intuition of the memoryless property.", 'start': 476.872, 'duration': 1.961}, {'end': 482.255, 'text': 'We proved that it was true last time for the exponential distribution,', 'start': 479.393, 'duration': 2.862}, {'end': 486.237, 'text': "but now let's show that it's only the exponential distribution that has that property.", 'start': 482.255, 'duration': 3.982}], 'summary': 'Exponentials have memoryless property; only exponential distribution has that property.', 'duration': 21.736, 'max_score': 464.501, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8464501.jpg'}, {'end': 553.644, 'src': 'embed', 'start': 521.9, 'weight': 2, 'content': [{'end': 526.665, 'text': 'So the memoryless property is a property of the distribution, not of the random variable itself per se.', 'start': 521.9, 'duration': 4.765}, {'end': 532.61, 'text': 'But we would say the random variable has the memoryless property if its distribution has the memoryless property.', 'start': 527.005, 'duration': 5.605}, {'end': 537.375, 'text': 'Okay, and then the claim is just that it has to be exponential.', 'start': 534.153, 'duration': 3.222}, {'end': 543.458, 'text': "So it's exponential lambda for some lambda.", 'start': 540.797, 'duration': 2.661}, {'end': 553.644, 'text': 'So this is a characterization of the exponential distribution.', 'start': 550.262, 'duration': 3.382}], 'summary': 'The memoryless property characterizes the exponential distribution as exponential lambda for some lambda.', 'duration': 31.744, 'max_score': 521.9, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8521900.jpg'}, {'end': 647.202, 'src': 'embed', 'start': 616.784, 'weight': 3, 'content': [{'end': 625.685, 'text': 'Now, the memory list property says in terms of g is easy to write.', 'start': 616.784, 'duration': 8.901}, {'end': 635.973, 'text': "It's just the equation g of s plus t equals g of s g of t.", 'start': 628.607, 'duration': 7.366}, {'end': 642.599, 'text': "And we saw this, I'm not gonna repeat the argument for this because it's the same thing as we did last time for the specific case of the exponential.", 'start': 635.973, 'duration': 6.626}, {'end': 647.202, 'text': 'Just write down the memoryless property is defined in terms of a conditional probability.', 'start': 643.079, 'duration': 4.123}], 'summary': 'The memoryless property is defined in terms of a conditional probability.', 'duration': 30.418, 'max_score': 616.784, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8616784.jpg'}, {'end': 1006.568, 'src': 'heatmap', 'start': 920.778, 'weight': 5, 'content': [{'end': 934.184, 'text': "Let's say g of xt equals g of t to the x for just by taking the limit of rational numbers, we can get real numbers.", 'start': 920.778, 'duration': 13.406}, {'end': 942.028, 'text': "Okay, now we're basically done, because this is true for all x and t.", 'start': 936.285, 'duration': 5.743}, {'end': 944.729, 'text': "So to simplify it, let's just let t equal 1.", 'start': 942.028, 'duration': 2.701}, {'end': 954.734, 'text': 'And if we let t equal 1, this just says that g of x equals g of 1 to the x.', 'start': 944.729, 'duration': 10.005}, {'end': 959.187, 'text': 'That looks like an exponential function.', 'start': 957.426, 'duration': 1.761}, {'end': 962.71, 'text': "In particular, let's write it in terms of base e.", 'start': 959.207, 'duration': 3.503}, {'end': 973.318, 'text': "That's the same thing as e to the x log g So this thing, now g is a probability.", 'start': 962.71, 'duration': 10.608}, {'end': 976.42, 'text': 'So g is clearly between 0 and 1.', 'start': 973.418, 'duration': 3.002}, {'end': 980.663, 'text': "If you take the log of a number between 0 and 1, you'll just get some negative real number.", 'start': 976.42, 'duration': 4.243}, {'end': 982.644, 'text': 'So this is just some negative number here.', 'start': 980.883, 'duration': 1.761}, {'end': 984.806, 'text': 'So we could call this thing minus lambda.', 'start': 982.965, 'duration': 1.841}, {'end': 988.02, 'text': 'where lambda is a positive number.', 'start': 986.479, 'duration': 1.541}, {'end': 993.062, 'text': "This is just a constant, right? So I'm just calling it minus lambda.", 'start': 989.901, 'duration': 3.161}, {'end': 994.643, 'text': 'It happens to be a negative constant.', 'start': 993.082, 'duration': 1.561}, {'end': 1000.625, 'text': "So that's just e to the minus lambda x, which is 1 minus the CDF for the exponential.", 'start': 995.283, 'duration': 5.342}, {'end': 1006.568, 'text': "So that's the only possibility, okay? So exponential is the only continuous memoryless distribution.", 'start': 1001.026, 'duration': 5.542}], 'summary': 'Exponential is the only continuous memoryless distribution, with g(x) = e^(-λx) and λ as a positive constant.', 'duration': 38.409, 'max_score': 920.778, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8920778.jpg'}], 'start': 349.268, 'title': 'Memoryless property', 'summary': "Discusses the significance of memoryless property in applications like science, economics, and the weibull distribution, emphasizing the exponential distribution's practical use. it also explores solving functional equations to prove the memoryless property and demonstrates that only exponential functions can satisfy the identity.", 'chapters': [{'end': 553.644, 'start': 349.268, 'title': 'Memoryless property in applications', 'summary': 'Discusses the memoryless property in applications such as science, economics, and the weibull distribution, emphasizing the significance of the exponential distribution and its memoryless property in practical use.', 'duration': 204.376, 'highlights': ['The Weibull distribution is the most popular distribution used in practice for survival time, obtained by taking an exponential to a power. The Weibull distribution is widely used in practice for survival time, obtained by taking an exponential to a power, which is a crucial building block in practical applications.', "The exponential distribution is a crucial building block and is memoryless, making it extremely useful in practical cases. The exponential distribution's memoryless property makes it a crucial building block and extremely useful in practical cases, despite not always being exactly true.", 'The memoryless property is a characteristic of the exponential distribution, where X is a continuous random variable with a positive value from 0 to infinity. The memoryless property is a characteristic of the exponential distribution, specifically for a continuous random variable with a positive value from 0 to infinity, and it must be exponential.']}, {'end': 1006.568, 'start': 555.105, 'title': 'Solving functional equation for memoryless property', 'summary': 'Discusses solving a functional equation to prove the memoryless property, deriving properties of the function g and demonstrating that only exponential functions can satisfy the identity.', 'duration': 451.463, 'highlights': ['The chapter discusses solving a functional equation to prove the memoryless property, deriving properties of the function g and demonstrating that only exponential functions can satisfy the identity.', 'The memoryless property is defined in terms of a conditional probability, and the equation g(s + t) = g(s) * g(t) is a key focus of the discussion.', 'The function g is shown to have specific properties such as g(kt) = g(t)^k for positive integers k, and g(t/k) = g(t)^(1/k) for positive integers k, which are important in establishing the unique nature of exponential functions as the only ones to satisfy the given equation.']}], 'duration': 657.3, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ8349268.jpg', 'highlights': ['The Weibull distribution is widely used in practice for survival time, obtained by taking an exponential to a power, which is a crucial building block in practical applications.', "The exponential distribution's memoryless property makes it a crucial building block and extremely useful in practical cases, despite not always being exactly true.", 'The memoryless property is a characteristic of the exponential distribution, specifically for a continuous random variable with a positive value from 0 to infinity, and it must be exponential.', 'The chapter discusses solving a functional equation to prove the memoryless property, deriving properties of the function g and demonstrating that only exponential functions can satisfy the identity.', 'The memoryless property is defined in terms of a conditional probability, and the equation g(s + t) = g(s) * g(t) is a key focus of the discussion.', 'The function g is shown to have specific properties such as g(kt) = g(t)^k for positive integers k, and g(t/k) = g(t)^(1/k) for positive integers k, which are important in establishing the unique nature of exponential functions as the only ones to satisfy the given equation.']}, {'end': 1376.26, 'segs': [{'end': 1126.777, 'src': 'heatmap', 'start': 1068.112, 'weight': 0, 'content': [{'end': 1087.939, 'text': 'So the definition is that, A random variable X has MGF M of T equals the expected value of E to the TX.', 'start': 1068.112, 'duration': 19.827}, {'end': 1092.464, 'text': 'This is as a function of T.', 'start': 1091.222, 'duration': 1.242}, {'end': 1109.123, 'text': 'And we say that it exists, this is not a useful concept unless this thing is actually finite on some interval around 0.', 'start': 1099.598, 'duration': 9.525}, {'end': 1122.574, 'text': "So we would just say if this is finite on some interval, let's say minus a to a, where a is greater than 0.", 'start': 1109.123, 'duration': 13.451}, {'end': 1126.777, 'text': 'It could be that this thing is finite for all numbers t, which is great.', 'start': 1122.574, 'duration': 4.203}], 'summary': 'A random variable x has mgf m of t equals the expected value of e to the tx, finite on interval around 0.', 'duration': 54.462, 'max_score': 1068.112, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81068112.jpg'}, {'end': 1318.032, 'src': 'embed', 'start': 1271.592, 'weight': 2, 'content': [{'end': 1276.076, 'text': 'Now intuitively at this point, we wanna swap the e and the sum.', 'start': 1271.592, 'duration': 4.484}, {'end': 1280.91, 'text': "And that's where some technical conditions come in.", 'start': 1277.889, 'duration': 3.021}, {'end': 1286.712, 'text': "In particular, that's where this fact matters that we have this interval, n equals 0 to infinity.", 'start': 1281.75, 'duration': 4.962}, {'end': 1293.134, 'text': 'So just suppose for a second that we can swap this sum and the e.', 'start': 1287.912, 'duration': 5.222}, {'end': 1298.955, 'text': 'Then we would get this thing, e, expected value of x to the n, t to the n over n factorial.', 'start': 1293.134, 'duration': 5.821}, {'end': 1304.697, 'text': 'This thing here, e of x to the n, is called the nth moment.', 'start': 1301.556, 'duration': 3.141}, {'end': 1307.088, 'text': 'So the first moment is the mean.', 'start': 1305.708, 'duration': 1.38}, {'end': 1311.49, 'text': 'The second moment is not the variance, unless the mean is 0,', 'start': 1307.609, 'duration': 3.881}, {'end': 1315.391, 'text': 'but the second moment and the first moment are what we need to compute the variance right?', 'start': 1311.49, 'duration': 3.901}, {'end': 1318.032, 'text': "And then there are higher moments, so that's called the nth moment.", 'start': 1315.611, 'duration': 2.421}], 'summary': 'Swapping e and the sum gives e of x to the nth moment.', 'duration': 46.44, 'max_score': 1271.592, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81271592.jpg'}, {'end': 1364.674, 'src': 'embed', 'start': 1339.539, 'weight': 1, 'content': [{'end': 1345.102, 'text': "So that's why it's called the moment generating function, cuz you see, all the moments are just sitting there in the Taylor series.", 'start': 1339.539, 'duration': 5.563}, {'end': 1349.885, 'text': 'As far as showing why you can swap the e and the sum.', 'start': 1346.843, 'duration': 3.042}, {'end': 1354.169, 'text': 'If this were a finite sum, that would just be immediately true by linearity, right?', 'start': 1350.427, 'duration': 3.742}, {'end': 1357.751, 'text': "Since it's an infinite sum that requires more justification.", 'start': 1354.589, 'duration': 3.162}, {'end': 1364.674, 'text': 'And for that kind of justification, either you need to take a certain real analysis course or stat 210.', 'start': 1358.371, 'duration': 6.303}], 'summary': 'The moment generating function contains all moments in its taylor series.', 'duration': 25.135, 'max_score': 1339.539, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81339539.jpg'}], 'start': 1007.248, 'title': 'Moment generating function and taylor series', 'summary': 'Explains moment generating function (mgf) and its purpose for describing a distribution, emphasizing the requirement for finiteness around 0. it also discusses its relationship with taylor series, highlighting the computation of variance and the technical conditions for swapping the sum and e.', 'chapters': [{'end': 1244.243, 'start': 1007.248, 'title': 'Moment generating function', 'summary': "Discusses the concept of moment generating function (mgf) and its definition, its purpose as a way to describe a distribution, and the interpretation of the dummy variable 't', emphasizing the requirement for finiteness on some interval around 0 which is essential for its usefulness.", 'duration': 236.995, 'highlights': ["The moment generating function (MGF), denoted as M of T, is defined as the expected value of E to the TX, serving as an alternative way to describe a distribution, with the requirement for finiteness on some interval around 0, such as minus a to a, where 'a' is greater than 0.", "The variable 'T' in MGF is a dummy variable, conventionally denoted as 'T', which serves as a placeholder and is essentially a well-defined function used for bookkeeping to keep track of the moments of a distribution.", "The MGF is a well-defined function of 'T' that allows the computation of its expectation for any number 't', providing a way to study and analyze the distribution, albeit it may be infinite for some values of 't'."]}, {'end': 1376.26, 'start': 1245.264, 'title': 'Moment generating function and taylor series', 'summary': 'Discusses moment generating function and taylor series, highlighting the concept of nth moment, its significance for computing variance, and the taylor series capturing all the moments of x. it also touches upon the technical conditions for swapping the sum and e, and the need for further analysis and math to justify the validity of the process.', 'duration': 130.996, 'highlights': ['The nth moment, e of x to the n, is discussed, emphasizing its role in computing variance, as well as its utility for various reasons.', "The chapter introduces the concept of moment generating function, highlighting how all the moments of x are captured in the Taylor series, providing a comprehensive representation of the distribution's properties.", 'The need for technical conditions and further analysis to justify the process of swapping the sum and e is emphasized, indicating the requirement for additional mathematical justification and real analysis.']}], 'duration': 369.012, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81007248.jpg', 'highlights': ["The moment generating function (MGF), denoted as M of T, is defined as the expected value of E to the TX, serving as an alternative way to describe a distribution, with the requirement for finiteness on some interval around 0, such as minus a to a, where 'a' is greater than 0.", "The chapter introduces the concept of moment generating function, highlighting how all the moments of x are captured in the Taylor series, providing a comprehensive representation of the distribution's properties.", 'The nth moment, e of x to the n, is discussed, emphasizing its role in computing variance, as well as its utility for various reasons.', 'The need for technical conditions and further analysis to justify the process of swapping the sum and e is emphasized, indicating the requirement for additional mathematical justification and real analysis.', "The variable 'T' in MGF is a dummy variable, conventionally denoted as 'T', which serves as a placeholder and is essentially a well-defined function used for bookkeeping to keep track of the moments of a distribution."]}, {'end': 1883.631, 'segs': [{'end': 1485.892, 'src': 'embed', 'start': 1402.213, 'weight': 1, 'content': [{'end': 1408.458, 'text': "I mean usually we might care about the mean and the variance, but we haven't yet worried that much about higher moments than that.", 'start': 1402.213, 'duration': 6.245}, {'end': 1414.022, 'text': 'So let me just tell you three reasons why the MGF is important.', 'start': 1409.899, 'duration': 4.123}, {'end': 1431.493, 'text': "Okay, so why is the MGF important? Well, the first reason is the moments, that's what we just talked about, cuz sometimes we do want the moments.", 'start': 1418.405, 'duration': 13.088}, {'end': 1444.243, 'text': "So we're gonna let X have MGF capital M of t.", 'start': 1432.454, 'duration': 11.789}, {'end': 1451.428, 'text': "If necessary, for clarity, we might subscript the x here, but right now we're just talking about one random variable and here's its MGF.", 'start': 1444.243, 'duration': 7.185}, {'end': 1452.909, 'text': "so I don't need a subscript.", 'start': 1451.428, 'duration': 1.481}, {'end': 1466.522, 'text': "The nth moment, that is E is, there's two ways to think of it.", 'start': 1456.437, 'duration': 10.085}, {'end': 1478.088, 'text': "The nicer way is that it's the coefficient of t to the n over n factorial in the Taylor expansion.", 'start': 1467.623, 'duration': 10.465}, {'end': 1485.892, 'text': 'So Maclaurin series, if you like.', 'start': 1483.711, 'duration': 2.181}], 'summary': 'Mgf is important for moments, as it helps to find nth moment and taylor expansion.', 'duration': 83.679, 'max_score': 1402.213, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81402213.jpg'}, {'end': 1599.475, 'src': 'embed', 'start': 1542.916, 'weight': 0, 'content': [{'end': 1550.16, 'text': 'So to get the nth moment, we could take the nth derivative of the MGF, evaluate it at 0.', 'start': 1542.916, 'duration': 7.244}, {'end': 1556.424, 'text': "But as we'll see, sometimes it's a lot easier to just directly work out the Taylor series by some other method.", 'start': 1550.16, 'duration': 6.264}, {'end': 1560.167, 'text': 'For example, we already know the Taylor series for e to the x right?', 'start': 1556.565, 'duration': 3.602}, {'end': 1564.17, 'text': 'Rather than going through like take derivative, take derivative, take derivative.', 'start': 1560.287, 'duration': 3.883}, {'end': 1565.851, 'text': 'Just write down the Taylor series.', 'start': 1564.59, 'duration': 1.261}, {'end': 1567.772, 'text': "Okay, so that's the first reason.", 'start': 1566.691, 'duration': 1.081}, {'end': 1573.506, 'text': 'Second reason, which is probably even more important.', 'start': 1568.743, 'duration': 4.763}, {'end': 1577.269, 'text': "the other two reasons are important, even if we don't care about moments, okay?", 'start': 1573.506, 'duration': 3.763}, {'end': 1580.771, 'text': 'The second reason is that the MGF determines the distribution.', 'start': 1577.669, 'duration': 3.102}, {'end': 1598.094, 'text': 'So another way to say that would be If you have two random variables, X and Y, and they both have the same MGF,', 'start': 1589.057, 'duration': 9.037}, {'end': 1599.475, 'text': 'then they must have the same distribution.', 'start': 1598.094, 'duration': 1.381}], 'summary': 'Mgf can be used to find nth moment and determine distribution equivalence between random variables.', 'duration': 56.559, 'max_score': 1542.916, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81542916.jpg'}, {'end': 1710.615, 'src': 'embed', 'start': 1673.998, 'weight': 2, 'content': [{'end': 1678.559, 'text': 'But if we have access to MGFs, things are a lot easier.', 'start': 1673.998, 'duration': 4.561}, {'end': 1684.461, 'text': "Convolution, you have to do this convolution sum or convolution integral, which we'll deal with somewhat later.", 'start': 1678.939, 'duration': 5.522}, {'end': 1685.861, 'text': "We've done a little bit of it before.", 'start': 1684.501, 'duration': 1.36}, {'end': 1687.161, 'text': "It's complicated.", 'start': 1686.401, 'duration': 0.76}, {'end': 1689.562, 'text': 'But suppose we have MGFs.', 'start': 1688.182, 'duration': 1.38}, {'end': 1697.004, 'text': "So let's say if X has MGF M sub X.", 'start': 1689.882, 'duration': 7.122}, {'end': 1700.505, 'text': 'And if Y has MGF M sub Y.', 'start': 1697.004, 'duration': 3.501}, {'end': 1710.615, 'text': "And they're independent, then we want the MGF of the sum.", 'start': 1703.07, 'duration': 7.545}], 'summary': 'Access to mgfs simplifies calculations for finding the mgf of the sum of independent random variables x and y.', 'duration': 36.617, 'max_score': 1673.998, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81673998.jpg'}], 'start': 1379.421, 'title': 'Importance of moment generating function', 'summary': 'Discusses the significance of moment generating function (mgf) in representing moments, taylor series, and simplifying computation of higher moments, emphasizing its relevance in probability and statistics. it also highlights the importance of moment generating functions for determining distributions, simplifying the handling of sums, and providing a convenient method for finding the mgf of the sum of independent random variables.', 'chapters': [{'end': 1565.851, 'start': 1379.421, 'title': 'Importance of moment generating function', 'summary': 'Discusses the importance of moment generating function (mgf) in representing moments, taylor series, and simplifying computation of higher moments, highlighting its relevance in probability and statistics.', 'duration': 186.43, 'highlights': ['Moment generating function (MGF) represents the nth moment of a random variable X as the coefficient of t to the n over n factorial in the Taylor expansion. The nth moment of a random variable X is represented by the coefficient of t to the n over n factorial in the Taylor expansion, providing a concise way to compute moments.', 'MGF allows for the computation of higher moments by taking nth derivatives and evaluating at 0, but it may be easier to directly work out the Taylor series by other methods for some functions. The MGF simplifies the computation of higher moments by taking nth derivatives and evaluating at 0, although for some functions, such as e to the x, it may be more efficient to directly write down the Taylor series.', 'Understanding the MGF is important for representing moments, Taylor series, and simplifying computation of higher moments in probability and statistics. The understanding of MGF is crucial for representing moments, Taylor series, and simplifying the computation of higher moments, which is significant in the field of probability and statistics.']}, {'end': 1883.631, 'start': 1566.691, 'title': 'Importance of moment generating functions', 'summary': 'Highlights the importance of moment generating functions for determining distributions, simplifying the handling of sums, and providing a convenient method for finding the mgf of the sum of independent random variables.', 'duration': 316.94, 'highlights': ['Determining Distributions The MGF determines the distribution of random variables, where having the same MGF implies the same distribution, making it a useful tool for recognizing and concluding distributions.', 'Handling Sums MGFs make sums of independent random variables much easier to handle by providing a simple method to find the MGF of the sum through multiplication, avoiding complicated convolutions or integrals.', 'Finding MGF of Sum The MGF of the sum of independent random variables can be found by multiplying their individual MGFs, providing a convenient method for dealing with sums without the need for complex integrals or convolutions.']}], 'duration': 504.21, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81379421.jpg', 'highlights': ['The MGF simplifies the computation of higher moments by taking nth derivatives and evaluating at 0, although for some functions, such as e to the x, it may be more efficient to directly write down the Taylor series.', 'Understanding the MGF is crucial for representing moments, Taylor series, and simplifying the computation of higher moments, which is significant in the field of probability and statistics.', 'The MGF of the sum of independent random variables can be found by multiplying their individual MGFs, providing a convenient method for dealing with sums without the need for complex integrals or convolutions.', 'MGFs make sums of independent random variables much easier to handle by providing a simple method to find the MGF of the sum through multiplication, avoiding complicated convolutions or integrals.', 'The MGF determines the distribution of random variables, where having the same MGF implies the same distribution, making it a useful tool for recognizing and concluding distributions.', 'The nth moment of a random variable X is represented by the coefficient of t to the n over n factorial in the Taylor expansion, providing a concise way to compute moments.']}, {'end': 2219.061, 'segs': [{'end': 1945.811, 'src': 'embed', 'start': 1884.832, 'weight': 0, 'content': [{'end': 1900.624, 'text': 'But because of that now we can immediately get the MGF of a binomial, because, Because, well, if we write down the definition,', 'start': 1884.832, 'duration': 15.792}, {'end': 1902.544, 'text': "then we're gonna have to do some big lotus thing.", 'start': 1900.624, 'duration': 1.92}, {'end': 1911.447, 'text': "But we don't have to do that, because if we think of the binomial as a sum of iid, Bernoulli, p's, and you use fact three there,", 'start': 1902.564, 'duration': 8.883}, {'end': 1920.449, 'text': 'then we immediately know that the M of t equals the MGF of Bernoulli to the nth power.', 'start': 1911.447, 'duration': 9.002}, {'end': 1925.79, 'text': 'So we just write it down immediately just by taking the nth power of that.', 'start': 1921.709, 'duration': 4.081}, {'end': 1936.124, 'text': 'Okay, so for practice, you could check, remember the binomial has, the mean of the binomial Np is Np, and the variance is Npq.', 'start': 1926.857, 'duration': 9.267}, {'end': 1939.607, 'text': 'And if you wanna check that statement, one there is true.', 'start': 1936.725, 'duration': 2.882}, {'end': 1945.811, 'text': "you could take this thing and check that the first derivative evaluated at zero, you'll get the mean.", 'start': 1939.607, 'duration': 6.204}], 'summary': 'The mgf of a binomial can be immediately derived as the mgf of bernoulli to the nth power, with mean np and variance npq.', 'duration': 60.979, 'max_score': 1884.832, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81884832.jpg'}, {'end': 2017.918, 'src': 'embed', 'start': 1989.063, 'weight': 2, 'content': [{'end': 1991.565, 'text': "So let's just talk about the standard normal first, cuz that's simpler.", 'start': 1989.063, 'duration': 2.502}, {'end': 1995.007, 'text': 'And we wanna compute the MGF.', 'start': 1993.486, 'duration': 1.521}, {'end': 2004.127, 'text': 'So by Lotus, this is 1 over square root of 2 pi integral minus infinity to infinity.', 'start': 1997.922, 'duration': 6.205}, {'end': 2006.809, 'text': 'So we want the expected value of e to the tz.', 'start': 2004.267, 'duration': 2.542}, {'end': 2009.612, 'text': "So by Lotus, it's e to the tz.", 'start': 2007.61, 'duration': 2.002}, {'end': 2015.877, 'text': 'And then we just have to write down the standard normal density, which is e to the minus z squared over 2.', 'start': 2010.072, 'duration': 5.805}, {'end': 2017.918, 'text': 'I already put the normalizing constant out there.', 'start': 2015.877, 'duration': 2.041}], 'summary': 'Compute the mgf for the standard normal, using lotus method.', 'duration': 28.855, 'max_score': 1989.063, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81989063.jpg'}, {'end': 2219.061, 'src': 'embed', 'start': 2196.243, 'weight': 3, 'content': [{'end': 2205.603, 'text': "We re-centered it at t, we didn't change the variance, okay? So we're integrating a normal density, we must get 1.", 'start': 2196.243, 'duration': 9.36}, {'end': 2211.511, 'text': "So, just by recognizing that that's a normal, except centered at t rather than centered at 0, we immediately know that's 1,", 'start': 2205.603, 'duration': 5.908}, {'end': 2213.614, 'text': 'so we get e to the t squared over 2..', 'start': 2211.511, 'duration': 2.103}, {'end': 2219.061, 'text': 'And from this, you can derive the non-standard normal as well, which you should do for practice.', 'start': 2213.614, 'duration': 5.447}], 'summary': 'Normal density centered at t gives e to the t squared over 2', 'duration': 22.818, 'max_score': 2196.243, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82196243.jpg'}], 'start': 1884.832, 'title': 'Mgf and completing the square for integration', 'summary': 'Covers the moment generating function (mgf) for binomial and normal distributions, emphasizing the mgf of binomial as the nth power of the mgf of bernoulli, and the mgf of standard normal distribution as the expected value of e to the tz, providing insights into the mean and variance. additionally, it discusses completing the square for integration, resulting in a normal distribution with a mean of 0 and variance of 1, yielding the value of e to the power of t squared over 2.', 'chapters': [{'end': 2044.594, 'start': 1884.832, 'title': 'Mgf of binomial and normal distribution', 'summary': 'Explains the moment generating function (mgf) for binomial and normal distributions, highlighting the mgf of binomial as the nth power of the mgf of bernoulli, and the mgf of standard normal distribution as the expected value of e to the tz, providing insights into the mean and variance of the binomial distribution.', 'duration': 159.762, 'highlights': ['The MGF of binomial is the nth power of the MGF of Bernoulli, simplifying the calculation process. By using fact three and regarding the binomial as a sum of iid Bernoulli distributions with probability p, we can immediately obtain the MGF of the binomial as the nth power of the MGF of Bernoulli.', 'Insights into the mean and variance of the binomial distribution. The mean of the binomial distribution is Np and the variance is Npq. This can be verified by evaluating the first and second derivatives of the MGF at zero to obtain the mean and variance.', 'The MGF of the standard normal distribution is computed using the expected value of e to the tz. The MGF of the standard normal distribution is calculated using the expected value of e to the tz by integrating the standard normal density, providing a deeper understanding of the standard normal distribution.']}, {'end': 2219.061, 'start': 2044.594, 'title': 'Completing the square for integration', 'summary': 'Discusses the process of completing the square to convert a function into a quadratic form, removing the linear term, and simplifying integration, resulting in a normal distribution with a mean of 0 and variance of 1, ultimately yielding the value of e to the power of t squared over 2.', 'duration': 174.467, 'highlights': ['By completing the square, the function is transformed into a quadratic form, eliminating the linear term and simplifying integration, resulting in a normal distribution with a mean of 0 and variance of 1, yielding the value of e to the power of t squared over 2.', 'Recognizing the transformed function as a normal distribution centered at t rather than 0, allows immediate determination of the integral as 1, resulting in e to the power of t squared over 2, and providing a basis for deriving non-standard normal distributions.']}], 'duration': 334.229, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ81884832.jpg', 'highlights': ['The MGF of binomial is the nth power of the MGF of Bernoulli, simplifying the calculation process.', 'Insights into the mean and variance of the binomial distribution: mean=Np, variance=Npq.', 'The MGF of the standard normal distribution is computed using the expected value of e to the tz.', 'By completing the square, the function is transformed into a quadratic form, resulting in a normal distribution with a mean of 0 and variance of 1.']}, {'end': 3042.413, 'segs': [{'end': 2252.753, 'src': 'embed', 'start': 2220.399, 'weight': 0, 'content': [{'end': 2226.561, 'text': "Okay, so we're gonna come back to MGFs later and use them.", 'start': 2220.399, 'duration': 6.162}, {'end': 2234.263, 'text': "But I want to do one more, kind of like famous probability example that's not exactly related to MGFs,", 'start': 2227.641, 'duration': 6.622}, {'end': 2239.705, 'text': "but which will be useful for the homework and for what we're doing next.", 'start': 2234.263, 'duration': 5.442}, {'end': 2245.087, 'text': "And that's called Laplace's Rule of Succession.", 'start': 2240.665, 'duration': 4.422}, {'end': 2252.753, 'text': 'Famous old problem, Laplace was a great mathematician and physicist.', 'start': 2247.83, 'duration': 4.923}], 'summary': "Introduction to laplace's rule of succession for probability examples and homework.", 'duration': 32.354, 'max_score': 2220.399, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82220399.jpg'}, {'end': 2474.401, 'src': 'embed', 'start': 2437.701, 'weight': 1, 'content': [{'end': 2444.414, 'text': "Well, the Bayesian point of view Is to say well, since it's unknown,", 'start': 2437.701, 'duration': 6.713}, {'end': 2449.797, 'text': "we're gonna quantify its uncertainty by treating it as a random variable that has some distribution.", 'start': 2444.414, 'duration': 5.383}, {'end': 2453.599, 'text': 'okay?. A distribution is just a reflection of our uncertainty.', 'start': 2449.797, 'duration': 3.802}, {'end': 2459.883, 'text': 'So the Bayesian approach is treat p as a random variable.', 'start': 2454.36, 'duration': 5.523}, {'end': 2474.401, 'text': "And then the reason it's called Bayesian is because then we can use Bayes' rule to say what's the distribution of p, given all the evidence we have,", 'start': 2463.894, 'duration': 10.507}], 'summary': "Bayesian approach quantifies uncertainty using random variable distribution and bayes' rule.", 'duration': 36.7, 'max_score': 2437.701, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82437701.jpg'}, {'end': 2634.109, 'src': 'embed', 'start': 2590.724, 'weight': 2, 'content': [{'end': 2594.225, 'text': "And so the sum of iid Bernoulli p's we know is binomial np.", 'start': 2590.724, 'duration': 3.501}, {'end': 2600.026, 'text': 'But p itself is random with the uniform distribution.', 'start': 2596.085, 'duration': 3.941}, {'end': 2609.249, 'text': "That's the structure, okay? And now the problem is defined.", 'start': 2605.228, 'duration': 4.021}, {'end': 2613.248, 'text': 'First of all, find the posterior distribution.', 'start': 2610.506, 'duration': 2.742}, {'end': 2619.394, 'text': 'By definition, posterior distribution means the distribution after we collect the data.', 'start': 2614.129, 'duration': 5.265}, {'end': 2624.538, 'text': 'This part is the prior, and the posterior is P given Sn.', 'start': 2620.114, 'duration': 4.424}, {'end': 2634.109, 'text': 'As we assume we observe s.', 'start': 2630.726, 'duration': 3.383}], 'summary': 'The problem involves finding the posterior distribution after observing data, with p being random and uniformly distributed.', 'duration': 43.385, 'max_score': 2590.724, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82590724.jpg'}, {'end': 2766.24, 'src': 'embed', 'start': 2741.481, 'weight': 3, 'content': [{'end': 2751.034, 'text': "The sun has risen on K of the last n days, okay? Given that information, how do we update our uncertainty about P? It's just Bayes' rule.", 'start': 2741.481, 'duration': 9.553}, {'end': 2757.316, 'text': "It's just a form that we haven't seen before, because on the one hand, these are PDFs.", 'start': 2751.054, 'duration': 6.262}, {'end': 2766.24, 'text': 'PDFs are not probabilities, okay? But PDFs, we can think of it kind of intuitively as a probability, or at least if we multiply by something.', 'start': 2757.597, 'duration': 8.643}], 'summary': "Using bayes' rule to update uncertainty about p based on sun rising on k of the last n days.", 'duration': 24.759, 'max_score': 2741.481, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82741481.jpg'}, {'end': 2837.997, 'src': 'embed', 'start': 2812.626, 'weight': 4, 'content': [{'end': 2818.229, 'text': "So you can make up some new notation if you want, but it's easier to just think about what things mean.", 'start': 2812.626, 'duration': 5.603}, {'end': 2823.273, 'text': "So this given p means we're just treating p as a known constant, even though it's a random variable.", 'start': 2818.61, 'duration': 4.663}, {'end': 2829.137, 'text': "So Bayes' rule, we swap these things, times f of p, that's the prior.", 'start': 2823.993, 'duration': 5.144}, {'end': 2837.997, 'text': "And it's also just equal to 1 because we used a uniform prior, so that makes it easy.", 'start': 2834.136, 'duration': 3.861}], 'summary': "Bayes' rule swaps things, times f of p, with a uniform prior making it easy.", 'duration': 25.371, 'max_score': 2812.626, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82812626.jpg'}, {'end': 3037.251, 'src': 'heatmap', 'start': 2919.314, 'weight': 0.731, 'content': [{'end': 2929.88, 'text': "We're gonna ignore the denominator because it doesn't depend on p, right? And for the numerator, that's just from the binomial.", 'start': 2919.314, 'duration': 10.566}, {'end': 2932.262, 'text': "That's n choose k.", 'start': 2930.281, 'duration': 1.981}, {'end': 2935.264, 'text': "n choose k is also a constant as it doesn't depend on p.", 'start': 2932.262, 'duration': 3.002}, {'end': 2937.765, 'text': "So I'm gonna ignore the n choose k.", 'start': 2935.264, 'duration': 2.501}, {'end': 2938.766, 'text': "That's just p to the k.", 'start': 2937.765, 'duration': 1.001}, {'end': 2949.745, 'text': "Q 1- p to the n- k, okay? So that part's 1, so that's actually easy.", 'start': 2941.58, 'duration': 8.165}, {'end': 2955.789, 'text': "To get the constant in front, then we'd have to integrate this thing, and we're gonna do that much later in the course.", 'start': 2950.746, 'duration': 5.043}, {'end': 2962.774, 'text': "But now let's just do the easier case, f of p given Sn equals n.", 'start': 2956.33, 'duration': 6.444}, {'end': 2965.876, 'text': "So that's the case where the sun did rise for the last n days.", 'start': 2962.774, 'duration': 3.102}, {'end': 2968.278, 'text': 'Now this is just p to the n.', 'start': 2966.437, 'duration': 1.841}, {'end': 2981.485, 'text': "Now this one is easy to normalize, right? Because the integral of this from 0 to 1 is just p to the n plus 1 over n plus 1, so it's 1 over n plus 1.", 'start': 2970.515, 'duration': 10.97}, {'end': 2983.767, 'text': 'So to normalize it, we just stick an n plus 1 there.', 'start': 2981.485, 'duration': 2.282}, {'end': 2987.171, 'text': 'Now this is a valid PDF.', 'start': 2986.009, 'duration': 1.162}, {'end': 2992.897, 'text': 'Okay, and now lastly to get this thing, so in other words, we got this thing without evaluating the denominator.', 'start': 2988.274, 'duration': 4.623}, {'end': 2998.481, 'text': 'And then lastly, if we want P equals 1 given Sn equals n.', 'start': 2993.338, 'duration': 5.143}, {'end': 3005.406, 'text': 'Well, just think of it this way.', 'start': 3003.425, 'duration': 1.981}, {'end': 3014.649, 'text': 'We want the expected value of a random variable by the fundamental bridge.', 'start': 3008.827, 'duration': 5.822}, {'end': 3017.81, 'text': 'We just want the expected value of a random variable with this distribution.', 'start': 3014.669, 'duration': 3.141}, {'end': 3024.873, 'text': "So that's just gonna be integral 0 to 1 n plus 1 p times p to the n dp.", 'start': 3017.83, 'duration': 7.043}, {'end': 3032.869, 'text': 'Integral p to the n plus 1 is p to the n plus 2 over n plus 2.', 'start': 3028.007, 'duration': 4.862}, {'end': 3037.251, 'text': 'So we get n plus 1 over n plus 2.', 'start': 3032.869, 'duration': 4.382}], 'summary': 'Derivation of a probability distribution with n and p constants.', 'duration': 117.937, 'max_score': 2919.314, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82919314.jpg'}], 'start': 2220.399, 'title': 'Probability calculation', 'summary': "Introduces laplace's rule of succession and discusses the calculation of the probability that the sun will rise tomorrow based on the number of consecutive days the sun has risen. it also explains the application of bayes' rule to find the probability of the sun rising based on past observations, with a specific example of calculating the probability of the sun rising after it has risen for the last n days. the sub summary also touches on the philosophical debates between bayesians and frequentists in dealing with unknown probabilities.", 'chapters': [{'end': 2634.109, 'start': 2220.399, 'title': "Laplace's rule of succession", 'summary': "Introduces laplace's rule of succession, discussing the calculation of the probability that the sun will rise tomorrow based on the number of consecutive days the sun has risen, and the philosophical debates between bayesians and frequentists in dealing with unknown probabilities.", 'duration': 413.71, 'highlights': ["Laplace's Rule of Succession is introduced, discussing the probability of the sun rising tomorrow based on consecutive days it has risen, leading to philosophical debates between Bayesians and frequentists. Introduction of Laplace's Rule of Succession, discussion of the probability of the sun rising tomorrow based on consecutive days it has risen, and philosophical debates between Bayesians and frequentists.", "Laplace's Rule of Succession involves treating the probability of the sun rising, denoted as p, as an unknown variable, leading to the Bayesian approach of quantifying its uncertainty by treating it as a random variable with a distribution. Treating the probability of the sun rising as an unknown variable, Bayesian approach of quantifying its uncertainty by treating it as a random variable with a distribution.", "The structure of the problem involves the sum of independent and identically distributed (iid) Bernoulli p's, known as binomial np, with p itself being random and having a uniform distribution. Structure of the problem involving the sum of iid Bernoulli p's, known as binomial np, with p having a uniform distribution."]}, {'end': 3042.413, 'start': 2634.109, 'title': "Bayes' rule for sun rising probability", 'summary': "Explains the application of bayes' rule to find the probability of the sun rising based on past observations, with a specific example of calculating the probability of the sun rising after it has risen for the last n days.", 'duration': 408.304, 'highlights': ["The chapter explains the application of Bayes' rule to find the probability of the sun rising based on past observations, with a specific example of calculating the probability of the sun rising after it has risen for the last n days.", "The chapter discusses the process of updating uncertainty about the probability of the sun rising using Bayes' rule, where the prior distribution is treated as uniform and updated to a conditional PDF, especially focused on the case where the sun has risen for the last n days.", "The chapter introduces the notation and concept of treating p as a known constant, even though it's a random variable, to apply Bayes' rule and swap the conditional and prior probabilities, emphasizing the continuous version of the law of total probability for a continuous random variable.", "The chapter presents the calculation of the posterior distribution, specifically focusing on the case where the sun has risen for the last n days, simplifying the normalization process to derive a valid PDF and calculating the expected value of the random variable using integral calculus and Laplace's principle."]}], 'duration': 822.014, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/N8O6zd6vTZ8/pics/N8O6zd6vTZ82220399.jpg', 'highlights': ["Introduction of Laplace's Rule of Succession, discussion of the probability of the sun rising tomorrow based on consecutive days it has risen, and philosophical debates between Bayesians and frequentists.", 'Treating the probability of the sun rising as an unknown variable, Bayesian approach of quantifying its uncertainty by treating it as a random variable with a distribution.', "Structure of the problem involving the sum of iid Bernoulli p's, known as binomial np, with p having a uniform distribution.", "The chapter explains the application of Bayes' rule to find the probability of the sun rising based on past observations, with a specific example of calculating the probability of the sun rising after it has risen for the last n days.", "The chapter introduces the notation and concept of treating p as a known constant, even though it's a random variable, to apply Bayes' rule and swap the conditional and prior probabilities, emphasizing the continuous version of the law of total probability for a continuous random variable.", "The chapter presents the calculation of the posterior distribution, specifically focusing on the case where the sun has risen for the last n days, simplifying the normalization process to derive a valid PDF and calculating the expected value of the random variable using integral calculus and Laplace's principle."]}], 'highlights': ['The exponential distribution is the only memoryless distribution in continuous time, while in discrete time, we have the geometric distribution, which is its discrete analog.', 'The relationship between the exponential and geometric distributions is very close, with the exponential being the continuous analog of the geometric.', 'The Weibull distribution is widely used in practice for survival time, obtained by taking an exponential to a power, which is a crucial building block in practical applications.', "The exponential distribution's memoryless property makes it a crucial building block and extremely useful in practical cases, despite not always being exactly true.", 'The MGF simplifies the computation of higher moments by taking nth derivatives and evaluating at 0, although for some functions, such as e to the x, it may be more efficient to directly write down the Taylor series.', 'The MGF of the sum of independent random variables can be found by multiplying their individual MGFs, providing a convenient method for dealing with sums without the need for complex integrals or convolutions.', "Introduction of Laplace's Rule of Succession, discussion of the probability of the sun rising tomorrow based on consecutive days it has risen, and philosophical debates between Bayesians and frequentists.", "The chapter explains the application of Bayes' rule to find the probability of the sun rising based on past observations, with a specific example of calculating the probability of the sun rising after it has risen for the last n days."]}