title

Lecture 24: Gamma distribution and Poisson process | Statistics 110

description

We introduce the Gamma distribution and discuss the connection between the Gamma distribution and Poisson processes.

detail

{'title': 'Lecture 24: Gamma distribution and Poisson process | Statistics 110', 'heatmap': [{'end': 557.065, 'start': 527.822, 'weight': 0.735}, {'end': 1180.364, 'start': 1112.186, 'weight': 0.701}, {'end': 1381.873, 'start': 1342.514, 'weight': 0.8}, {'end': 1590.609, 'start': 1490.136, 'weight': 0.824}, {'end': 1675.724, 'start': 1598.097, 'weight': 0.967}, {'end': 1907.094, 'start': 1814.343, 'weight': 0.856}, {'end': 2464.734, 'start': 2421.485, 'weight': 0.731}], 'summary': "The lecture covers number sequence and arithmetic puzzles, factorials, stirling's formula, gamma function and distribution, connections with normal, exponential, poisson, beta distributions, proving success through arrival summation, and mean and variance of the gamma function, providing insights into their properties and applications in mathematics and statistics.", 'chapters': [{'end': 186.574, 'segs': [{'end': 34.396, 'src': 'embed', 'start': 0.879, 'weight': 1, 'content': [{'end': 3.581, 'text': "Thought we'd start with a little puzzle I'm trying to solve.", 'start': 0.879, 'duration': 2.702}, {'end': 6.103, 'text': 'Maybe you can help me with the puzzle.', 'start': 3.781, 'duration': 2.322}, {'end': 10.807, 'text': 'Okay, let me make it a little less volume, thanks.', 'start': 6.804, 'duration': 4.003}, {'end': 13.389, 'text': "I don't need to echo that much.", 'start': 11.887, 'duration': 1.502}, {'end': 15.67, 'text': "All right, so here's the puzzle.", 'start': 13.809, 'duration': 1.861}, {'end': 21.415, 'text': "I thought this was a pretty interesting one, and it may be related to what we're gonna do.", 'start': 17.011, 'duration': 4.404}, {'end': 34.396, 'text': "Okay, so the question is, what's the next number in the sequence? 0, 1, 2.", 'start': 21.815, 'duration': 12.581}], 'summary': "Puzzle sequence: 0, 1, 2 - what's the next number?", 'duration': 33.517, 'max_score': 0.879, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY879.jpg'}, {'end': 186.574, 'src': 'embed', 'start': 74.399, 'weight': 0, 'content': [{'end': 77.922, 'text': 'Is there any majority opinion on this? Three.', 'start': 74.399, 'duration': 3.523}, {'end': 79.943, 'text': 'Three, nothing.', 'start': 78.842, 'duration': 1.101}, {'end': 87.77, 'text': 'Why did you say three? Integers.', 'start': 81.084, 'duration': 6.686}, {'end': 89.251, 'text': "Integers, okay, zero's an integer.", 'start': 88.03, 'duration': 1.221}, {'end': 102.329, 'text': "Why three? Yeah? It's values that the Poisson could take, okay? But I don't have to list them in order, though.", 'start': 90.312, 'duration': 12.017}, {'end': 104.27, 'text': 'I could have listed them out of order.', 'start': 102.349, 'duration': 1.921}, {'end': 112.015, 'text': "All right, well, okay, a lot of you seem to like the number three, but I haven't yet heard a good reason why.", 'start': 105.971, 'duration': 6.044}, {'end': 113.816, 'text': "I'm kind of interested.", 'start': 112.035, 'duration': 1.781}, {'end': 115.837, 'text': "It's an arithmetic sequence.", 'start': 113.836, 'duration': 2.001}, {'end': 117.758, 'text': "It's an arithmetic sequence.", 'start': 116.297, 'duration': 1.461}, {'end': 119.119, 'text': "I hadn't noticed that.", 'start': 118.298, 'duration': 0.821}, {'end': 123.231, 'text': "So I guess you're pointing out that It's an arithmetic, hey, that's pretty cool.", 'start': 119.319, 'duration': 3.912}, {'end': 125.273, 'text': 'So you go from here, add one, add one.', 'start': 123.251, 'duration': 2.022}, {'end': 127.615, 'text': 'Arithmetic sequence, okay.', 'start': 126.494, 'duration': 1.121}, {'end': 131.479, 'text': 'That might have been what the puzzle writer had in mind.', 'start': 127.896, 'duration': 3.583}, {'end': 133.941, 'text': "I've been trying to solve this for the last couple days, I got stuck.", 'start': 131.539, 'duration': 2.402}, {'end': 137.084, 'text': "I hadn't thought of three yet.", 'start': 134.762, 'duration': 2.322}, {'end': 139.186, 'text': 'The best answer I thought of so far was 720 factorial.', 'start': 137.144, 'duration': 2.042}, {'end': 157.365, 'text': 'So we can debate a little bit, is that more correct or is three more correct? Assuming it does go this way, anyone wanna guess the next term? Three.', 'start': 143.775, 'duration': 13.59}, {'end': 158.866, 'text': 'Three? Okay.', 'start': 157.385, 'duration': 1.481}, {'end': 165.992, 'text': 'All right, yeah, I think you got it.', 'start': 164.731, 'duration': 1.261}, {'end': 170.555, 'text': 'All right, so let me tell you why am I talking about this sequence.', 'start': 166.792, 'duration': 3.763}, {'end': 175.672, 'text': "Well, it's partly just like I love puzzles since the time I was little,", 'start': 170.811, 'duration': 4.861}, {'end': 181.393, 'text': 'but I hated this kind of puzzle because there never seems to be a principle for coming up with an answer.', 'start': 175.672, 'duration': 5.721}, {'end': 184.254, 'text': 'I make up one thing and the answer is supposed to be something else.', 'start': 181.853, 'duration': 2.401}, {'end': 186.574, 'text': "Well, what's the measure of what's a better right?", 'start': 184.754, 'duration': 1.82}], 'summary': 'Discussion on finding the next term in a sequence, with emphasis on the number three and its arithmetic properties.', 'duration': 112.175, 'max_score': 74.399, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY74399.jpg'}], 'start': 0.879, 'title': 'Number and arithmetic sequence puzzles', 'summary': 'Discusses number sequence puzzle with varied responses and no consensus, where three is the most popular choice. it also explores an arithmetic sequence puzzle, focusing on the challenge of finding the next term and the frustration of not having a clear principle for solving it.', 'chapters': [{'end': 112.015, 'start': 0.879, 'title': 'Number sequence puzzle', 'summary': 'Discusses a number sequence puzzle where participants attempt to identify the next number in the sequence, resulting in a variety of responses and no consensus, with the number three being the most popular choice.', 'duration': 111.136, 'highlights': ['Participants attempt to solve a number sequence puzzle, resulting in various responses and no consensus, with the number three being the most popular choice.', 'The sequence puzzle involves the numbers 0, 1, 2, leading to diverse suggestions such as 4, 3, 5, and mathematical constants like 42 and 1/E.', 'Despite a lack of agreement, the number three emerges as the most favored choice, although no compelling rationale for this choice is presented.']}, {'end': 186.574, 'start': 112.035, 'title': 'Arithmetic sequence puzzle', 'summary': 'Discusses an arithmetic sequence puzzle, with a focus on the challenge of finding the next term and the frustration of not having a clear principle for solving it.', 'duration': 74.539, 'highlights': ['The chapter discusses the frustration of solving a puzzle with no clear principle for finding an answer.', 'The speaker is intrigued by the arithmetic sequence puzzle and is impressed by the concept.', 'The speaker is puzzled and frustrated by the lack of a clear principle for solving the arithmetic sequence puzzle.']}], 'duration': 185.695, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY879.jpg', 'highlights': ['Participants attempt to solve a number sequence puzzle, resulting in various responses and no consensus, with the number three being the most popular choice.', 'The sequence puzzle involves the numbers 0, 1, 2, leading to diverse suggestions such as 4, 3, 5, and mathematical constants like 42 and 1/E.', 'Despite a lack of agreement, the number three emerges as the most favored choice, although no compelling rationale for this choice is presented.', 'The chapter discusses the frustration of solving a puzzle with no clear principle for finding an answer.', 'The speaker is puzzled and frustrated by the lack of a clear principle for solving the arithmetic sequence puzzle.', 'The speaker is intrigued by the arithmetic sequence puzzle and is impressed by the concept.']}, {'end': 407.601, 'segs': [{'end': 216.835, 'src': 'embed', 'start': 187.114, 'weight': 0, 'content': [{'end': 192.655, 'text': "I mean you need some measure of maybe the simplest answer, but will two people always agree on what's simpler??", 'start': 187.114, 'duration': 5.541}, {'end': 194.096, 'text': 'No, right?', 'start': 193.376, 'duration': 0.72}, {'end': 199.197, 'text': 'You need some complexity measure or something to actually make that into a valid, well-defined problem.', 'start': 194.136, 'duration': 5.061}, {'end': 205.332, 'text': 'In a sense, this is just as good an answer as 3.', 'start': 200.751, 'duration': 4.581}, {'end': 209.313, 'text': "Some of you had in mind an arithmetic sequence, that's a good suggestion.", 'start': 205.332, 'duration': 3.981}, {'end': 216.835, 'text': 'But I was thinking of 0, well, 0 is 0, 1.', 'start': 210.614, 'duration': 6.221}], 'summary': 'Measuring complexity is essential for well-defined problems.', 'duration': 29.721, 'max_score': 187.114, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY187114.jpg'}, {'end': 281.54, 'src': 'embed', 'start': 246.031, 'weight': 3, 'content': [{'end': 249.953, 'text': "But anyway, that means that's doing 3 factorial, 4 factorial, etc.", 'start': 246.031, 'duration': 3.922}, {'end': 253.935, 'text': "All right, so that's what one interpretation could be.", 'start': 250.293, 'duration': 3.642}, {'end': 263.474, 'text': 'So the general question is If we have a sequence of numbers, how do we extend it?', 'start': 256.356, 'duration': 7.118}, {'end': 270.577, 'text': 'And I was thinking of this example cuz I was thinking about factorials and how I wanna explain extending factorials.', 'start': 265.035, 'duration': 5.542}, {'end': 281.54, 'text': "So if you ever wondered what's pi factorial, we're gonna talk about questions like that.", 'start': 271.137, 'duration': 10.403}], 'summary': 'Exploring extending sequences of numbers, such as factorials, and considering the example of pi factorial.', 'duration': 35.509, 'max_score': 246.031, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY246031.jpg'}, {'end': 363.628, 'src': 'embed', 'start': 332.811, 'weight': 1, 'content': [{'end': 341.396, 'text': 'It says that n factorial is approximately square root of 2 pi n times n over e to the n.', 'start': 332.811, 'duration': 8.585}, {'end': 346.304, 'text': 'This is actually an extremely good approximation.', 'start': 343.823, 'duration': 2.481}, {'end': 352.725, 'text': "Even if you try this out on a calculator where n is 12 or 20, it's actually very, very good.", 'start': 346.484, 'duration': 6.241}, {'end': 363.628, 'text': 'Not only that, but this approximation is so good that if you take the ratio of this divided by this, it will converge to 1 as n goes to infinity.', 'start': 354.446, 'duration': 9.182}], 'summary': 'N factorial approximates square root of 2 pi n times n over e to the n, converging to 1 as n goes to infinity.', 'duration': 30.817, 'max_score': 332.811, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY332811.jpg'}], 'start': 187.114, 'title': "Factorials and stirling's formula", 'summary': "Explores extending factorials and sequences, with specific examples such as 0 factorial being 1, and introduces stirling's formula, which provides a very good approximation for factorials and converges to 1 as n goes to infinity.", 'chapters': [{'end': 303.394, 'start': 187.114, 'title': 'Extending factorials and sequences', 'summary': 'Discusses the concept of extending factorials and sequences, exploring the factorial function and the interpretation of sequences with increasing factorials, with specific examples such as 0 factorial being 1, 1 factorial being 1, 2 factorial being 2, 3 factorial being 6, and 4 factorial being 24.', 'duration': 116.28, 'highlights': ['The factorial function and the interpretation of sequences with increasing factorials are explored, with specific examples such as 0 factorial being 1, 1 factorial being 1, 2 factorial being 2, 3 factorial being 6, and 4 factorial being 24.', "The chapter delves into the concept of extending factorials and sequences, presenting the idea of extending factorials to values like pi factorial and discussing the factorial function's graphical representation.", 'The need for a complexity measure to define a valid problem in extending factorials and sequences is highlighted, emphasizing the challenge of determining simplicity and the role of complexity measures in resolving such challenges.']}, {'end': 407.601, 'start': 303.394, 'title': "Stirling's formula for factorials", 'summary': "Discusses the rapid growth of factorials and introduces stirling's formula, which provides a very good approximation for factorials and converges to 1 as n goes to infinity.", 'duration': 104.207, 'highlights': ["Stirling's formula provides an approximation for factorials, stating that n factorial is approximately square root of 2 pi n times n over e to the n, and it converges to 1 as n goes to infinity.", 'The formula is both beautiful and useful, and provides an extremely good approximation, even for large values of n such as 12 or 20.', 'The main driving factor in the rapid growth of factorials is n over e to the n, which dominates as n goes to infinity.']}], 'duration': 220.487, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY187114.jpg', 'highlights': ['The need for a complexity measure to define a valid problem in extending factorials and sequences is highlighted, emphasizing the challenge of determining simplicity and the role of complexity measures in resolving such challenges.', 'The formula is both beautiful and useful, and provides an extremely good approximation, even for large values of n such as 12 or 20.', 'The main driving factor in the rapid growth of factorials is n over e to the n, which dominates as n goes to infinity.', "The chapter delves into the concept of extending factorials and sequences, presenting the idea of extending factorials to values like pi factorial and discussing the factorial function's graphical representation.", "Stirling's formula provides an approximation for factorials, stating that n factorial is approximately square root of 2 pi n times n over e to the n, and it converges to 1 as n goes to infinity.", 'The factorial function and the interpretation of sequences with increasing factorials are explored, with specific examples such as 0 factorial being 1, 1 factorial being 1, 2 factorial being 2, 3 factorial being 6, and 4 factorial being 24.']}, {'end': 863.378, 'segs': [{'end': 481.384, 'src': 'embed', 'start': 451.29, 'weight': 0, 'content': [{'end': 456.852, 'text': "We need the gamma function because we're going to be introducing something called the gamma distribution.", 'start': 451.29, 'duration': 5.562}, {'end': 459.17, 'text': "We're not done with beta.", 'start': 458.089, 'duration': 1.081}, {'end': 464.093, 'text': "We were doing beta last time, and hopefully it wasn't too scary, despite being Halloween.", 'start': 459.47, 'duration': 4.623}, {'end': 466.034, 'text': "We'll come back to the beta.", 'start': 465.154, 'duration': 0.88}, {'end': 469.296, 'text': 'The beta turns out to be extremely closely connected with the gamma.', 'start': 466.054, 'duration': 3.242}, {'end': 473.319, 'text': "So to understand the beta, we need to do the gamma, which we're about to do.", 'start': 469.697, 'duration': 3.622}, {'end': 481.384, 'text': 'The gamma distribution is named the gamma distribution because it stems from the gamma function,', 'start': 474.022, 'duration': 7.362}], 'summary': 'Introduction to gamma function and its connection to beta distribution.', 'duration': 30.094, 'max_score': 451.29, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY451290.jpg'}, {'end': 557.065, 'src': 'heatmap', 'start': 496.389, 'weight': 1, 'content': [{'end': 501.39, 'text': "I've seen people make lists of the ten most famous constants, but not the ten most famous functions.", 'start': 496.389, 'duration': 5.001}, {'end': 508.978, 'text': "But I'm sure someone has, and if they haven't, or if they have and they did a good job, they would put the gamma function on the list okay?", 'start': 502.097, 'duration': 6.881}, {'end': 514.82, 'text': "So it's very, very important in math, not just in probability and statistics, but it's also important in statistics,", 'start': 509.358, 'duration': 5.462}, {'end': 516.14, 'text': 'which is why we need it right now.', 'start': 514.82, 'duration': 1.32}, {'end': 519.2, 'text': 'But it has its own life outside of this subject.', 'start': 516.26, 'duration': 2.94}, {'end': 521.961, 'text': "All right, so here's the definition of the gamma function.", 'start': 519.861, 'duration': 2.1}, {'end': 526.782, 'text': "That's just a capital letter gamma.", 'start': 521.981, 'duration': 4.801}, {'end': 531.785, 'text': 'of a is defined by the following integral.', 'start': 527.822, 'duration': 3.963}, {'end': 542.834, 'text': 'Integral 0 to infinity, x to the a, e to the minus x, dx over x.', 'start': 532.286, 'duration': 10.548}, {'end': 548.638, 'text': "Usually you'll see it written at the next to the a minus 1 here, which of course is the same thing, cuz you can cancel out one of the x's.", 'start': 542.834, 'duration': 5.804}, {'end': 555.083, 'text': "But for certain reasons, it's convenient keeping kind of one x on hold over here.", 'start': 549.219, 'duration': 5.864}, {'end': 557.065, 'text': 'Just a different way to write it, but the same thing.', 'start': 555.464, 'duration': 1.601}], 'summary': 'The gamma function is important in math, especially in probability and statistics, and has a specific integral definition.', 'duration': 52.249, 'max_score': 496.389, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY496389.jpg'}, {'end': 760.495, 'src': 'embed', 'start': 692.912, 'weight': 2, 'content': [{'end': 698.614, 'text': 'Gamma of n is n-1 factorial, so you just have to be a little bit careful.', 'start': 692.912, 'duration': 5.702}, {'end': 702.575, 'text': "That's just for historical reasons, pretty much.", 'start': 699.614, 'duration': 2.961}, {'end': 708.457, 'text': 'This is for any n a positive integer.', 'start': 704.936, 'duration': 3.521}, {'end': 721.583, 'text': "So as I said, there's other ways you could extend factorials to the positive real numbers.", 'start': 716.462, 'duration': 5.121}, {'end': 728.305, 'text': 'But this one turns out to be the one that has at least found the most different applications in math.', 'start': 721.663, 'duration': 6.642}, {'end': 732.226, 'text': 'So this one is the most natural and best one known in a certain sense.', 'start': 728.385, 'duration': 3.841}, {'end': 741.169, 'text': 'So why is this true? Well, let me write down another identity for gamma.', 'start': 734.127, 'duration': 7.042}, {'end': 750.206, 'text': "Gamma of x plus 1 equals, So there's this recursive formula for gamma.", 'start': 741.749, 'duration': 8.457}, {'end': 755.09, 'text': "You don't need to know much about the gamma function for this course.", 'start': 750.426, 'duration': 4.664}, {'end': 760.495, 'text': 'Depending on what kind of math you do later, you might need a lot of gamma stuff.', 'start': 756.051, 'duration': 4.444}], 'summary': 'Gamma function has n-1 factorial, found many applications in math.', 'duration': 67.583, 'max_score': 692.912, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY692912.jpg'}], 'start': 407.861, 'title': 'Gamma function and distribution', 'summary': 'Introduces the importance of the gamma function, its role in connecting non-negative integers, its relevance to the beta function, and its significance in mathematics and statistics. it also explains the definition and domain of the gamma function, its connection with factorials, and its extension to positive real numbers.', 'chapters': [{'end': 516.14, 'start': 407.861, 'title': 'Introduction to gamma function and distribution', 'summary': 'Introduces the importance of the gamma function and its role in connecting non-negative integers, its relevance to the beta function, and its significance in mathematics and statistics.', 'duration': 108.279, 'highlights': ['The gamma function is crucial for interpolating non-negative integers and is the basis for introducing the gamma distribution, which is closely connected to the beta function.', 'The gamma function is regarded as one of the most famous functions in mathematics and holds significant importance in both probability and statistics.', 'Understanding the gamma function is essential as it plays a vital role in connecting non-negative integers and has significant relevance to the beta function.']}, {'end': 690.755, 'start': 516.26, 'title': 'Definition of the gamma function', 'summary': 'Explains the definition and domain of the gamma function, which is defined by an integral and exists for any real a greater than 0, extending the factorial function.', 'duration': 174.495, 'highlights': ['The gamma function is defined by the integral: Integral 0 to infinity, x to the a, e to the minus x, dx over x, and exists for any real a greater than 0.', 'The gamma function is an extension of the factorial function, and it would be more convenient if gamma of a is a factorial.', 'When x goes to infinity, e to the minus x is the dominant part, causing the integral to converge for any real a greater than 0.', 'When x goes near 0, integrating x to the a minus 1 is fine as long as a is greater than 0, showing that the integral converges in this case.']}, {'end': 863.378, 'start': 692.912, 'title': 'Gamma function and factorial extension', 'summary': 'Discusses the gamma function, its recursive formula, and its relationship with factorials, emphasizing its extension to positive real numbers and its significance in mathematics.', 'duration': 170.466, 'highlights': ['The gamma function extends factorials to positive real numbers and has found numerous applications in mathematics.', 'The gamma function satisfies a recursive formula: gamma of x+1 = x * gamma of x.', 'The gamma function is closely related to factorials and shares the same recursive formula, implying their equivalence.', 'The gamma function of 1 equals 1, and the recursive formula for gamma leads to gamma of 2 equaling 1 and gamma of 3 equaling 2.']}], 'duration': 455.517, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY407861.jpg', 'highlights': ['The gamma function is crucial for interpolating non-negative integers and is the basis for introducing the gamma distribution, closely connected to the beta function.', 'The gamma function is regarded as one of the most famous functions in mathematics and holds significant importance in both probability and statistics.', 'The gamma function extends factorials to positive real numbers and has found numerous applications in mathematics.', 'The gamma function is defined by the integral: Integral 0 to infinity, x to the a, e to the minus x, dx over x, and exists for any real a greater than 0.', 'The gamma function satisfies a recursive formula: gamma of x+1 = x * gamma of x.', 'Understanding the gamma function is essential as it plays a vital role in connecting non-negative integers and has significant relevance to the beta function.']}, {'end': 1300.945, 'segs': [{'end': 899.443, 'src': 'embed', 'start': 865.188, 'weight': 2, 'content': [{'end': 871.551, 'text': "Okay, so that's the gamma function, and that's what happens for factorials.", 'start': 865.188, 'duration': 6.363}, {'end': 879.556, 'text': "One other thing about the gamma function that's worth knowing is what's gamma of 1 half?", 'start': 873.372, 'duration': 6.184}, {'end': 887.48, 'text': "Just in case you ever wondered what's the factorial of negative 1 half and what does that mean?", 'start': 881.857, 'duration': 5.623}, {'end': 890.602, 'text': 'Well, gamma of 1 half.', 'start': 888.261, 'duration': 2.341}, {'end': 896.282, 'text': "So when it's an integer, we just get factorials which are integers.", 'start': 891.82, 'duration': 4.462}, {'end': 899.443, 'text': 'Gamma of 1 half is square root of pi.', 'start': 896.822, 'duration': 2.621}], 'summary': 'The gamma function of 1/2 is the square root of pi.', 'duration': 34.255, 'max_score': 865.188, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY865188.jpg'}, {'end': 1180.364, 'src': 'heatmap', 'start': 1037.924, 'weight': 0, 'content': [{'end': 1045.547, 'text': 'So you could either think of it as if you knew this fact, this would give you another way to get the normalizing constant of the normal.', 'start': 1037.924, 'duration': 7.623}, {'end': 1053.589, 'text': 'Or the way we actually have it is we already know from a different method the normalizing constant of the normal, then it can prove this fact.', 'start': 1046.087, 'duration': 7.502}, {'end': 1058.151, 'text': 'So once we have gamma of 1 half, then we know gamma of 3 halves.', 'start': 1054.99, 'duration': 3.161}, {'end': 1066.036, 'text': '3 halves is 1 plus 1 half.', 'start': 1062.413, 'duration': 3.623}, {'end': 1070.279, 'text': "So if we use this formula, it just says it's 1 half gamma of 1 half.", 'start': 1066.576, 'duration': 3.703}, {'end': 1075.163, 'text': 'So that would be square root of pi divided by 2, and so on.', 'start': 1070.699, 'duration': 4.464}, {'end': 1080.607, 'text': 'So then we could get gamma of 5 halves, gamma of 7 halves, and get all the half integers that way.', 'start': 1075.563, 'duration': 5.044}, {'end': 1084.649, 'text': "All right, so that's just a quick introduction to the gamma function.", 'start': 1082.148, 'duration': 2.501}, {'end': 1088.871, 'text': "What's on this board is all you need to know about the gamma function.", 'start': 1085.73, 'duration': 3.141}, {'end': 1094.194, 'text': "But now let's talk about the gamma distribution to connect it back to probability.", 'start': 1089.412, 'duration': 4.782}, {'end': 1105.86, 'text': "Okay, so here's kind of a simple, naive way to create a PDF.", 'start': 1095.475, 'duration': 10.385}, {'end': 1111.786, 'text': 'based on the gamma function.', 'start': 1110.345, 'duration': 1.441}, {'end': 1113.967, 'text': 'So we have this integral here.', 'start': 1112.186, 'duration': 1.781}, {'end': 1119.169, 'text': "This is just the definition of that's just the definition of gamma, of A, okay?", 'start': 1114.207, 'duration': 4.962}, {'end': 1123.992, 'text': 'And suppose we wanna somehow guess at some kind of PDF.', 'start': 1119.289, 'duration': 4.703}, {'end': 1131.435, 'text': "that's gonna be related to the gamma function and we want a valid PDF.", 'start': 1123.992, 'duration': 7.443}, {'end': 1138.219, 'text': "And to have a valid PDF, all we need is something that's non-negative and integrates to 1.", 'start': 1132.636, 'duration': 5.583}, {'end': 1156.403, 'text': 'What would you do? Normalize it, how? How would you normalize this thing? Just divide by that value.', 'start': 1138.219, 'duration': 18.184}, {'end': 1161.145, 'text': "So if we divide both sides by gamma of A, that's a PDF.", 'start': 1156.823, 'duration': 4.322}, {'end': 1171.449, 'text': 'So 1 equals integral 0 to infinity, 1 over gamma of A, x to the A, e to the minus x, dx over x.', 'start': 1162.505, 'duration': 8.944}, {'end': 1180.364, 'text': "That's all just like a simple naive trick that we have that integral, just divide by gamma of A, now we have a PDF.", 'start': 1173.599, 'duration': 6.765}], 'summary': 'Quick introduction to the gamma function and its connection to probability, deriving a valid pdf using the gamma function.', 'duration': 86.068, 'max_score': 1037.924, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1037924.jpg'}], 'start': 865.188, 'title': 'Gamma function and distribution', 'summary': 'Explores the gamma function, emphasizing gamma of 1 half as the square root of pi and its connection to the normalizing constant in the normal distribution. it also introduces the gamma function and its relation to the normalizing constant of the normal distribution, before exploring its connection to the exponential distribution.', 'chapters': [{'end': 930.085, 'start': 865.188, 'title': 'Gamma function and square root of pi', 'summary': 'Explores the gamma function, highlighting the value of gamma of 1 half as the square root of pi and its connection to the normalizing constant in the normal distribution.', 'duration': 64.897, 'highlights': ["The gamma function's value for 1 half is the square root of pi, which is a key connection to the normalizing constant in the normal distribution.", 'The gamma function yields factorials for integers, providing a clear understanding of its behavior for integer inputs.']}, {'end': 1300.945, 'start': 931.126, 'title': 'Introduction to gamma function and distribution', 'summary': 'Introduces the gamma function and its relation to the normalizing constant of the normal distribution, before exploring the naive method of creating a pdf using the gamma function and its connection to the exponential distribution.', 'duration': 369.819, 'highlights': ['The chapter introduces the gamma function and its relation to the normalizing constant of the normal distribution. It discusses the use of the gamma function to obtain the normalizing constant of the normal distribution, showcasing the connection between gamma of 1 half and gamma of 3 halves.', 'The naive method of creating a PDF using the gamma function and its connection to the exponential distribution. It explains a simple, naive trick to create a valid PDF related to the gamma function, by dividing the integral by gamma of A, and then discusses the relationship between gamma and exponential distribution with a scale parameter.']}], 'duration': 435.757, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY865188.jpg', 'highlights': ["The gamma function's value for 1 half is the square root of pi, which is a key connection to the normalizing constant in the normal distribution.", 'The chapter introduces the gamma function and its relation to the normalizing constant of the normal distribution. It discusses the use of the gamma function to obtain the normalizing constant of the normal distribution, showcasing the connection between gamma of 1 half and gamma of 3 halves.', 'The gamma function yields factorials for integers, providing a clear understanding of its behavior for integer inputs.', 'The naive method of creating a PDF using the gamma function and its connection to the exponential distribution. It explains a simple, naive trick to create a valid PDF related to the gamma function, by dividing the integral by gamma of A, and then discusses the relationship between gamma and exponential distribution with a scale parameter.']}, {'end': 1907.094, 'segs': [{'end': 1383.273, 'src': 'heatmap', 'start': 1342.514, 'weight': 0, 'content': [{'end': 1348.197, 'text': "So I'm just replacing x by lambda y to the a, e to the minus lambda y.", 'start': 1342.514, 'duration': 5.683}, {'end': 1355.3, 'text': '1 over x, and x is lambda y times lambda.', 'start': 1348.217, 'duration': 7.083}, {'end': 1358.301, 'text': 'So that lambda cancels that lambda.', 'start': 1356.48, 'duration': 1.821}, {'end': 1361.883, 'text': "And that's what it looks like.", 'start': 1360.202, 'duration': 1.681}, {'end': 1365.326, 'text': 'I should emphasize the possible values.', 'start': 1362.825, 'duration': 2.501}, {'end': 1367.727, 'text': 'This is for x greater than 0.', 'start': 1365.366, 'duration': 2.361}, {'end': 1370.108, 'text': 'This one is for y greater than 0.', 'start': 1367.727, 'duration': 2.381}, {'end': 1381.873, 'text': 'So, like the exponential, the gamma is a continuous distribution on the positive real numbers, okay?', 'start': 1370.108, 'duration': 11.765}, {'end': 1383.273, 'text': 'All right.', 'start': 1381.893, 'duration': 1.38}], 'summary': 'Gamma distribution is continuous on positive real numbers.', 'duration': 35.056, 'max_score': 1342.514, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1342514.jpg'}, {'end': 1497.604, 'src': 'embed', 'start': 1461.893, 'weight': 1, 'content': [{'end': 1464.395, 'text': "But as you'll see later, gamma is closely related to normal.", 'start': 1461.893, 'duration': 2.502}, {'end': 1470.479, 'text': "So once we have the gamma and the normal, we can create the rest of the distributions we'll need.", 'start': 1464.675, 'duration': 5.804}, {'end': 1483.97, 'text': 'OK, so all right, so to talk about the gamma and exponential connection, We need to think about a Poisson process.', 'start': 1472.281, 'duration': 11.689}, {'end': 1497.604, 'text': "Okay, so we've only talked a little bit about Poisson processes, which is, if you take Stat 171, which is stochastic processes,", 'start': 1490.136, 'duration': 7.468}], 'summary': 'Gamma is related to normal, used to create distributions. poisson process discussed for stochastic processes.', 'duration': 35.711, 'max_score': 1461.893, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1461893.jpg'}, {'end': 1590.609, 'src': 'heatmap', 'start': 1490.136, 'weight': 0.824, 'content': [{'end': 1497.604, 'text': "Okay, so we've only talked a little bit about Poisson processes, which is, if you take Stat 171, which is stochastic processes,", 'start': 1490.136, 'duration': 7.468}, {'end': 1499.186, 'text': 'you can do a lot of Poisson process stuff.', 'start': 1497.604, 'duration': 1.582}, {'end': 1503.205, 'text': 'For our purposes, we just need to know a couple basic facts.', 'start': 1499.462, 'duration': 3.743}, {'end': 1507.548, 'text': "First of all, what is a Poisson process? You've seen them before.", 'start': 1503.705, 'duration': 3.843}, {'end': 1509.209, 'text': 'Just to quickly remind you,', 'start': 1508.009, 'duration': 1.2}, {'end': 1525.502, 'text': "we're imagining emails arriving in an inbox where the number of emails in some time interval let's say n sub t equals number of emails up to time t.", 'start': 1509.209, 'duration': 16.293}, {'end': 1526.763, 'text': 'We just have this timeline here.', 'start': 1525.502, 'duration': 1.261}, {'end': 1540.475, 'text': "And the Poisson process is called a Poisson process because we're assuming that that's Poisson with rate lambda times t right?", 'start': 1530.808, 'duration': 9.667}, {'end': 1546.039, 'text': 'So in a time interval of length t this is true.', 'start': 1540.655, 'duration': 5.384}, {'end': 1548.6, 'text': "I stated this as you're going from 0 to t.", 'start': 1546.099, 'duration': 2.501}, {'end': 1553.404, 'text': "But we're actually assuming that this is true in any time interval of length t.", 'start': 1548.6, 'duration': 4.804}, {'end': 1554.805, 'text': "It's gonna be Poisson lambda t.", 'start': 1553.404, 'duration': 1.401}, {'end': 1558.301, 'text': "And there's only one other assumption for a Poisson process.", 'start': 1555.58, 'duration': 2.721}, {'end': 1565.243, 'text': 'That is that the number of arrivals in disjoint intervals are independent.', 'start': 1558.781, 'duration': 6.462}, {'end': 1572.845, 'text': 'That is, the number of emails you get from this time to this time, is independent of the number of emails you get from this time to this time,', 'start': 1566.403, 'duration': 6.442}, {'end': 1573.385, 'text': 'for example.', 'start': 1572.845, 'duration': 0.54}, {'end': 1578.787, 'text': 'So there are only those two assumptions.', 'start': 1575.966, 'duration': 2.821}, {'end': 1590.609, 'text': 'And Okay, so a few weeks ago we saw what the connection with the exponential distribution is.', 'start': 1582.608, 'duration': 8.001}], 'summary': 'Poisson process models emails arriving with rate lambda times t, and assumes independence of arrivals in disjoint intervals.', 'duration': 100.473, 'max_score': 1490.136, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1490136.jpg'}, {'end': 1675.724, 'src': 'heatmap', 'start': 1598.097, 'weight': 0.967, 'content': [{'end': 1612.528, 'text': "So we're just drawing, Draw an x every time you get an email, right? That kind of picture, that's the picture we have in mind.", 'start': 1598.097, 'duration': 14.431}, {'end': 1623.356, 'text': 'Okay, so what we proved a few weeks ago was that if we call this one t1, t1 is the time of the first email.', 'start': 1612.908, 'duration': 10.448}, {'end': 1626.079, 'text': "that's exponential lambda, okay?", 'start': 1623.356, 'duration': 2.723}, {'end': 1636.98, 'text': "And the proof for that is just to say, just to remind you, What's the probability that t1 is greater than t?", 'start': 1626.619, 'duration': 10.361}, {'end': 1640.862, 'text': "That's the same thing as saying at time t.", 'start': 1637.92, 'duration': 2.942}, {'end': 1642.484, 'text': "I haven't yet gotten any email.", 'start': 1640.862, 'duration': 1.622}, {'end': 1647.167, 'text': "That's the same thing as saying that n sub t equals 0, right?", 'start': 1642.984, 'duration': 4.183}, {'end': 1649.589, 'text': "Cuz, it's just the same thing, right?", 'start': 1647.327, 'duration': 2.262}, {'end': 1658.555, 'text': "No emails up until time t, but that's just given by a Poisson e to the minus lambda t.", 'start': 1649.649, 'duration': 8.906}, {'end': 1660.897, 'text': 'The rest of the Poisson stuff is just one.', 'start': 1658.555, 'duration': 2.342}, {'end': 1663.899, 'text': "So it's just this one line calculation.", 'start': 1661.397, 'duration': 2.502}, {'end': 1675.724, 'text': "That shows that, and then you say, that's just 1 minus the exponential lambda CDF, so that's true, okay? So this first time is exponential.", 'start': 1664.281, 'duration': 11.443}], 'summary': 'Proof that first email arrival time is exponential with parameter lambda.', 'duration': 77.627, 'max_score': 1598.097, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1598097.jpg'}, {'end': 1771.807, 'src': 'embed', 'start': 1734.529, 'weight': 3, 'content': [{'end': 1743.175, 'text': "Rather than stating it with Poissons, we could say that the inter-arrival times, that is just these distances between x's in this timeline.", 'start': 1734.529, 'duration': 8.646}, {'end': 1753.822, 'text': 'Inter-arrival times are iid exponential lambda.', 'start': 1743.915, 'duration': 9.907}, {'end': 1761.159, 'text': 'So those are the inter-arrival times.', 'start': 1758.177, 'duration': 2.982}, {'end': 1771.807, 'text': 'But what if we wanna know the actual times? So we call this T2 and T3, T4, T5, and so on.', 'start': 1762.02, 'duration': 9.787}], 'summary': 'Inter-arrival times are iid exponential lambda, while actual times are denoted as t2, t3, t4, t5, and so on.', 'duration': 37.278, 'max_score': 1734.529, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1734529.jpg'}, {'end': 1907.094, 'src': 'heatmap', 'start': 1814.343, 'weight': 0.856, 'content': [{'end': 1820.266, 'text': "where these xj's are, these inter-arrival times, which we just said are exponential?", 'start': 1814.343, 'duration': 5.923}, {'end': 1821.787, 'text': 'lambda iid.', 'start': 1820.266, 'duration': 1.521}, {'end': 1831.973, 'text': 'Okay This is the most important story for the gamma distribution, at least for the integer case.', 'start': 1824.088, 'duration': 7.885}, {'end': 1833.695, 'text': "Right now, I'm assuming n is an integer.", 'start': 1832.033, 'duration': 1.662}, {'end': 1836.036, 'text': 'Over here, a did not need to be an integer.', 'start': 1834.175, 'duration': 1.861}, {'end': 1845.803, 'text': 'But if we assume that it is an integer, then this is gonna be gamma of n lambda.', 'start': 1836.597, 'duration': 9.206}, {'end': 1852.188, 'text': "We haven't shown this yet.", 'start': 1850.867, 'duration': 1.321}, {'end': 1854.71, 'text': "So that's something we need to prove.", 'start': 1852.969, 'duration': 1.741}, {'end': 1859.059, 'text': 'that this kind of a sum of iid exponentials will be gamma.', 'start': 1855.117, 'duration': 3.942}, {'end': 1860.5, 'text': "That's what we're gonna show next.", 'start': 1859.119, 'duration': 1.381}, {'end': 1865.142, 'text': 'But we can see the analogy.', 'start': 1862.521, 'duration': 2.621}, {'end': 1868.184, 'text': 'remember the negative binomial and the geometric?', 'start': 1865.142, 'duration': 3.042}, {'end': 1871.886, 'text': "The geometric you're waiting for one success in discrete time, right?", 'start': 1868.584, 'duration': 3.302}, {'end': 1876.689, 'text': "Negative binomial you're waiting for, however, many seven successes, say right?", 'start': 1872.406, 'duration': 4.283}, {'end': 1879.63, 'text': 'And you can think of the negative binomial as the sum of geometrics.', 'start': 1876.709, 'duration': 2.921}, {'end': 1882.832, 'text': 'So you wait for your first success, then you wait for your second, and then third.', 'start': 1879.65, 'duration': 3.182}, {'end': 1885.561, 'text': 'And so on, right, in discrete time.', 'start': 1883.24, 'duration': 2.321}, {'end': 1890.304, 'text': 'The exponential is the continuous time analog of the geometric.', 'start': 1885.862, 'duration': 4.442}, {'end': 1895.367, 'text': 'So this is a continuous time analog of negative binomial.', 'start': 1891.545, 'duration': 3.822}, {'end': 1900.43, 'text': 'So how long do you have to wait for n successes in continuous time?', 'start': 1895.747, 'duration': 4.683}, {'end': 1904.773, 'text': 'Where, in this case, success just means getting an email.', 'start': 1902.131, 'duration': 2.642}, {'end': 1907.094, 'text': "so that's a pretty weak definition of success.", 'start': 1904.773, 'duration': 2.321}], 'summary': 'The sum of iid exponentials will be gamma, an analogy to negative binomial and geometric distributions.', 'duration': 92.751, 'max_score': 1814.343, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1814343.jpg'}, {'end': 1917.574, 'src': 'embed', 'start': 1885.862, 'weight': 5, 'content': [{'end': 1890.304, 'text': 'The exponential is the continuous time analog of the geometric.', 'start': 1885.862, 'duration': 4.442}, {'end': 1895.367, 'text': 'So this is a continuous time analog of negative binomial.', 'start': 1891.545, 'duration': 3.822}, {'end': 1900.43, 'text': 'So how long do you have to wait for n successes in continuous time?', 'start': 1895.747, 'duration': 4.683}, {'end': 1904.773, 'text': 'Where, in this case, success just means getting an email.', 'start': 1902.131, 'duration': 2.642}, {'end': 1907.094, 'text': "so that's a pretty weak definition of success.", 'start': 1904.773, 'duration': 2.321}, {'end': 1917.574, 'text': "But anyway, that's what's going on, okay? Success is however you wanted to define it, but in this case we just mean the arrivals.", 'start': 1907.154, 'duration': 10.42}], 'summary': 'Exponential is the continuous time analog of the geometric. it represents waiting time for n successes in continuous time.', 'duration': 31.712, 'max_score': 1885.862, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1885862.jpg'}], 'start': 1301.805, 'title': 'Gamma distribution, poisson process, and exponential distribution', 'summary': 'Discusses the transformation of functions into distributions, particularly the gamma distribution, and its relationship with exponential, poisson, normal, and beta distributions, as well as the concept of poisson process, inter-arrival times, and the connection with exponential distribution, highlighting the properties and the connection with the gamma distribution.', 'chapters': [{'end': 1483.97, 'start': 1301.805, 'title': 'Gamma distribution and its relationship', 'summary': 'Discusses the transformation of functions into distributions, particularly the gamma distribution, and its relationship with exponential, poisson, normal, and beta distributions, providing insights into the connections between these distributions and their significance in statistical analysis.', 'duration': 182.165, 'highlights': ['Gamma distribution is a continuous distribution on the positive real numbers The gamma distribution is continuous on the positive real numbers, similar to the exponential distribution.', 'Gamma distribution relates to exponential, Poisson, normal, and beta distributions The gamma distribution is closely related to the exponential, Poisson, normal, and beta distributions, providing insights into their connections and variations.', 'Remaining distributions are variations of gammas or normals The remaining distributions are variations of gamma or possibly with normals, highlighting the significance of gamma and its close relationship with normal distribution.']}, {'end': 1907.094, 'start': 1490.136, 'title': 'Poisson process and exponential distribution', 'summary': 'Explains the concept of poisson process, inter-arrival times, and the connection with exponential distribution, highlighting the properties and the connection with the gamma distribution.', 'duration': 416.958, 'highlights': ['The inter-arrival times in a Poisson process are iid exponential lambda, and the time of the nth arrival can be defined as the sum of these inter-arrival times.', 'The connection between the Poisson process and the gamma distribution, specifically the gamma of n lambda, is highlighted, although the proof for this connection is yet to be demonstrated.', 'The analogy between the negative binomial and the geometric distributions is discussed, emphasizing the continuous time analog of negative binomial as the exponential distribution.']}], 'duration': 605.289, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1301805.jpg', 'highlights': ['Gamma distribution is continuous on the positive real numbers, similar to the exponential distribution.', 'Gamma distribution is closely related to the exponential, Poisson, normal, and beta distributions, providing insights into their connections and variations.', 'Remaining distributions are variations of gamma or possibly with normals, highlighting the significance of gamma and its close relationship with normal distribution.', 'The inter-arrival times in a Poisson process are iid exponential lambda, and the time of the nth arrival can be defined as the sum of these inter-arrival times.', 'The connection between the Poisson process and the gamma distribution, specifically the gamma of n lambda, is highlighted.', 'The analogy between the negative binomial and the geometric distributions is discussed, emphasizing the continuous time analog of negative binomial as the exponential distribution.']}, {'end': 2540.114, 'segs': [{'end': 2040.821, 'src': 'embed', 'start': 1981.936, 'weight': 0, 'content': [{'end': 1983.998, 'text': 'And then take that thing and convolve it with x4.', 'start': 1981.936, 'duration': 2.062}, {'end': 1986.461, 'text': 'Well, you get pretty tired of it pretty soon.', 'start': 1984.479, 'duration': 1.982}, {'end': 1991.846, 'text': 'But hopefully before you got tired of it, you would see a pattern, and you could prove it by induction.', 'start': 1986.961, 'duration': 4.885}, {'end': 1994.485, 'text': "But I don't wanna do it that way.", 'start': 1993.384, 'duration': 1.101}, {'end': 2003.311, 'text': "We have a much better method available that is MGFs, okay? MGF, right, because we're adding up IID things.", 'start': 1994.525, 'duration': 8.786}, {'end': 2009.435, 'text': 'And in class, we already derived the MGF of an exponential.', 'start': 2003.411, 'duration': 6.024}, {'end': 2016.56, 'text': "It's a nice looking thing, and so we may as well use that, okay? So that's what we're gonna prove right now.", 'start': 2009.496, 'duration': 7.064}, {'end': 2032.336, 'text': "Proof that, let's say t equals the sum xj j equals 1 to n with xj iid exponential lambda.", 'start': 2021.089, 'duration': 11.247}, {'end': 2040.821, 'text': "Let's do exponential 1 first, is gamma of n1.", 'start': 2035.998, 'duration': 4.823}], 'summary': 'Using mgfs to prove the sum of iid exponential random variables equals gamma of n', 'duration': 58.885, 'max_score': 1981.936, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1981936.jpg'}, {'end': 2132.343, 'src': 'embed', 'start': 2062.451, 'weight': 2, 'content': [{'end': 2066.433, 'text': "So I'd rather just set it equal to 1, and then we can always multiply or divide by lambda later.", 'start': 2062.451, 'duration': 3.982}, {'end': 2067.974, 'text': "It's not a big deal.", 'start': 2066.612, 'duration': 1.362}, {'end': 2071.755, 'text': "All right, so let's do that using the MGF.", 'start': 2069.554, 'duration': 2.201}, {'end': 2074.577, 'text': 'So the MGF of Xj.', 'start': 2073.116, 'duration': 1.461}, {'end': 2083.723, 'text': "The MGF of X1, since they're IID, they all have the same MGF.", 'start': 2079.241, 'duration': 4.482}, {'end': 2088.385, 'text': "It's the same as MGF of X2, and all XJs.", 'start': 2083.783, 'duration': 4.602}, {'end': 2096.408, 'text': "It's just 1 over 1 minus t.", 'start': 2090.025, 'duration': 6.383}, {'end': 2105.092, 'text': 'Which we can also think of as a geometric series, right? So we used this before to get all the moments of X1, all at once using the geometric series.', 'start': 2096.408, 'duration': 8.684}, {'end': 2109.983, 'text': 'And this is valid for t less than 1.', 'start': 2105.532, 'duration': 4.451}, {'end': 2112.445, 'text': 'Okay, so we did that before.', 'start': 2109.983, 'duration': 2.462}, {'end': 2124.376, 'text': 'Therefore, We can write down the MGF of T.', 'start': 2115.108, 'duration': 9.268}, {'end': 2126.938, 'text': "I'll call it Tn just as a reminder that there's n of them.", 'start': 2124.376, 'duration': 2.562}, {'end': 2132.343, 'text': 'MGF of Tn, easy, just raise this to the nth power.', 'start': 2127.499, 'duration': 4.844}], 'summary': 'Using mgf, we find mgf of xj = 1/(1-t); mgf of tn = (1/(1-t))^n.', 'duration': 69.892, 'max_score': 2062.451, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2062451.jpg'}, {'end': 2196.027, 'src': 'embed', 'start': 2160.558, 'weight': 4, 'content': [{'end': 2167.822, 'text': 'Well, to answer that question, we need to look at the MGF of a gamma distribution right?', 'start': 2160.558, 'duration': 7.264}, {'end': 2175.466, 'text': "So now I'll make up a new notation, just so that I'm not doing circular reasoning.", 'start': 2168.142, 'duration': 7.324}, {'end': 2180.069, 'text': "Well, let's just say y.", 'start': 2176.067, 'duration': 4.002}, {'end': 2186.653, 'text': "Let y be gamma of n1, and let's find its MGF okay?", 'start': 2180.069, 'duration': 6.584}, {'end': 2196.027, 'text': "And we said before that there's no masquerading distributions that pretend to have a certain MGF and aren't actually what they appear to be, right?", 'start': 2187.033, 'duration': 8.994}], 'summary': 'The speaker discusses finding the mgf of a gamma distribution and introduces a new notation.', 'duration': 35.469, 'max_score': 2160.558, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2160558.jpg'}, {'end': 2380.121, 'src': 'embed', 'start': 2346.081, 'weight': 5, 'content': [{'end': 2351.944, 'text': 'But what you do need to be able to do is pattern recognition and see, well, this looks like a gamma integral.', 'start': 2346.081, 'duration': 5.863}, {'end': 2354.065, 'text': "It's just that I have this extra constant in here.", 'start': 2352.184, 'duration': 1.881}, {'end': 2355.867, 'text': "That's really no big deal, right?", 'start': 2354.366, 'duration': 1.501}, {'end': 2361.91, 'text': 'Just make a change of variables just to make this e to the minus x again, right?', 'start': 2355.887, 'duration': 6.023}, {'end': 2380.121, 'text': "So we're just gonna let x equal 1, minus ty, so that dx equals 1, minus t dy.", 'start': 2361.99, 'duration': 18.131}], 'summary': 'Recognize patterns and make change of variables for gamma integral.', 'duration': 34.04, 'max_score': 2346.081, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2346081.jpg'}, {'end': 2464.734, 'src': 'heatmap', 'start': 2421.485, 'weight': 0.731, 'content': [{'end': 2432.071, 'text': 'And y to the n is just, Well, y is x over 1 minus t, so we are gonna get 1 minus t to the minus n.', 'start': 2421.485, 'duration': 10.586}, {'end': 2436.515, 'text': "That looks pretty promising, right? Cuz that's actually exactly what we wanted over there.", 'start': 2432.071, 'duration': 4.444}, {'end': 2442.1, 'text': "And then we're gonna have x to the n, e to the minus x.", 'start': 2437.996, 'duration': 4.104}, {'end': 2452.948, 'text': 'One convenient thing about kind of keeping this as dy over y instead of canceling it, is that this multiplicative thing is gonna cancel.', 'start': 2444.964, 'duration': 7.984}, {'end': 2456.53, 'text': 'So dy over y is actually the same thing as dx over x.', 'start': 2453.228, 'duration': 3.302}, {'end': 2458.171, 'text': 'That actually makes this kind of thing easier.', 'start': 2456.53, 'duration': 1.641}, {'end': 2464.734, 'text': 'And that is again gamma of n, not gamma of a, or just let a equal n.', 'start': 2460.412, 'duration': 4.322}], 'summary': 'Using y = x/(1-t), and x^n * e^(-x) simplifies the equation, making it easier to solve. this results in gamma of n, not gamma of a, or just let a equal n.', 'duration': 43.249, 'max_score': 2421.485, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2421485.jpg'}, {'end': 2540.114, 'src': 'embed', 'start': 2504.711, 'weight': 6, 'content': [{'end': 2507.932, 'text': 'And well, another thing we need to do is get the moments.', 'start': 2504.711, 'duration': 3.221}, {'end': 2517.216, 'text': "So again, that sounds like it could be a nasty problem, but actually it's not bad at all.", 'start': 2508.913, 'duration': 8.303}, {'end': 2519.817, 'text': 'once you see the pattern, okay?', 'start': 2517.216, 'duration': 2.601}, {'end': 2521.058, 'text': "So let's let X be.", 'start': 2519.897, 'duration': 1.161}, {'end': 2540.114, 'text': 'And by the way, this calculation here actually in this last line, here this part, I actually never assumed that n was actually an integer.', 'start': 2527.725, 'duration': 12.389}], 'summary': 'The process involves getting moments, which is not as difficult as it may seem, and does not require n to be an integer.', 'duration': 35.403, 'max_score': 2504.711, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2504711.jpg'}], 'start': 1907.154, 'title': 'Proving success through arrival summation', 'summary': 'Discusses methods for proving arrival success by summing up arrivals, introducing convolution and mgfs, and deriving the mgf of an exponential distribution. it also covers the mgf of the gamma distribution, demonstrating the simplification of the integral and its validity, and explains the process of making a change of variables to simplify the integral, resembling a gamma function and its connection with the exponential, along with the approach for moment calculation.', 'chapters': [{'end': 2062.411, 'start': 1907.154, 'title': 'Proving success through arrival summation', 'summary': 'Discusses proving the arrival success by summing up arrivals, introducing two methods for proving it: convolution and mgfs, and deriving the mgf of an exponential distribution.', 'duration': 155.257, 'highlights': ['The chapter discusses two methods for proving arrival success: convolution and MGFs, with the latter being preferred for summing up IID exponential distributions.', 'The chapter introduces the use of Moment Generating Functions (MGFs) for proving the sum of IID exponential distributions, providing a more efficient method than convolution.', 'The chapter discusses deriving the MGF of an exponential distribution, specifically gamma of n1, and extends it to general lambda by scaling.']}, {'end': 2345.68, 'start': 2062.451, 'title': 'Mgf of gamma distribution', 'summary': 'Discusses the moment-generating function (mgf) of the gamma distribution, demonstrating how to find mgf of a gamma distribution and explaining the simplification of the integral to find the mgf, ultimately showing the validity of mgf for the distribution.', 'duration': 283.229, 'highlights': ['Demonstrating the MGF of Xj and simplifying it to 1 over 1 minus t, valid for t less than 1. The MGF of Xj is simplified to 1 over 1 minus t, valid for t less than 1, demonstrating the simplicity of the MGF for the distribution.', 'Explaining the process of finding the MGF of Tn by raising it to the nth power, resulting in 1 over 1 minus t to the n, for t less than 1. The process of finding the MGF of Tn by raising it to the nth power is explained, resulting in 1 over 1 minus t to the n, for t less than 1, showcasing the straightforward approach to obtaining the MGF.', 'Illustrating the simplification of the integral to find the MGF of a gamma distribution, showcasing the familiarity of the integral with the gamma function. The simplification of the integral to find the MGF of a gamma distribution is illustrated, showcasing the familiarity of the integral with the gamma function and demonstrating the process of obtaining the MGF.']}, {'end': 2540.114, 'start': 2346.081, 'title': 'Gamma integral and moment calculation', 'summary': 'Explains the process of making a change of variables to simplify the integral to resemble a gamma function and discusses the connection with the exponential, along with the approach for moment calculation.', 'duration': 194.033, 'highlights': ['The process of making a change of variables to simplify the integral to resemble a gamma function and discussing the connection with the exponential.', "The statement 'dy over y is the same as dx over x' simplifies the calculation and makes it easier with the algebra.", 'The approach for moment calculation, which is not as difficult as it seems once the pattern is recognized.']}], 'duration': 632.96, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY1907154.jpg', 'highlights': ['The chapter introduces the use of Moment Generating Functions (MGFs) for proving the sum of IID exponential distributions, providing a more efficient method than convolution.', 'The chapter discusses deriving the MGF of an exponential distribution, specifically gamma of n1, and extends it to general lambda by scaling.', 'Demonstrating the MGF of Xj and simplifying it to 1 over 1 minus t, valid for t less than 1, showcasing the simplicity of the MGF for the distribution.', 'Explaining the process of finding the MGF of Tn by raising it to the nth power, resulting in 1 over 1 minus t to the n, for t less than 1, showcasing the straightforward approach to obtaining the MGF.', 'Illustrating the simplification of the integral to find the MGF of a gamma distribution, showcasing the familiarity of the integral with the gamma function and demonstrating the process of obtaining the MGF.', 'The process of making a change of variables to simplify the integral to resemble a gamma function and discussing the connection with the exponential.', 'The approach for moment calculation, which is not as difficult as it seems once the pattern is recognized.']}, {'end': 2927.212, 'segs': [{'end': 2874.863, 'src': 'embed', 'start': 2843.1, 'weight': 0, 'content': [{'end': 2849.265, 'text': 'So it says that gamma a1 has mean a and variance a.', 'start': 2843.1, 'duration': 6.165}, {'end': 2852.527, 'text': 'Sounds a little like the Poisson where the mean equals the variance.', 'start': 2849.265, 'duration': 3.262}, {'end': 2857.892, 'text': 'But this is kind of a special, for the Poisson the mean is always equal to the variance.', 'start': 2853.088, 'duration': 4.804}, {'end': 2862.233, 'text': 'And this is kind of just because I let lambda equal 1.', 'start': 2858.312, 'duration': 3.921}, {'end': 2874.863, 'text': 'So now if we bring back the lambda, gamma of A lambda, well, to get that, we define that just by rescaling it, by dividing by lambda.', 'start': 2862.233, 'duration': 12.63}], 'summary': 'Gamma distribution with mean and variance a; poisson comparison; rescaling by lambda.', 'duration': 31.763, 'max_score': 2843.1, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2843100.jpg'}, {'end': 2925.568, 'src': 'embed', 'start': 2895.182, 'weight': 1, 'content': [{'end': 2897.743, 'text': 'So gamma A lambda has this mean and this variance.', 'start': 2895.182, 'duration': 2.561}, {'end': 2907.568, 'text': "But notice that it was just easier to do this thing with lambda equals 1 first, as long as we don't forget at the end to bring back the lambda.", 'start': 2898.004, 'duration': 9.564}, {'end': 2916.692, 'text': 'All right, so this is just kind of like an introduction to the gamma distribution and the gamma function, how it connects with the exponential.', 'start': 2908.668, 'duration': 8.024}, {'end': 2920.614, 'text': "Next time I'll show you how it connects to the beta and the normal.", 'start': 2917.512, 'duration': 3.102}, {'end': 2925.568, 'text': "A little bit early today, cuz I don't wanna scare you again with the beta till next time.", 'start': 2921.578, 'duration': 3.99}], 'summary': 'Intro to gamma distribution, connecting with exponential, beta, and normal distributions.', 'duration': 30.386, 'max_score': 2895.182, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2895182.jpg'}], 'start': 2540.454, 'title': 'Mean and variance of gamma function', 'summary': 'Discusses the calculation of the mean and variance of the gamma function using lotus and mgf, revealing that the mean and variance of gamma function a are both a, irrespective of whether a is an integer.', 'chapters': [{'end': 2927.212, 'start': 2540.454, 'title': 'Mean and variance of gamma function', 'summary': 'Discusses the calculation of the mean and variance of the gamma function using lotus and mgf, revealing that the mean and variance of gamma function a are both a, irrespective of whether a is an integer.', 'duration': 386.758, 'highlights': ['The mean and variance of gamma function a are both a, irrespective of whether a is an integer. This is demonstrated through the direct use of LOTUS and MGF, showing that the expected value and variance of the gamma function a are both equal to a, regardless of whether a is an integer or not.', 'The gamma function has very nice properties, including the simplification of gamma of a plus 1 to a gamma a. The gamma function exhibits favorable properties, such as the simplification of gamma of a plus 1 to a gamma a, which facilitates the calculation of the mean and variance in the context of the gamma function.', 'Introduction to the gamma distribution and its connection with the exponential, beta, and normal distributions. The chapter serves as an introduction to the gamma distribution and its relationship with the exponential, beta, and normal distributions, providing a foundation for future discussions on these topics.']}], 'duration': 386.758, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Qjeswpm0cWY/pics/Qjeswpm0cWY2540454.jpg', 'highlights': ['The mean and variance of gamma function a are both a, irrespective of whether a is an integer. This is demonstrated through the direct use of LOTUS and MGF, showing that the expected value and variance of the gamma function a are both equal to a, regardless of whether a is an integer or not.', 'The gamma function has very nice properties, including the simplification of gamma of a plus 1 to a gamma a. The gamma function exhibits favorable properties, such as the simplification of gamma of a plus 1 to a gamma a, which facilitates the calculation of the mean and variance in the context of the gamma function.', 'Introduction to the gamma distribution and its connection with the exponential, beta, and normal distributions. The chapter serves as an introduction to the gamma distribution and its relationship with the exponential, beta, and normal distributions, providing a foundation for future discussions on these topics.']}], 'highlights': ['The gamma function is crucial for interpolating non-negative integers and is the basis for introducing the gamma distribution, closely connected to the beta function.', 'The gamma function is regarded as one of the most famous functions in mathematics and holds significant importance in both probability and statistics.', 'The gamma function extends factorials to positive real numbers and has found numerous applications in mathematics.', 'The gamma function is defined by the integral: Integral 0 to infinity, x to the a, e to the minus x, dx over x, and exists for any real a greater than 0.', 'The gamma function satisfies a recursive formula: gamma of x+1 = x * gamma of x.', 'The mean and variance of gamma function a are both a, irrespective of whether a is an integer. This is demonstrated through the direct use of LOTUS and MGF, showing that the expected value and variance of the gamma function a are both equal to a, regardless of whether a is an integer or not.', 'The need for a complexity measure to define a valid problem in extending factorials and sequences is highlighted, emphasizing the challenge of determining simplicity and the role of complexity measures in resolving such challenges.', 'The formula is both beautiful and useful, and provides an extremely good approximation, even for large values of n such as 12 or 20.', 'The main driving factor in the rapid growth of factorials is n over e to the n, which dominates as n goes to infinity.', "The chapter delves into the concept of extending factorials and sequences, presenting the idea of extending factorials to values like pi factorial and discussing the factorial function's graphical representation.", "Stirling's formula provides an approximation for factorials, stating that n factorial is approximately square root of 2 pi n times n over e to the n, and it converges to 1 as n goes to infinity.", 'The factorial function and the interpretation of sequences with increasing factorials are explored, with specific examples such as 0 factorial being 1, 1 factorial being 1, 2 factorial being 2, 3 factorial being 6, and 4 factorial being 24.', 'The chapter introduces the use of Moment Generating Functions (MGFs) for proving the sum of IID exponential distributions, providing a more efficient method than convolution.', 'The chapter discusses deriving the MGF of an exponential distribution, specifically gamma of n1, and extends it to general lambda by scaling.', 'Demonstrating the MGF of Xj and simplifying it to 1 over 1 minus t, valid for t less than 1, showcasing the simplicity of the MGF for the distribution.', 'Explaining the process of finding the MGF of Tn by raising it to the nth power, resulting in 1 over 1 minus t to the n, for t less than 1, showcasing the straightforward approach to obtaining the MGF.', 'Illustrating the simplification of the integral to find the MGF of a gamma distribution, showcasing the familiarity of the integral with the gamma function and demonstrating the process of obtaining the MGF.', 'The process of making a change of variables to simplify the integral to resemble a gamma function and discussing the connection with the exponential.', 'The approach for moment calculation, which is not as difficult as it seems once the pattern is recognized.', 'The chapter introduces the gamma function and its relation to the normalizing constant of the normal distribution. It discusses the use of the gamma function to obtain the normalizing constant of the normal distribution, showcasing the connection between gamma of 1 half and gamma of 3 halves.', 'The gamma function yields factorials for integers, providing a clear understanding of its behavior for integer inputs.', 'The naive method of creating a PDF using the gamma function and its connection to the exponential distribution. It explains a simple, naive trick to create a valid PDF related to the gamma function, by dividing the integral by gamma of A, and then discusses the relationship between gamma and exponential distribution with a scale parameter.', 'Gamma distribution is continuous on the positive real numbers, similar to the exponential distribution.', 'Gamma distribution is closely related to the exponential, Poisson, normal, and beta distributions, providing insights into their connections and variations.', 'Remaining distributions are variations of gamma or possibly with normals, highlighting the significance of gamma and its close relationship with normal distribution.', 'The inter-arrival times in a Poisson process are iid exponential lambda, and the time of the nth arrival can be defined as the sum of these inter-arrival times.', 'The connection between the Poisson process and the gamma distribution, specifically the gamma of n lambda, is highlighted.', 'The analogy between the negative binomial and the geometric distributions is discussed, emphasizing the continuous time analog of negative binomial as the exponential distribution.', 'Introduction to the gamma distribution and its connection with the exponential, beta, and normal distributions. The chapter serves as an introduction to the gamma distribution and its relationship with the exponential, beta, and normal distributions, providing a foundation for future discussions on these topics.', 'The speaker is puzzled and frustrated by the lack of a clear principle for solving the arithmetic sequence puzzle.', 'The speaker is intrigued by the arithmetic sequence puzzle and is impressed by the concept.', 'Despite a lack of agreement, the number three emerges as the most favored choice, although no compelling rationale for this choice is presented.', 'The chapter discusses the frustration of solving a puzzle with no clear principle for finding an answer.', 'Participants attempt to solve a number sequence puzzle, resulting in various responses and no consensus, with the number three being the most popular choice.', 'The sequence puzzle involves the numbers 0, 1, 2, leading to diverse suggestions such as 4, 3, 5, and mathematical constants like 42 and 1/E.', 'The chapter introduces the gamma function and its relation to the normalizing constant of the normal distribution. It discusses the use of the gamma function to obtain the normalizing constant of the normal distribution, showcasing the connection between gamma of 1 half and gamma of 3 halves.', 'The gamma function yields factorials for integers, providing a clear understanding of its behavior for integer inputs.', 'The naive method of creating a PDF using the gamma function and its connection to the exponential distribution. It explains a simple, naive trick to create a valid PDF related to the gamma function, by dividing the integral by gamma of A, and then discusses the relationship between gamma and exponential distribution with a scale parameter.']}