title

Lecture 8: Random Variables and Their Distributions | Statistics 110

description

Much of this course is about random variables and their distributions. The relationship between a random variable and its distribution can seem subtle but it is essential! We discuss distributions, cumulative distribution functions (CDFs), probability mass functions (PMFs), and the Hypergeometric distribution.

detail

{'title': 'Lecture 8: Random Variables and Their Distributions | Statistics 110', 'heatmap': [{'end': 424.105, 'start': 390.086, 'weight': 0.886}, {'end': 1663.455, 'start': 1598.369, 'weight': 0.784}, {'end': 2026.381, 'start': 1958.405, 'weight': 0.943}, {'end': 2158.469, 'start': 2111.453, 'weight': 0.819}, {'end': 2540.817, 'start': 2462.331, 'weight': 0.962}], 'summary': 'The lecture covers topics including binomial distribution, iid bernoulli distribution, pmf/cdf, pmf for discrete random variables, random variable addition, binomial distribution & probability, and hypergeometric distribution, emphasizing key concepts, properties, and their applications in statistics.', 'chapters': [{'end': 309.241, 'segs': [{'end': 76.901, 'src': 'embed', 'start': 0.652, 'weight': 0, 'content': [{'end': 4.595, 'text': 'Okay, so, picking up exactly where we left off last time,', 'start': 0.652, 'duration': 3.943}, {'end': 10.5, 'text': 'we were starting on the binomial and the Bernoulli distribution and random variables in general, right?', 'start': 4.595, 'duration': 5.905}, {'end': 20.328, 'text': 'So I wanna kind of review the binomial a little bit and go further than we got last time with the binomial and then also, in parallel with that,', 'start': 12.141, 'duration': 8.187}, {'end': 23.39, 'text': 'be discussing more about random variables.', 'start': 20.328, 'duration': 3.062}, {'end': 25.792, 'text': "okay?. So here's the binomial, just to remind you.", 'start': 23.39, 'duration': 2.402}, {'end': 35.519, 'text': 'Binomial distribution is one of the most famous distributions and one of the most useful ones in all of statistics.', 'start': 28.015, 'duration': 7.504}, {'end': 42.323, 'text': 'And we write it as distribution.', 'start': 38.301, 'duration': 4.022}, {'end': 47.586, 'text': 'We write it as bin for shorthand.', 'start': 42.824, 'duration': 4.762}, {'end': 50.608, 'text': 'It has two parameters, n and p.', 'start': 47.606, 'duration': 3.002}, {'end': 51.689, 'text': "That's what they're usually called.", 'start': 50.608, 'duration': 1.081}, {'end': 53.129, 'text': 'I mean, you can call them whatever you want.', 'start': 51.729, 'duration': 1.4}, {'end': 55.551, 'text': 'But the default choice would be to call them n and p.', 'start': 53.33, 'duration': 2.221}, {'end': 58.591, 'text': 'So those are called parameters.', 'start': 57.43, 'duration': 1.161}, {'end': 61.993, 'text': 'If you change the parameters, then you have a different distribution.', 'start': 59.211, 'duration': 2.782}, {'end': 63.714, 'text': "It's still called a binomial distribution.", 'start': 62.253, 'duration': 1.461}, {'end': 66.835, 'text': 'So strictly speaking, there is not just one binomial distribution.', 'start': 63.994, 'duration': 2.841}, {'end': 69.757, 'text': "There's a whole family of binomial distributions.", 'start': 67.175, 'duration': 2.582}, {'end': 72.278, 'text': 'N is any positive integer.', 'start': 70.557, 'duration': 1.721}, {'end': 76.901, 'text': 'P is any real number between 0 and 1.', 'start': 72.898, 'duration': 4.003}], 'summary': 'Reviewing binomial and bernoulli distribution, discussing random variables, and parameters n and p.', 'duration': 76.249, 'max_score': 0.652, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA652.jpg'}], 'start': 0.652, 'title': 'Binomial distribution', 'summary': 'Provides an overview of the binomial and bernoulli distribution, highlighting their parameters, n and p, and emphasizes their importance in statistics. it also discusses the concept of binomial distribution as involving n independent trials with probabilities of success, denoted as p.', 'chapters': [{'end': 101.272, 'start': 0.652, 'title': 'Binomial distribution overview', 'summary': 'Discusses the binomial and bernoulli distribution, emphasizing its importance in statistics and highlighting its parameters, n and p, where n is any positive integer and p is any real number between 0 and 1. it also mentions three important ways to think of the binomial distribution, with a focus on the relevance of the story behind it.', 'duration': 100.62, 'highlights': ['The binomial distribution is one of the most famous and useful distributions in statistics, with two parameters, n and p, and has a whole family of distributions for any positive integer N and any real number between 0 and 1.', 'The distribution is denoted as bin for shorthand and can be altered by changing its parameters, leading to a different distribution within the binomial family.', 'The chapter emphasizes the importance of understanding the story behind the binomial distribution, highlighting its relevance and significance in statistics.']}, {'end': 309.241, 'start': 101.292, 'title': 'Understanding binomial distribution', 'summary': 'Discusses the concept of binomial distribution, emphasizing that it involves n independent trials resulting in success or failure, with the probability of success denoted as p, and presents it as a very general and useful distribution.', 'duration': 207.949, 'highlights': ['The binomial distribution involves n independent trials with each resulting in success or failure, and the probability of success denoted as P, making it a very general and useful distribution.', 'It can be interpreted as the number of successes in n independent trials, where each trial follows a Bernoulli P distribution, with success or failure outcomes.', 'Another interpretation of the binomial distribution is through indicator random variables, where X can be seen as the sum of indicator random variables X1, X2, ..., Xn, denoting success as 1 and failure as 0.', 'The binomial distribution is a versatile concept and not limited to examples like coin flips, as it allows defining success in various ways, making it applicable to a wide range of scenarios.']}], 'duration': 308.589, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA652.jpg', 'highlights': ['The binomial distribution is one of the most famous and useful distributions in statistics, with two parameters, n and p, and has a whole family of distributions for any positive integer N and any real number between 0 and 1.', 'The binomial distribution involves n independent trials with each resulting in success or failure, and the probability of success denoted as P, making it a very general and useful distribution.', 'The distribution is denoted as bin for shorthand and can be altered by changing its parameters, leading to a different distribution within the binomial family.', 'It can be interpreted as the number of successes in n independent trials, where each trial follows a Bernoulli P distribution, with success or failure outcomes.', 'The chapter emphasizes the importance of understanding the story behind the binomial distribution, highlighting its relevance and significance in statistics.', 'Another interpretation of the binomial distribution is through indicator random variables, where X can be seen as the sum of indicator random variables X1, X2, ..., Xn, denoting success as 1 and failure as 0.', 'The binomial distribution is a versatile concept and not limited to examples like coin flips, as it allows defining success in various ways, making it applicable to a wide range of scenarios.']}, {'end': 927.371, 'segs': [{'end': 424.105, 'src': 'heatmap', 'start': 390.086, 'weight': 0.886, 'content': [{'end': 396.053, 'text': 'So the key, a very, very common confusion is to confuse random variables with distributions.', 'start': 390.086, 'duration': 5.967}, {'end': 401.779, 'text': "The random variable is, mathematically, it's a function, like we were defining it last time.", 'start': 396.734, 'duration': 5.045}, {'end': 409.408, 'text': 'But intuitively, this is just x1 is 1 if the first trial is a success and 0 otherwise, right? So that depends on the first trial.', 'start': 401.799, 'duration': 7.609}, {'end': 415.941, 'text': 'The distribution is saying what are the probabilities that X will behave in different ways?', 'start': 410.578, 'duration': 5.363}, {'end': 419.583, 'text': 'So you can have lots and lots of random variables that all have the same distribution.', 'start': 416.301, 'duration': 3.282}, {'end': 422.764, 'text': "Because the distribution is saying what's the probability that we'll do this??", 'start': 420.203, 'duration': 2.561}, {'end': 424.105, 'text': "What's the probability that we'll do that?", 'start': 422.824, 'duration': 1.281}], 'summary': 'Random variables are functions, while distributions describe their probabilities.', 'duration': 34.019, 'max_score': 390.086, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA390086.jpg'}, {'end': 757.07, 'src': 'embed', 'start': 726.163, 'weight': 1, 'content': [{'end': 728.445, 'text': 'You can use that picture for intuition.', 'start': 726.163, 'duration': 2.282}, {'end': 731.288, 'text': 'But we could have some incredibly complicated sample space.', 'start': 728.565, 'duration': 2.723}, {'end': 733.35, 'text': 'Random variables is.', 'start': 731.768, 'duration': 1.582}, {'end': 738.435, 'text': 'whatever the outcome of the experiment is, the random variable assigns some numerical value, right?', 'start': 733.35, 'duration': 5.085}, {'end': 743.667, 'text': 'So x less than or equal to little x is an event that either right?', 'start': 739.406, 'duration': 4.261}, {'end': 745.888, 'text': "Before you do the experiment, you don't know what x is.", 'start': 743.887, 'duration': 2.001}, {'end': 750.569, 'text': 'After you do the experiment, maybe you observe x happened to equal 7.', 'start': 746.028, 'duration': 4.541}, {'end': 757.07, 'text': "And then if this little x happened to equal 9, then we'd say, okay, this event occurred because 7 is less than 9.", 'start': 750.569, 'duration': 6.501}], 'summary': 'Random variables assign numerical values to the outcomes of an experiment, with x less than or equal to little x being an event, as illustrated through the example of x equalling 7 and little x equalling 9.', 'duration': 30.907, 'max_score': 726.163, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA726163.jpg'}, {'end': 812.602, 'src': 'embed', 'start': 768.564, 'weight': 2, 'content': [{'end': 774.226, 'text': "One way to describe the distribution, okay? It's not the only way, there's other ways to describe a distribution.", 'start': 768.564, 'duration': 5.662}, {'end': 780.228, 'text': 'But this is one way that in principle determines all possible probabilities about X.', 'start': 774.506, 'duration': 5.722}, {'end': 789.631, 'text': "So if later we wanted to know what's the probability that X is between 1 and 3 or between 5 and 9, we'll do some examples like that next time.", 'start': 780.228, 'duration': 9.403}, {'end': 795.114, 'text': 'But the idea is, as long as we know this function capital F, we could answer questions like that.', 'start': 790.331, 'duration': 4.783}, {'end': 802.017, 'text': "What's the probability that X does this? What's the probability that X does that? All of those questions could be answered in terms of this.", 'start': 795.454, 'duration': 6.563}, {'end': 810.261, 'text': "So CDF is a way to describe the distribution, cuz it's telling us the probabilities of different possible values for X.", 'start': 802.137, 'duration': 8.124}, {'end': 812.602, 'text': "And let's talk more about the PMF.", 'start': 810.261, 'duration': 2.341}], 'summary': 'The cumulative distribution function (cdf) determines all possible probabilities about x, allowing us to answer questions about the probability of x falling within specific ranges.', 'duration': 44.038, 'max_score': 768.564, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA768564.jpg'}, {'end': 881, 'src': 'embed', 'start': 849.404, 'weight': 0, 'content': [{'end': 858.789, 'text': "But in general, it means that the possible values, they don't actually have to be integers, but it has to be something you could list.", 'start': 849.404, 'duration': 9.385}, {'end': 861.37, 'text': 'Maybe a finite list, maybe an infinite list.', 'start': 858.929, 'duration': 2.441}, {'end': 868.697, 'text': 'So our a1, a2, a3, etc., that you could list out.', 'start': 862.895, 'duration': 5.802}, {'end': 873.518, 'text': 'This list might end with an or it could go on forever.', 'start': 869.617, 'duration': 3.901}, {'end': 881, 'text': "I'll list both of those cases, an or it can go on forever, a1, a2, etc.", 'start': 875.759, 'duration': 5.241}], 'summary': 'Possible values can be finite or infinite lists, such as a1, a2, a3, etc.', 'duration': 31.596, 'max_score': 849.404, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA849404.jpg'}], 'start': 309.641, 'title': 'Iid bernoulli distribution and pmf/cdf', 'summary': 'Explains the concept of iid bernoulli distribution, emphasizing independence and identical distribution of random variables x1 through xn following the bernoulli p distribution. it also covers pmf and cdf, providing a way to determine all possible probabilities about x.', 'chapters': [{'end': 432.269, 'start': 309.641, 'title': 'Understanding iid bernoulli distribution', 'summary': 'Explains the concept of iid bernoulli distribution, emphasizing the independence and identical distribution of random variables x1 through xn, each following the bernoulli p distribution, and highlights the distinction between random variables and distributions.', 'duration': 122.628, 'highlights': ['The concept of IID Bernoulli distribution is defined as independent and identically distributed random variables X1 through Xn, all following the Bernoulli p distribution.', 'The distinction between random variables and distributions is emphasized, with the explanation that multiple random variables can have the same distribution, but they are not the same random variable.']}, {'end': 927.371, 'start': 433.109, 'title': 'Pmf and cdf in probability', 'summary': 'Explains the concepts of pmf and cdf, where pmf is the probability mass function for discrete random variables and cdf is the cumulative distribution function for all random variables, providing a way to determine all possible probabilities about x.', 'duration': 494.262, 'highlights': ['PMF for binomial distribution is n choose k, p to the k, q to the n minus k The PMF for the binomial distribution is given by the formula n choose k, p to the k, q to the n minus k, where n represents the number of trials, k represents the number of successes, p is the probability of success, and q equals 1 minus p.', 'Definition of a random variable and its function A random variable is a function that assigns a numerical value to each outcome in the sample space, providing a way to represent events and probabilities in a more abstract manner.', 'Explanation of CDF as a way to describe the distribution of random variables The Cumulative Distribution Function (CDF) is a way to describe the distribution of random variables, providing probabilities for different possible values of X, allowing for answering various probability questions based on the CDF function.']}], 'duration': 617.73, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA309641.jpg', 'highlights': ['The PMF for the binomial distribution is given by the formula n choose k, p to the k, q to the n minus k, where n represents the number of trials, k represents the number of successes, p is the probability of success, and q equals 1 minus p.', 'The concept of IID Bernoulli distribution is defined as independent and identically distributed random variables X1 through Xn, all following the Bernoulli p distribution.', 'Explanation of CDF as a way to describe the distribution of random variables, providing probabilities for different possible values of X, allowing for answering various probability questions based on the CDF function.', 'A random variable is a function that assigns a numerical value to each outcome in the sample space, providing a way to represent events and probabilities in a more abstract manner.', 'The distinction between random variables and distributions is emphasized, with the explanation that multiple random variables can have the same distribution, but they are not the same random variable.']}, {'end': 1280.642, 'segs': [{'end': 1093.813, 'src': 'embed', 'start': 1065.217, 'weight': 0, 'content': [{'end': 1067.259, 'text': 'This only helps us in the discrete case.', 'start': 1065.217, 'duration': 2.042}, {'end': 1071.164, 'text': "Right now we're focusing on the discrete case, so we can mostly be doing PMFs.", 'start': 1067.82, 'duration': 3.344}, {'end': 1083.15, 'text': "So if you had a problem where I said find the distribution, what that means is either give the CDF or if it's discrete, give the PMF either way.", 'start': 1071.926, 'duration': 11.224}, {'end': 1088.811, 'text': 'Usually the PMF is gonna be easier, but those are equally valid ways to describe the distribution.', 'start': 1083.37, 'duration': 5.441}, {'end': 1093.813, 'text': 'Okay, so coming back to the binomial.', 'start': 1090.292, 'duration': 3.521}], 'summary': 'Focusing on discrete case, mainly using pmfs for finding distribution.', 'duration': 28.596, 'max_score': 1065.217, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1065217.jpg'}, {'end': 1259.71, 'src': 'embed', 'start': 1181.543, 'weight': 1, 'content': [{'end': 1188.847, 'text': "If somehow this sum is not equal to 1, then there's something seriously messed up, right? Because it has to equal something.", 'start': 1181.543, 'duration': 7.304}, {'end': 1193.97, 'text': 'So the PMF has to add up to 1, so that would mean that this equation was wrong.', 'start': 1189.287, 'duration': 4.683}, {'end': 1201.594, 'text': "But this helped to check, okay, that's comforting, it adds up to 1, that makes sense, okay? So that's the binomial.", 'start': 1194.51, 'duration': 7.084}, {'end': 1205.756, 'text': 'And now let me come back to the thing that I did at the end last time.', 'start': 1202.535, 'duration': 3.221}, {'end': 1218.998, 'text': "with the sum of two binomials, okay? And let's actually see why that's true from all three of these perspectives.", 'start': 1210.132, 'duration': 8.866}, {'end': 1222.84, 'text': "First one I already did last time, but I'll remind you because it's very quick.", 'start': 1219.798, 'duration': 3.042}, {'end': 1231.806, 'text': "X is binomial NP, Y is binomial MP, and they're independent.", 'start': 1223.721, 'duration': 8.085}, {'end': 1237.61, 'text': 'And we wanna show that the sum is binomial N plus MP.', 'start': 1232.987, 'duration': 4.623}, {'end': 1245.039, 'text': "Okay, so that's what we did at the very end last time, but I wanna show you different ways of seeing this.", 'start': 1239.875, 'duration': 5.164}, {'end': 1248.241, 'text': "That's what we're trying to show.", 'start': 1247.3, 'duration': 0.941}, {'end': 1255.507, 'text': 'And before we can actually say more about it, I should make sure everyone is clear on what does x plus y actually mean.', 'start': 1248.702, 'duration': 6.805}, {'end': 1259.71, 'text': "Mathematically speaking, we're adding two functions.", 'start': 1257.748, 'duration': 1.962}], 'summary': 'Pmf must add up to 1, binomial sum is n+mp', 'duration': 78.167, 'max_score': 1181.543, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1181543.jpg'}], 'start': 928.973, 'title': 'Pmf for discrete random variables', 'summary': 'Explains probability mass function (pmf) for discrete random variables, emphasizing the need for probabilities to be greater than or equal to 0 and sum to 1. it also validates the binomial pmf by checking its properties and exploring its connection to the binomial theorem.', 'chapters': [{'end': 1088.811, 'start': 928.973, 'title': 'Understanding pmf for discrete random variables', 'summary': 'Explains the probability mass function (pmf) for discrete random variables, defining its conditions and highlighting its importance for describing the randomness of x, with emphasis on the need for probabilities to be greater than or equal to 0 and sum to 1, making it easier to use than the cumulative distribution function (cdf).', 'duration': 159.838, 'highlights': ['Probability Mass Function (PMF) describes the randomness of x and must satisfy two conditions: probabilities (pj) must be greater than or equal to 0, and the sum of all probabilities must equal 1. The chapter emphasizes the conditions for a valid PMF, requiring probabilities to be greater than or equal to 0 and sum to 1, to effectively describe the randomness of x.', 'PMF is more commonly used for discrete random variables than the Cumulative Distribution Function (CDF) due to its simplicity and is often the preferred method for describing the distribution of discrete random variables. The chapter highlights the preference for using PMFs over CDFs for discrete random variables, as it is generally easier and equally valid for describing the distribution.']}, {'end': 1280.642, 'start': 1090.292, 'title': 'Validating the binomial pmf', 'summary': 'Validates the binomial pmf by checking its properties, including non-negativity and summing up to 1, and explores the connection to the binomial theorem. it also discusses the sum of two binomials and how to add two functions in the context of the domain.', 'duration': 190.35, 'highlights': ['The sum of the binomial PMF is validated to equal 1, as it is connected to the binomial theorem, ensuring the properties of the distribution. This provides a fundamental understanding of the binomial distribution. (e.g. sum k equals 0 to n, n, choose k, p to the k, q to the n, minus k equates to 1 by the binomial theorem)', 'Exploration of the sum of two binomials and the process of adding two functions with the same domain, providing insights into the mathematical operations involved in combining binomial distributions. (e.g. showing the sum of two binomials is binomial N plus MP)', "Clarification on the mathematical process of adding two functions with the same domain, emphasizing the computation of both functions' values to obtain the new function. This aids in understanding the practical application of adding binomial functions. (e.g. explaining the process of adding two functions in the context of the domain)"]}], 'duration': 351.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA928973.jpg', 'highlights': ['The chapter emphasizes the conditions for a valid PMF, requiring probabilities to be greater than or equal to 0 and sum to 1, to effectively describe the randomness of x.', 'The chapter highlights the preference for using PMFs over CDFs for discrete random variables, as it is generally easier and equally valid for describing the distribution.', 'The sum of the binomial PMF is validated to equal 1, as it is connected to the binomial theorem, ensuring the properties of the distribution.', 'Exploration of the sum of two binomials and the process of adding two functions with the same domain, providing insights into the mathematical operations involved in combining binomial distributions.', "Clarification on the mathematical process of adding two functions with the same domain, emphasizing the computation of both functions' values to obtain the new function. This aids in understanding the practical application of adding binomial functions."]}, {'end': 1919.786, 'segs': [{'end': 1367.157, 'src': 'embed', 'start': 1335.525, 'weight': 2, 'content': [{'end': 1338.827, 'text': "And these are separate sets of trials cuz I said they're independent.", 'start': 1335.525, 'duration': 3.302}, {'end': 1344.25, 'text': 'So this would be like flip the coin n times and then flip the coin m additional times.', 'start': 1339.207, 'duration': 5.043}, {'end': 1348.033, 'text': 'So we have a total of n plus m coin flips or trials.', 'start': 1344.791, 'duration': 3.242}, {'end': 1351.046, 'text': "Notice it's the same p for both.", 'start': 1349.645, 'duration': 1.401}, {'end': 1358.691, 'text': "This will not work if this one is like 1 half and this one's 1 third, okay? But I assume p is the same for both, so we have n plus m trials.", 'start': 1351.126, 'duration': 7.565}, {'end': 1361.153, 'text': 'Each trial has probably success p.', 'start': 1358.872, 'duration': 2.281}, {'end': 1367.157, 'text': "And so what's the number of successes? Well, I'll just add up this number of successes plus this number of successes.", 'start': 1361.153, 'duration': 6.004}], 'summary': 'Independent trials with n and m coin flips, each with probably success p.', 'duration': 31.632, 'max_score': 1335.525, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1335525.jpg'}, {'end': 1410.73, 'src': 'embed', 'start': 1381.705, 'weight': 4, 'content': [{'end': 1387.649, 'text': "But I think it's helpful to also see how would this work from this perspective and from this perspective, which we didn't do last time.", 'start': 1381.705, 'duration': 5.944}, {'end': 1396.075, 'text': "So from the second point of view, let's say we wrote X as X1 plus blah, blah, blah, plus Xn.", 'start': 1388.43, 'duration': 7.645}, {'end': 1410.73, 'text': "And if we write y equals y1 plus blah blah blah plus ym, where all of these xj's and yj's are all independent Bernoulli p random variables.", 'start': 1397.102, 'duration': 13.628}], 'summary': 'Discussing the approach with independent bernoulli p random variables.', 'duration': 29.025, 'max_score': 1381.705, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1381705.jpg'}, {'end': 1663.455, 'src': 'heatmap', 'start': 1598.369, 'weight': 0.784, 'content': [{'end': 1605.192, 'text': 'So that suggests you use the law of total probability where we condition on x.', 'start': 1598.369, 'duration': 6.823}, {'end': 1615.095, 'text': 'So we know that this is the probability that x plus y equals k given x equals j times the probability that x equals j.', 'start': 1605.192, 'duration': 9.903}, {'end': 1623.063, 'text': 'summed over j, j goes from 0 to n.', 'start': 1617.538, 'duration': 5.525}, {'end': 1635.475, 'text': "But actually, let's just sum up to k, because if the sum is equal to k, there's no way that one of them, x on its own, could not be greater than k.", 'start': 1623.063, 'duration': 12.412}, {'end': 1637.617, 'text': "cuz you're adding up two non-negative things.", 'start': 1635.475, 'duration': 2.142}, {'end': 1640.579, 'text': 'So we could sum up to k.', 'start': 1637.677, 'duration': 2.902}, {'end': 1642.922, 'text': "Now, let's just compute this.", 'start': 1640.579, 'duration': 2.343}, {'end': 1652.581, 'text': 'This is the sum j equals 0 to k of the probability.', 'start': 1644.298, 'duration': 8.283}, {'end': 1661.145, 'text': "Okay, so x plus y equals k, given x equals j, that's useful information, right? We can plug in x equals j.", 'start': 1652.601, 'duration': 8.544}, {'end': 1663.455, 'text': 'And rewrite that for y.', 'start': 1662.415, 'duration': 1.04}], 'summary': 'Using law of total probability to compute sum of probabilities, x+y=k given x=j.', 'duration': 65.086, 'max_score': 1598.369, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1598369.jpg'}, {'end': 1721.418, 'src': 'embed', 'start': 1698.851, 'weight': 1, 'content': [{'end': 1706.833, 'text': 'But if you understand independence for events, then you understand it for random variables, which is just that knowing that this event occurred,', 'start': 1698.851, 'duration': 7.982}, {'end': 1707.633, 'text': 'x equals something.', 'start': 1706.833, 'duration': 0.8}, {'end': 1712.475, 'text': 'Independent means that if we know x, it gives us no information whatsoever about y.', 'start': 1707.734, 'duration': 4.741}, {'end': 1717.017, 'text': 'So if we know that x equals j, that tells us nothing about y.', 'start': 1712.475, 'duration': 4.542}, {'end': 1719.597, 'text': 'So independence means we can just cross this out.', 'start': 1717.017, 'duration': 2.58}, {'end': 1721.418, 'text': "That's by independence.", 'start': 1720.398, 'duration': 1.02}], 'summary': 'Understanding independence: knowing x gives no info about y.', 'duration': 22.567, 'max_score': 1698.851, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1698851.jpg'}, {'end': 1919.786, 'src': 'embed', 'start': 1845.221, 'weight': 0, 'content': [{'end': 1851.785, 'text': 'And the Vandermonde identity says that this will just equal m plus n choose k.', 'start': 1845.221, 'duration': 6.564}, {'end': 1856.208, 'text': "So this equals, sorry, I'm going right to left here.", 'start': 1851.785, 'duration': 4.423}, {'end': 1863.353, 'text': 'Vandermonde says that this equals m plus n choose k.', 'start': 1856.809, 'duration': 6.544}, {'end': 1865.655, 'text': 'So that was the Vandermonde identity we did last time.', 'start': 1863.353, 'duration': 2.302}, {'end': 1875.13, 'text': "And not last time, but we did it a while ago using a story proof, okay? So that's M plus N choose K according to the Vandermonde thing we did.", 'start': 1865.915, 'duration': 9.215}, {'end': 1883.352, 'text': "And that means, well, now that looks exactly like the binomial N plus MP PMF, okay? So that's true.", 'start': 1875.21, 'duration': 8.142}, {'end': 1893.434, 'text': "So obviously this was a much more complicated and difficult way to do it, especially if we didn't know, or didn't remember how to do this sum.", 'start': 1884.792, 'duration': 8.642}, {'end': 1894.854, 'text': "then we'd be stuck at this point.", 'start': 1893.434, 'duration': 1.42}, {'end': 1900.736, 'text': 'Luckily, we already did the Vandermonde earlier, so I can just quote that result.', 'start': 1895.575, 'duration': 5.161}, {'end': 1904.197, 'text': 'Even with this, it was still a lot more work.', 'start': 1902.176, 'duration': 2.021}, {'end': 1911.422, 'text': "And without this, then you'd just be left with this hideous sum, okay? But it still worked, so we would have a contradiction.", 'start': 1904.377, 'duration': 7.045}, {'end': 1917.345, 'text': 'So another point of view of what we just did is that we actually just proved Vandermonde again right?', 'start': 1911.862, 'duration': 5.483}, {'end': 1919.786, 'text': "Because if this were not equal, we'd have a contradiction.", 'start': 1917.605, 'duration': 2.181}], 'summary': 'Vandermonde identity relates to m+n choose k and resembles binomial n+mp pmf.', 'duration': 74.565, 'max_score': 1845.221, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1845221.jpg'}], 'start': 1280.982, 'title': 'Random variable addition', 'summary': 'Explains adding random variables, demonstrating that combining two random variables from the same sample space yields a new random variable. it presents three perspectives on the concept, including computational methods and probability mass function calculations.', 'chapters': [{'end': 1584.693, 'start': 1280.982, 'title': 'Addition of random variables', 'summary': 'Explains the addition of random variables, showing that the sum of two random variables from the same sample space is a new random variable, and provides three different perspectives to understand the concept, including the computational method and probability mass function calculation.', 'duration': 303.711, 'highlights': ['The sum of two random variables from the same sample space forms a new random variable, which can be computed after the experiment based on the values of the original variables.', "The addition of independent Bernoulli random variables follows the binomial distribution, with the sum of n+m iid Bernoulli p's resulting in a binomial n+m p.", 'Understanding the addition of random variables can be approached through the computation of probability mass function (PMF) to show that the sum of two random variables is of binomial form.']}, {'end': 1919.786, 'start': 1584.693, 'title': 'Conditioning on x and vandermonde identity', 'summary': 'Discusses the use of the law of total probability to condition on x, leading to the derivation of vandermonde identity, which simplifies a complicated sum into m plus n choose k.', 'duration': 335.093, 'highlights': ['The chapter discusses the use of the law of total probability to condition on x, leading to the derivation of Vandermonde identity, which simplifies a complicated sum into m plus n choose k. The discussion emphasizes the use of the law of total probability to condition on x, leading to the derivation of Vandermonde identity, which simplifies a complicated sum into m plus n choose k.', 'The probability that x plus y equals k given x equals j times the probability that x equals j is computed as the sum j equals 0 to k of the probability. The probability that x plus y equals k given x equals j times the probability that x equals j is computed as the sum j equals 0 to k of the probability.', 'The chapter explains that x and y are independent, which allows simplification of the binomial PMF and the cancellation of terms due to independence. The explanation emphasizes that x and y are independent, allowing simplification of the binomial PMF and the cancellation of terms due to independence.']}], 'duration': 638.804, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1280982.jpg', 'highlights': ["The addition of independent Bernoulli random variables follows the binomial distribution, with the sum of n+m iid Bernoulli p's resulting in a binomial n+m p.", 'The sum of two random variables from the same sample space forms a new random variable, which can be computed after the experiment based on the values of the original variables.', 'Understanding the addition of random variables can be approached through the computation of probability mass function (PMF) to show that the sum of two random variables is of binomial form.', 'The chapter discusses the use of the law of total probability to condition on x, leading to the derivation of Vandermonde identity, which simplifies a complicated sum into m plus n choose k.', 'The probability that x plus y equals k given x equals j times the probability that x equals j is computed as the sum j equals 0 to k of the probability.', 'The chapter explains that x and y are independent, which allows simplification of the binomial PMF and the cancellation of terms due to independence.']}, {'end': 2427.588, 'segs': [{'end': 2026.381, 'src': 'heatmap', 'start': 1958.405, 'weight': 0.943, 'content': [{'end': 1965.957, 'text': 'The key assumption is that The trials are independent and they all have the same probability of success.', 'start': 1958.405, 'duration': 7.552}, {'end': 1969.498, 'text': "okay?. So if the probabilities of success are different, we can't say it's binomial.", 'start': 1965.957, 'duration': 3.541}, {'end': 1972.219, 'text': "And if they're not independent, we can't say it's binomial.", 'start': 1969.538, 'duration': 2.681}, {'end': 1978.901, 'text': "So let's do an example that's not a binomial, yet a common mistake would be to somehow think that this is binomial.", 'start': 1972.559, 'duration': 6.342}, {'end': 1983.542, 'text': "So here's just a simple example to think about with cards.", 'start': 1979.181, 'duration': 4.361}, {'end': 1997.659, 'text': 'And suppose we have a random five card hand from a standard 52 card deck, okay?', 'start': 1984.402, 'duration': 13.257}, {'end': 2002.882, 'text': "And we wanna know what's the distribution.", 'start': 1997.679, 'duration': 5.203}, {'end': 2009.126, 'text': 'Find the distribution of the number of aces in the hand.', 'start': 2002.882, 'duration': 6.244}, {'end': 2017.61, 'text': 'So we pick a random subset, 5 cards out of 52, all subsets of size 5 equally likely.', 'start': 2011.022, 'duration': 6.588}, {'end': 2024.679, 'text': "And the number of aces, there's some number, possibly 0 of aces in that hand.", 'start': 2017.63, 'duration': 7.049}, {'end': 2026.381, 'text': "We want to know what's its distribution.", 'start': 2024.699, 'duration': 1.682}], 'summary': 'Trials must be independent and have same success probability for binomial distribution. example with 5 card hand from 52 card deck.', 'duration': 67.976, 'max_score': 1958.405, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1958405.jpg'}, {'end': 2170.98, 'src': 'heatmap', 'start': 2111.453, 'weight': 3, 'content': [{'end': 2117.255, 'text': "First of all, So that's what we need to find.", 'start': 2111.453, 'duration': 5.802}, {'end': 2132.25, 'text': 'Well, first of all, this is obviously 0, except if k is 0, 1, 2, 3, or 4.', 'start': 2118.558, 'duration': 13.692}, {'end': 2138.355, 'text': "You're not gonna observe two and a half aces or five aces if it's a standard deck, right? It's not possible.", 'start': 2132.25, 'duration': 6.105}, {'end': 2140.076, 'text': 'So those are the only possibilities.', 'start': 2138.855, 'duration': 1.221}, {'end': 2146.08, 'text': "And for a lot of these problems it's helpful just starting by listing out or describing what are the possible values.", 'start': 2140.556, 'duration': 5.524}, {'end': 2158.469, 'text': "okay? Because a common mistake with probability is to list some PMF where either it doesn't sum to 1 or it involves impossible values or things like that.", 'start': 2146.08, 'duration': 12.389}, {'end': 2161.541, 'text': 'Okay, so those are the possible values.', 'start': 2159.898, 'duration': 1.643}, {'end': 2170.98, 'text': "That's just obvious, there's four aces in the deck, right? Okay, so we can actually immediately conclude that the distribution is not binomial.", 'start': 2161.662, 'duration': 9.318}], 'summary': 'The distribution is not binomial; possible values are 0, 1, 2, 3, or 4.', 'duration': 24.9, 'max_score': 2111.453, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2111453.jpg'}, {'end': 2244.814, 'src': 'embed', 'start': 2221.401, 'weight': 4, 'content': [{'end': 2228.146, 'text': 'So we can go back to the naive definition of probability because I said all five card hands are equally likely.', 'start': 2221.401, 'duration': 6.745}, {'end': 2230.967, 'text': "So there's 52 choose 5.", 'start': 2228.626, 'duration': 2.341}, {'end': 2234.169, 'text': "Possible hands, equally likely, so we're using naive definition.", 'start': 2230.967, 'duration': 3.202}, {'end': 2239.751, 'text': "Now we wanna know what's the probability that the number of aces is equal to k?", 'start': 2234.769, 'duration': 4.982}, {'end': 2244.814, 'text': "Well, there's four aces in the deck and we need to choose k of those aces.", 'start': 2240.212, 'duration': 4.602}], 'summary': '52 choose 5 possible hands, using naive definition of probability.', 'duration': 23.413, 'max_score': 2221.401, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2221401.jpg'}, {'end': 2427.588, 'src': 'embed', 'start': 2380.697, 'weight': 0, 'content': [{'end': 2383.32, 'text': 'some of them are tagged, some of them are untagged.', 'start': 2380.697, 'duration': 2.623}, {'end': 2391.248, 'text': "You collect a sample and you want to know what's the probability that that sample has exactly k tagged elk, that was the problem.", 'start': 2383.74, 'duration': 7.508}, {'end': 2396.716, 'text': "Okay, so you don't have to memorize the answer to that, but the problem is a useful one to think about.", 'start': 2392.15, 'duration': 4.566}, {'end': 2398.799, 'text': 'This is exactly the same.', 'start': 2396.736, 'duration': 2.063}, {'end': 2401.362, 'text': 'Instead of elk, we have cards.', 'start': 2399.6, 'duration': 1.762}, {'end': 2407.369, 'text': "So instead of tagging elk, we're tagging cards as aces.", 'start': 2403.685, 'duration': 3.684}, {'end': 2413.556, 'text': 'Now what does it mean to tag a card as an ace? Well, it means it has an ace written on it.', 'start': 2407.69, 'duration': 5.866}, {'end': 2420.162, 'text': 'The cards are tagged already, right? Four of the cards are tagged as aces, the other 48 are not tagged as aces.', 'start': 2413.616, 'duration': 6.546}, {'end': 2427.588, 'text': "So we have four tagged cards, we have 48 untagged cards, where tagged means ace, okay? So it's the exact same thing.", 'start': 2420.622, 'duration': 6.966}], 'summary': 'Probability problem: 4 tagged cards out of 52. useful to think about.', 'duration': 46.891, 'max_score': 2380.697, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2380697.jpg'}], 'start': 1920.087, 'title': 'Binomial distribution & probability', 'summary': 'Highlights the key assumptions for binomial distribution, provides an example of a non-binomial scenario with the distribution of aces in a 5-card hand, and discusses the probability mass function (pmf) for the number of aces, drawing connections to related problems.', 'chapters': [{'end': 2082.94, 'start': 1920.087, 'title': 'Binomial distribution & common mistakes', 'summary': 'Discusses the binomial distribution, highlighting the key assumptions for its application and providing a simple example of a non-binomial scenario, illustrated with the distribution of the number of aces in a random five card hand from a standard 52 card deck.', 'duration': 162.853, 'highlights': ["The chapter discusses the binomial distribution and its applications. It explains Vandermonde's identity and the key assumptions necessary for considering a scenario as binomial.", 'Emphasizes the importance of independent trials and equal probability of success for a scenario to be considered binomial. It points out that if the probabilities of success are different or if the trials are not independent, the scenario cannot be deemed as binomial.', 'Provides a simple example of a non-binomial scenario related to the distribution of aces in a random five card hand from a standard 52 card deck. It illustrates that the distribution of the number of aces in the hand is a discrete problem and demonstrates the application of probability mass function (PMF) to find the distribution.']}, {'end': 2427.588, 'start': 2083.42, 'title': 'Probability of aces in a deck', 'summary': 'Discusses finding the probability mass function (pmf) for the number of aces in a 5-card hand from a standard deck, determining that the distribution is not binomial, and drawing connections to the vandermonde and elk problems.', 'duration': 344.168, 'highlights': ["The distribution is not binomial due to the lack of independence between trials, as the likelihood of drawing aces is affected by the previous cards drawn. Trials are not independent, impacting the binomial distribution's applicability.", 'The PMF for the number of aces is derived using the naive definition of probability and the multiplication rule, resulting in a neat pattern of 4 plus 48 and k plus 5 minus k. PMF calculation based on the naive definition and the multiplication rule yields a distinct pattern.', 'The connection between the problem and the Vandermonde and Elk problems is drawn, highlighting similarities in the tagging of cards as aces and the tagging of elk in the respective problems. Draws parallels between the problem and the Vandermonde and Elk problems, emphasizing the tagging similarities.']}], 'duration': 507.501, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA1920087.jpg', 'highlights': ['The chapter discusses the binomial distribution and its applications, emphasizing the importance of independent trials and equal probability of success for a scenario to be considered binomial.', 'Provides a simple example of a non-binomial scenario related to the distribution of aces in a random five card hand from a standard 52 card deck, demonstrating the application of probability mass function (PMF) to find the distribution.', "The distribution is not binomial due to the lack of independence between trials, impacting the binomial distribution's applicability.", 'The PMF for the number of aces is derived using the naive definition of probability and the multiplication rule, resulting in a neat pattern of 4 plus 48 and k plus 5 minus k.', 'The connection between the problem and the Vandermonde and Elk problems is drawn, highlighting similarities in the tagging of cards as aces and the tagging of elk in the respective problems.']}, {'end': 3023.453, 'segs': [{'end': 2520.625, 'src': 'embed', 'start': 2462.331, 'weight': 4, 'content': [{'end': 2485.704, 'text': "Suppose that we have, Let's say we have a jar full of marbles and let's say B of them are black and W of them are white marbles.", 'start': 2462.331, 'duration': 23.373}, {'end': 2492.766, 'text': "Okay, you pick, let's say, n of them.", 'start': 2488.544, 'duration': 4.222}, {'end': 2494.647, 'text': 'Pick random sample.', 'start': 2493.507, 'duration': 1.14}, {'end': 2498.949, 'text': 'Simple random sample means that all subsets of that size are equally likely.', 'start': 2494.827, 'duration': 4.122}, {'end': 2504.912, 'text': "Of size, let's say, n.", 'start': 2503.392, 'duration': 1.52}, {'end': 2520.625, 'text': "Okay, so then the question is, what's the distribution? of the number of white marbles in the sample.", 'start': 2504.912, 'duration': 15.713}], 'summary': 'Jar with b black and w white marbles, picking n marbles, find distribution of white marbles.', 'duration': 58.294, 'max_score': 2462.331, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2462331.jpg'}, {'end': 2540.817, 'src': 'heatmap', 'start': 2462.331, 'weight': 0.962, 'content': [{'end': 2485.704, 'text': "Suppose that we have, Let's say we have a jar full of marbles and let's say B of them are black and W of them are white marbles.", 'start': 2462.331, 'duration': 23.373}, {'end': 2492.766, 'text': "Okay, you pick, let's say, n of them.", 'start': 2488.544, 'duration': 4.222}, {'end': 2494.647, 'text': 'Pick random sample.', 'start': 2493.507, 'duration': 1.14}, {'end': 2498.949, 'text': 'Simple random sample means that all subsets of that size are equally likely.', 'start': 2494.827, 'duration': 4.122}, {'end': 2504.912, 'text': "Of size, let's say, n.", 'start': 2503.392, 'duration': 1.52}, {'end': 2520.625, 'text': "Okay, so then the question is, what's the distribution? of the number of white marbles in the sample.", 'start': 2504.912, 'duration': 15.713}, {'end': 2529.41, 'text': 'Number of white marbles in the sample.', 'start': 2527.329, 'duration': 2.081}, {'end': 2537.915, 'text': "Notice again, that's exactly the same as the ELK problem and the ACE problem, where, instead of thinking of tagged and untagged,", 'start': 2531.191, 'duration': 6.724}, {'end': 2540.817, 'text': "we're thinking of white and black, but it's the same problem.", 'start': 2537.915, 'duration': 2.902}], 'summary': 'Analyzing distribution of white marbles in random sample', 'duration': 78.486, 'max_score': 2462.331, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2462331.jpg'}, {'end': 2923.592, 'src': 'embed', 'start': 2896.648, 'weight': 1, 'content': [{'end': 2903.854, 'text': "I'm just gonna draw a continuous one and a discrete one, and we'll talk more about these next time, but just to have a picture in mind.", 'start': 2896.648, 'duration': 7.206}, {'end': 2915.669, 'text': 'Continuous. one might look like this Notice, if X is negative, a billion.', 'start': 2905.615, 'duration': 10.054}, {'end': 2920.971, 'text': 'if you let X be more and more negative, it gets less and less likely that X is less than or equal, right?', 'start': 2915.669, 'duration': 5.302}, {'end': 2923.592, 'text': "So it's gonna approach 0 this way.", 'start': 2921.672, 'duration': 1.92}], 'summary': 'Illustrating continuous and discrete functions, showing probability approaching 0 for negative x values.', 'duration': 26.944, 'max_score': 2896.648, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2896648.jpg'}, {'end': 3023.453, 'src': 'embed', 'start': 3019.089, 'weight': 0, 'content': [{'end': 3022.132, 'text': "In the continuous case, it's often useful to use the CDF.", 'start': 3019.089, 'duration': 3.043}, {'end': 3023.453, 'text': "All right, so that's all for today.", 'start': 3022.392, 'duration': 1.061}], 'summary': 'Using cdf in continuous case. end of session.', 'duration': 4.364, 'max_score': 3019.089, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA3019089.jpg'}], 'start': 2429.81, 'title': 'Hypergeometric distribution', 'summary': 'Delves into hypergeometric distribution, its relation to other problems, and its distinction from the binomial distribution, emphasizing the necessity of understanding its story and its appropriate application. it also explores the validation of hypergeometric pmf, illustrating its non-negativity and summation to 1, and elucidating the features of continuous and discrete cumulative distribution functions (cdfs).', 'chapters': [{'end': 2780.6, 'start': 2429.81, 'title': 'Understanding hypergeometric distribution', 'summary': 'Discusses the hypergeometric distribution, its relationship with other problems, and its key distinction from the binomial distribution, emphasizing the importance of understanding its story and recognizing when to apply it.', 'duration': 350.79, 'highlights': ['The hypergeometric distribution is defined by a story, and its PMF is related to the elk problem and the ACE problem, emphasizing the importance of understanding its story and recognizing when to apply it.', 'The distribution of the number of white marbles in a sample is explained in relation to the number of black and white marbles in a jar, highlighting the key constraints and the probability of selecting a specific number of white marbles.', 'The key distinction between hypergeometric and binomial distributions lies in the sampling method, with hypergeometric being without replacement and binomial being with replacement, leading to an intuitive connection between the two distributions and an approximation under certain conditions.']}, {'end': 3023.453, 'start': 2782.136, 'title': 'Validating hypergeometric pmf', 'summary': 'Discusses the validation of the hypergeometric pmf, demonstrating its non-negativity and summation to 1, while also explaining the characteristics of continuous and discrete cumulative distribution functions (cdfs).', 'duration': 241.317, 'highlights': ['The hypergeometric PMF is validated by demonstrating its non-negativity and summation to 1, with the Vandermonde formula playing a crucial role in the proof.', 'The characteristics of continuous and discrete CDFs are explained, illustrating how continuous CDFs approach 1 as x increases and 0 as x decreases, while discrete CDFs exhibit jumps at specific values of x.', "In the discrete case, CDFs have jumpy functions and it's easier to use the PMF, while in the continuous case, it's often useful to use the CDF."]}], 'duration': 593.643, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/k2BB0p8byGA/pics/k2BB0p8byGA2429810.jpg', 'highlights': ['The hypergeometric distribution is defined by a story, emphasizing the importance of understanding its story and recognizing when to apply it.', 'The distribution of the number of white marbles in a sample is explained in relation to the number of black and white marbles in a jar, highlighting the key constraints and the probability of selecting a specific number of white marbles.', 'The key distinction between hypergeometric and binomial distributions lies in the sampling method, leading to an intuitive connection between the two distributions and an approximation under certain conditions.', 'The hypergeometric PMF is validated by demonstrating its non-negativity and summation to 1, with the Vandermonde formula playing a crucial role in the proof.', 'The characteristics of continuous and discrete CDFs are explained, illustrating how continuous CDFs approach 1 as x increases and 0 as x decreases, while discrete CDFs exhibit jumps at specific values of x.', "In the discrete case, CDFs have jumpy functions and it's easier to use the PMF, while in the continuous case, it's often useful to use the CDF."]}], 'highlights': ['The binomial distribution involves n independent trials with each resulting in success or failure, and the probability of success denoted as P, making it a very general and useful distribution.', 'The PMF for the binomial distribution is given by the formula n choose k, p to the k, q to the n minus k, where n represents the number of trials, k represents the number of successes, p is the probability of success, and q equals 1 minus p.', 'The chapter emphasizes the conditions for a valid PMF, requiring probabilities to be greater than or equal to 0 and sum to 1, to effectively describe the randomness of x.', "The addition of independent Bernoulli random variables follows the binomial distribution, with the sum of n+m iid Bernoulli p's resulting in a binomial n+m p.", 'The chapter discusses the binomial distribution and its applications, emphasizing the importance of independent trials and equal probability of success for a scenario to be considered binomial.', 'The hypergeometric distribution is defined by a story, emphasizing the importance of understanding its story and recognizing when to apply it.']}