title

Lecture 11: The Poisson distribution | Statistics 110

description

We introduce the Poisson distribution, which is arguably the most important discrete distribution in all of statistics. We explore its uses as an approximate distribution and its connections with the Binomial.

detail

{'title': 'Lecture 11: The Poisson distribution | Statistics 110', 'heatmap': [{'end': 1309.437, 'start': 1254.189, 'weight': 0.807}, {'end': 2336.664, 'start': 2286.234, 'weight': 0.885}], 'summary': 'The lecture delves into understanding and transforming random variables, the significance and applications of poisson distribution, poisson approximation, convergence, pmf convergence, connection between binomial and poisson distributions, and poisson approximation for triple birthday matches, with a focus on the implications and practical applications in probability theory and statistics.', 'chapters': [{'end': 355.291, 'segs': [{'end': 85.264, 'src': 'embed', 'start': 24.917, 'weight': 0, 'content': [{'end': 29.097, 'text': "It's trying to understand the difference between a distribution and a random variable, right?", 'start': 24.917, 'duration': 4.18}, {'end': 34.438, 'text': "That's what we've been talking about, but it's subtle at first, so you have to keep practicing that.", 'start': 29.137, 'duration': 5.301}, {'end': 38.139, 'text': 'So this is what I call sympathetic magic.', 'start': 35.218, 'duration': 2.921}, {'end': 50.451, 'text': 'Sympathetic magic is what I call the mistake of confusing a random variable with its distribution.', 'start': 41.866, 'duration': 8.585}, {'end': 63.64, 'text': "So an example of that is that's come up on some of the homeworks.", 'start': 54.374, 'duration': 9.266}, {'end': 77.157, 'text': 'is you have a sum of two random variables and kind of just blindly saying the PMF of the sum is the sum of the PMFs or something like that?', 'start': 63.64, 'duration': 13.517}, {'end': 85.264, 'text': "That is, you're supposed to be adding random variables, and if you instead add PMFs, it just makes no sense at all.", 'start': 77.477, 'duration': 7.787}], 'summary': 'Understanding the difference between distribution and random variable is crucial in probability theory and statistics.', 'duration': 60.347, 'max_score': 24.917, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY24917.jpg'}, {'end': 264.74, 'src': 'embed', 'start': 232.229, 'weight': 4, 'content': [{'end': 233.61, 'text': "That's the map, that's the territory.", 'start': 232.229, 'duration': 1.381}, {'end': 235.311, 'text': "We don't make that mistake.", 'start': 234.11, 'duration': 1.201}, {'end': 244.306, 'text': "Here, though, when we're talking about random variables and distributions, it's more mathematical, it's a more abstract thing,", 'start': 237.301, 'duration': 7.005}, {'end': 246.427, 'text': 'and people do make that mistake all the time.', 'start': 244.306, 'duration': 2.121}, {'end': 252.712, 'text': "So the mistake is completely analogous, it's just in this context that somehow the human mind easily does that.", 'start': 246.447, 'duration': 6.265}, {'end': 258.476, 'text': "And in the map territory context, it's not such a big deal.", 'start': 253.272, 'duration': 5.204}, {'end': 264.74, 'text': "Here's an analogy along these lines that I like even more.", 'start': 260.156, 'duration': 4.584}], 'summary': 'Mistake in understanding random variables and distributions is common due to abstract nature.', 'duration': 32.511, 'max_score': 232.229, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY232229.jpg'}, {'end': 316.143, 'src': 'embed', 'start': 291.556, 'weight': 6, 'content': [{'end': 297.718, 'text': "It's the same thing, and I like this analogy even more, because now we can think of if you have one blueprint,", 'start': 291.556, 'duration': 6.162}, {'end': 301.539, 'text': 'you could build many houses from the same blueprint, right?', 'start': 297.718, 'duration': 3.821}, {'end': 304.06, 'text': 'Just use the blueprint and build in different locations.', 'start': 301.579, 'duration': 2.481}, {'end': 308.581, 'text': 'So you can have as many random variables as you want, all with the same distribution.', 'start': 304.46, 'duration': 4.121}, {'end': 313.923, 'text': "They could be iid, which would mean they're independent with their same distribution, or they could be dependent,", 'start': 308.621, 'duration': 5.302}, {'end': 316.143, 'text': 'but they could have all the same distribution.', 'start': 313.923, 'duration': 2.22}], 'summary': 'Using the same blueprint, you can build multiple houses in different locations, representing random variables with the same distribution.', 'duration': 24.587, 'max_score': 291.556, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY291556.jpg'}], 'start': 0.249, 'title': 'Understanding and transforming random variables', 'summary': 'Discusses the distinction between random variables and their distributions, highlighting the implications of confusing them through examples and analogies, emphasizing the importance of clarity to avoid errors in probability theory and statistics.', 'chapters': [{'end': 171.55, 'start': 0.249, 'title': 'Understanding distribution and random variables', 'summary': 'Discusses the fundamental mistake of confusing a random variable with its distribution, demonstrated through the example of adding pmfs, emphasizing the importance of distinguishing between the two concepts and the implications of this error.', 'duration': 171.301, 'highlights': ['The mistake of confusing a random variable with its distribution is highlighted, particularly in the context of adding PMFs, emphasizing the importance of understanding the distinction and its implications.', 'The concept of sympathetic magic is introduced as the mistake of confusing a random variable with its distribution, which can lead to errors in probability calculations.', 'The example of adding PMFs is used to illustrate the confusion between random variables and their distributions, emphasizing the invalidity of blindly adding PMFs and the potential for probabilities to exceed 1.', 'The chapter addresses the misconception of equating the sum of PMFs to the sum of random variables, highlighting the incorrect application and the implications of such errors in probability calculations.']}, {'end': 355.291, 'start': 172.09, 'title': 'Transforming random variables', 'summary': 'Explains the common mistake of confusing random variables with their distributions, using analogies of maps and territories and blueprints and houses, emphasizing the importance of understanding the distinction to avoid errors in probability theory and statistics.', 'duration': 183.201, 'highlights': ['The mistake of confusing random variables with their distributions is common in probability theory and statistics, as emphasized through analogies of maps and territories and blueprints and houses. This is important because understanding this distinction is crucial in probability theory and statistics to avoid errors. The analogy of maps and territories and blueprints and houses is used to illustrate the point.', "The analogy of maps and territories is used to emphasize the distinction between random variables and their distributions, highlighting the mathematical and abstract nature of the mistake. The analogy of maps and territories is used to illustrate the mistake's abstract and mathematical nature, contrasting it with the more tangible nature of map and territory confusion.", 'The analogy of blueprints for houses is introduced to further explain the relationship between random variables and their distributions, highlighting the concept of multiple random variables sharing the same distribution. The analogy of blueprints for houses is used to explain the concept of multiple random variables sharing the same distribution and the possibility of building many houses from the same blueprint.']}], 'duration': 355.042, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY249.jpg', 'highlights': ['The mistake of confusing a random variable with its distribution is highlighted, particularly in the context of adding PMFs, emphasizing the importance of understanding the distinction and its implications.', 'The concept of sympathetic magic is introduced as the mistake of confusing a random variable with its distribution, which can lead to errors in probability calculations.', 'The example of adding PMFs is used to illustrate the confusion between random variables and their distributions, emphasizing the invalidity of blindly adding PMFs and the potential for probabilities to exceed 1.', 'The chapter addresses the misconception of equating the sum of PMFs to the sum of random variables, highlighting the incorrect application and the implications of such errors in probability calculations.', 'The mistake of confusing random variables with their distributions is common in probability theory and statistics, as emphasized through analogies of maps and territories and blueprints and houses. This is important because understanding this distinction is crucial in probability theory and statistics to avoid errors.', 'The analogy of maps and territories is used to emphasize the distinction between random variables and their distributions, highlighting the mathematical and abstract nature of the mistake.', 'The analogy of blueprints for houses is introduced to further explain the relationship between random variables and their distributions, highlighting the concept of multiple random variables sharing the same distribution.']}, {'end': 924.297, 'segs': [{'end': 412.344, 'src': 'embed', 'start': 383.336, 'weight': 5, 'content': [{'end': 385.038, 'text': "so that's the main topic for today.", 'start': 383.336, 'duration': 1.702}, {'end': 388.501, 'text': 'I wanna show you, first of all, what is it?', 'start': 386.139, 'duration': 2.362}, {'end': 390.743, 'text': 'Secondly, why is it important?', 'start': 388.721, 'duration': 2.022}, {'end': 396.188, 'text': 'Arguably, the Poisson is the most important discrete distribution in all of statistics.', 'start': 391.644, 'duration': 4.544}, {'end': 401.834, 'text': 'Depends how you define importance, but I think you can make a pretty good argument for that.', 'start': 397.85, 'duration': 3.984}, {'end': 412.344, 'text': "It's named after Poisson, who was a famous, brilliant French mathematician, who was the first,", 'start': 404.182, 'duration': 8.162}], 'summary': 'The poisson distribution is important in statistics, named after the french mathematician poisson.', 'duration': 29.008, 'max_score': 383.336, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY383336.jpg'}, {'end': 497.832, 'src': 'embed', 'start': 438.666, 'weight': 0, 'content': [{'end': 446.272, 'text': 'So unlike the binomial, which is bounded between 0 and n, Vassan could take any non-negative integer value.', 'start': 438.666, 'duration': 7.606}, {'end': 458.922, 'text': 'And the PMF is just e to the minus lambda, lambda to the k over k factorial, where k, as I said, is a non-negative integer, 0 otherwise.', 'start': 446.332, 'duration': 12.59}, {'end': 464.066, 'text': 'And lambda is the parameter.', 'start': 460.423, 'duration': 3.643}, {'end': 472.246, 'text': 'And I call it a rate parameter for reasons that will become clear at some point.', 'start': 466.184, 'duration': 6.062}, {'end': 478.689, 'text': 'But for now, just think of it as this is a one parameter distribution with parameter traditionally called lambda.', 'start': 472.386, 'duration': 6.303}, {'end': 483.63, 'text': "But you could call it whatever you want, that's just the parameter of the distribution.", 'start': 478.709, 'duration': 4.921}, {'end': 486.651, 'text': 'But lambda is the most common name for that particular parameter.', 'start': 483.65, 'duration': 3.001}, {'end': 491.173, 'text': 'So lambda is a positive constant.', 'start': 487.092, 'duration': 4.081}, {'end': 493.714, 'text': 'Lambda could be any positive real number.', 'start': 491.353, 'duration': 2.361}, {'end': 497.832, 'text': "That's the parameter of the Poisson distribution.", 'start': 494.451, 'duration': 3.381}], 'summary': 'Poisson distribution has a rate parameter lambda that can take any positive real number.', 'duration': 59.166, 'max_score': 438.666, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY438666.jpg'}, {'end': 585.539, 'src': 'embed', 'start': 552.852, 'weight': 4, 'content': [{'end': 564.203, 'text': 'so basically all we did was take one term from the Taylor series for e to the lambda and then put a constant in front so that they add up to 1..', 'start': 552.852, 'duration': 11.351}, {'end': 565.284, 'text': 'So that is a PMF.', 'start': 564.203, 'duration': 1.081}, {'end': 572.37, 'text': "And while we're doing a calculation like this, we may as well compute the expected value also.", 'start': 567.386, 'duration': 4.984}, {'end': 585.539, 'text': "So let's find the mean e And as notation we would write X is plus on lambda, and we'll usually just abbreviate that to POIS, okay?", 'start': 573.291, 'duration': 12.248}], 'summary': 'Using a taylor series, we derived a pmf with a mean of lambda, written as pois.', 'duration': 32.687, 'max_score': 552.852, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY552852.jpg'}, {'end': 775.768, 'src': 'embed', 'start': 740.482, 'weight': 3, 'content': [{'end': 743.143, 'text': "Okay, so that's the expected value of a Poisson.", 'start': 740.482, 'duration': 2.661}, {'end': 753.169, 'text': 'Now why do we care about the Poisson though? So let me just mention a few examples where the Poisson is used.', 'start': 743.664, 'duration': 9.505}, {'end': 764.86, 'text': "It's the most widely, in practice, it's the single most widely used distribution as a model for discrete data in the real world.", 'start': 754.032, 'duration': 10.828}, {'end': 775.768, 'text': 'So often used for applications like, Well, let me just say what the general application is.', 'start': 765.44, 'duration': 10.328}], 'summary': 'Poisson distribution widely used for modeling discrete data in real world.', 'duration': 35.286, 'max_score': 740.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY740482.jpg'}], 'start': 355.291, 'title': 'Poisson distribution', 'summary': 'Discusses the significance of poisson distribution, emphasizing its pmf formula, rate parameter lambda, and applications, e.g., modeling email counts.', 'chapters': [{'end': 497.832, 'start': 355.291, 'title': 'Poisson distribution: importance and parameters', 'summary': "Discusses the importance of poisson distribution in statistics, highlighting its key attributes and relevance, particularly its pmf formula and parameter lambda's significance as a rate parameter.", 'duration': 142.541, 'highlights': ['The Poisson distribution is highlighted as the most important discrete distribution in statistics, attributed to its relevance and applications.', 'The PMF of the Poisson distribution is expressed as e^(-lambda) * (lambda^k) / k!, where k is a non-negative integer and lambda is the parameter.', 'Lambda is emphasized as the rate parameter in the Poisson distribution, representing a positive constant and playing a crucial role in defining the distribution.']}, {'end': 924.297, 'start': 498.332, 'title': 'Poisson distribution and expected value', 'summary': 'Discusses the importance of the poisson distribution, its expected value, and its wide applications, including in modeling the count of occurrences with small probabilities, such as the number of emails received in an hour.', 'duration': 425.965, 'highlights': ['The Poisson distribution is validated as a probability mass function by ensuring that its non-negative values sum to 1, which is proven using the Taylor series for e to the lambda and the calculation of the expected value. (Importance of Poisson distribution and validation)', 'The expected value of a Poisson distribution is found to be its parameter, lambda, and it is emphasized as a useful and easy-to-remember result, with a brief explanation of its practical significance. (Expected value of Poisson distribution and its practical significance)', 'The Poisson distribution is highlighted as the most widely used distribution for modeling discrete data in real-world applications, particularly in scenarios where the count of occurrences with small probabilities, such as the number of emails received in an hour, needs to be modeled. (Wide applications of Poisson distribution)']}], 'duration': 569.006, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY355291.jpg', 'highlights': ['The PMF of the Poisson distribution is expressed as e^(-lambda) * (lambda^k) / k!, where k is a non-negative integer and lambda is the parameter.', 'Lambda is emphasized as the rate parameter in the Poisson distribution, representing a positive constant and playing a crucial role in defining the distribution.', 'The expected value of a Poisson distribution is found to be its parameter, lambda, and it is emphasized as a useful and easy-to-remember result, with a brief explanation of its practical significance.', 'The Poisson distribution is highlighted as the most widely used distribution for modeling discrete data in real-world applications, particularly in scenarios where the count of occurrences with small probabilities, such as the number of emails received in an hour, needs to be modeled.', 'The Poisson distribution is validated as a probability mass function by ensuring that its non-negative values sum to 1, which is proven using the Taylor series for e to the lambda and the calculation of the expected value.', 'The Poisson distribution is highlighted as the most important discrete distribution in statistics, attributed to its relevance and applications.']}, {'end': 1254.089, 'segs': [{'end': 966.958, 'src': 'embed', 'start': 924.537, 'weight': 2, 'content': [{'end': 935.597, 'text': 'Well, imagine that there are a large number of people who potentially could email you in that one hour block of time, okay?', 'start': 924.537, 'duration': 11.06}, {'end': 944.102, 'text': "But for any individual person, unless you have someone who's constantly emailing you every few seconds all day in that particular block of time,", 'start': 935.797, 'duration': 8.305}, {'end': 947.624, 'text': "it's fairly unlikely that any specific person will email you.", 'start': 944.102, 'duration': 3.522}, {'end': 952.606, 'text': "But if there are a lot of people who could email you, then that's balancing it out right?", 'start': 947.984, 'duration': 4.622}, {'end': 957.17, 'text': "So we're defining success to be for each person, whether or not they email you in that hour.", 'start': 952.626, 'duration': 4.544}, {'end': 960.493, 'text': 'There are a lot of people who could email you, but each one is fairly unlikely.', 'start': 957.19, 'duration': 3.303}, {'end': 962.374, 'text': "That's the setup.", 'start': 960.773, 'duration': 1.601}, {'end': 965.076, 'text': 'Change email to phone calls or whatever you want here.', 'start': 962.414, 'duration': 2.662}, {'end': 966.958, 'text': 'You can make up as many examples as you want.', 'start': 965.116, 'duration': 1.842}], 'summary': 'Success is defined by whether people email you in a busy hour, balancing the likelihood of individual emails.', 'duration': 42.421, 'max_score': 924.537, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY924537.jpg'}, {'end': 1159.818, 'src': 'embed', 'start': 1116.462, 'weight': 0, 'content': [{'end': 1119.443, 'text': "And that's called the Poisson paradigm.", 'start': 1116.462, 'duration': 2.981}, {'end': 1129.585, 'text': 'Or just more simply, just call it Poisson approximation, Poisson paradigm.', 'start': 1124.144, 'duration': 5.441}, {'end': 1144.348, 'text': "But just to say that in words, let's say we have events A1, A2, blah, blah, blah, through An.", 'start': 1129.805, 'duration': 14.543}, {'end': 1157.096, 'text': 'Suppose we have a lot of events, right? And they are with p of aj equals pj.', 'start': 1145.97, 'duration': 11.126}, {'end': 1159.818, 'text': 'So n is large.', 'start': 1157.757, 'duration': 2.061}], 'summary': 'The poisson paradigm approximates a large number of events with p(aj) = pj and n is large.', 'duration': 43.356, 'max_score': 1116.462, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1116462.jpg'}, {'end': 1221.524, 'src': 'embed', 'start': 1190.116, 'weight': 1, 'content': [{'end': 1194.677, 'text': 'But actually this result is even more general, so they could be weakly dependent.', 'start': 1190.116, 'duration': 4.561}, {'end': 1198.499, 'text': 'And we have a mathematical definition of independence.', 'start': 1195.258, 'duration': 3.241}, {'end': 1202.84, 'text': "It's very difficult to mathematically define weakly dependent.", 'start': 1199.419, 'duration': 3.421}, {'end': 1211.88, 'text': 'The intuition is just that independence comes in degrees to some extent.', 'start': 1205.618, 'duration': 6.262}, {'end': 1221.524, 'text': 'And that independence says that learning, for example learning A1 through A3, whether they happened or not, gives us independence,', 'start': 1211.94, 'duration': 9.584}], 'summary': 'The concept of weak dependence is difficult to mathematically define, but it suggests that independence comes in degrees.', 'duration': 31.408, 'max_score': 1190.116, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1190116.jpg'}], 'start': 924.537, 'title': 'Balancing email probability and poisson paradigm', 'summary': 'Explains the balancing of email probability and defines success in receiving emails, while also discussing the poisson paradigm, weak dependence, and its applications in various scenarios.', 'chapters': [{'end': 966.958, 'start': 924.537, 'title': 'Balancing email probability', 'summary': 'Explains how the probability of receiving emails in a one-hour time block is balanced out by the large number of potential senders, making it unlikely for any specific person to email you, and defines success based on whether each person emails you in that hour.', 'duration': 42.421, 'highlights': ['The probability of receiving emails is balanced out by the large number of potential senders.', "It's fairly unlikely that any specific person will email you in that one-hour time block.", 'Success is defined based on whether each person emails you in that hour.']}, {'end': 1254.089, 'start': 968.199, 'title': 'Poisson paradigm and weak dependence', 'summary': 'Discusses the poisson paradigm and weak dependence in events, explaining the approximation and its application in various scenarios, such as the number of chocolate chips in a cookie and the number of earthquakes in a year, while also delving into the concept of weak dependence.', 'duration': 285.89, 'highlights': ['The Poisson paradigm is an approximation used in various scenarios, such as the number of chocolate chips in a cookie and the number of earthquakes in a year, where events are unlikely but can occur in numerous possible locations, making it a useful approximation in many problems. Poisson paradigm as an approximation for unlikely events occurring in numerous possible locations, its application in scenarios like the number of chocolate chips in a cookie and the number of earthquakes in a year, and its usefulness in problem-solving.', 'The concept of weak dependence is discussed, where events may not be fully independent but still exhibit varying degrees of independence, leading to a mathematical challenge in defining weak dependence and its implications on event occurrences. Explanation of weak dependence in events, challenges in mathematically defining weak dependence, and the implications of weak dependence on event occurrences.']}], 'duration': 329.552, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY924537.jpg', 'highlights': ['The Poisson paradigm is an approximation used in various scenarios, such as the number of chocolate chips in a cookie and the number of earthquakes in a year, where events are unlikely but can occur in numerous possible locations, making it a useful approximation in many problems.', 'The concept of weak dependence is discussed, where events may not be fully independent but still exhibit varying degrees of independence, leading to a mathematical challenge in defining weak dependence and its implications on event occurrences.', 'The probability of receiving emails is balanced out by the large number of potential senders.', 'Success is defined based on whether each person emails you in that hour.', "It's fairly unlikely that any specific person will email you in that one-hour time block."]}, {'end': 1492.294, 'segs': [{'end': 1361.728, 'src': 'heatmap', 'start': 1254.189, 'weight': 0, 'content': [{'end': 1258.692, 'text': "So it's slight deviations from independence, that's what we're talking about.", 'start': 1254.189, 'duration': 4.503}, {'end': 1273.997, 'text': 'Then the claim is that the number of events that occur, number of the AJs that occur is approximately Poisson.', 'start': 1261.371, 'duration': 12.626}, {'end': 1281.461, 'text': 'And we can figure out right now what lambda would have to be for that to make sense.', 'start': 1274.918, 'duration': 6.543}, {'end': 1286.258, 'text': 'We just showed that lambda is the expected value of the Poisson.', 'start': 1283.356, 'duration': 2.902}, {'end': 1291.963, 'text': 'So it makes sense that lambda should be the expected number of how many of these events occur.', 'start': 1286.859, 'duration': 5.104}, {'end': 1299.229, 'text': "But by linearity, even if they're dependent, by linearity, the expected number of events that occur is the sum of the pj's.", 'start': 1292.423, 'duration': 6.806}, {'end': 1301.671, 'text': "So lambda should be the sum of the pj's.", 'start': 1299.709, 'duration': 1.962}, {'end': 1309.437, 'text': 'Okay, this is approximate.', 'start': 1307.215, 'duration': 2.222}, {'end': 1313.06, 'text': 'So this is also called the Poisson approximation.', 'start': 1310.658, 'duration': 2.402}, {'end': 1316.233, 'text': "That's the same thing as a Poisson paradigm.", 'start': 1314.171, 'duration': 2.062}, {'end': 1329.161, 'text': "So one case that we've looked at a lot is the case where all the events are independent and all the pj's are the same.", 'start': 1321.016, 'duration': 8.145}, {'end': 1331.383, 'text': "They're all equal to the same p.", 'start': 1329.782, 'duration': 1.601}, {'end': 1334.085, 'text': 'In that case, we just have Bernoulli trials with the same p.', 'start': 1331.383, 'duration': 2.702}, {'end': 1338.428, 'text': 'In that case, we know that the number of events that occur is exactly binomial np.', 'start': 1334.085, 'duration': 4.343}, {'end': 1349.165, 'text': "And in a minute we're gonna prove that in fact the binomial NP does converge to a Poisson when we get N get large and P get small.", 'start': 1340.982, 'duration': 8.183}, {'end': 1350.345, 'text': 'in a certain way, okay?', 'start': 1349.165, 'duration': 1.18}, {'end': 1358.647, 'text': 'But this is much, much more general than just for the binomial, because this is saying that Poisson is gonna work well even if the Ps are different,', 'start': 1350.725, 'duration': 7.922}, {'end': 1361.728, 'text': 'right?. Remember, for the binomial we had independent trials with the same P.', 'start': 1358.647, 'duration': 3.081}], 'summary': 'The poisson approximation for number of events, lambda, can be calculated as the sum of the probabilities of the events occurring, even when the events are dependent or have different probabilities.', 'duration': 74.869, 'max_score': 1254.189, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1254189.jpg'}, {'end': 1501.956, 'src': 'embed', 'start': 1475.771, 'weight': 3, 'content': [{'end': 1480.072, 'text': 'And we can do more general versions than this, but this is a natural one to start with.', 'start': 1475.771, 'duration': 4.301}, {'end': 1484.033, 'text': 'Because remember, the expected value of a binomial NP is n times p.', 'start': 1480.192, 'duration': 3.841}, {'end': 1487.553, 'text': 'The expected value of a Poisson lambda we just showed is lambda.', 'start': 1484.033, 'duration': 3.52}, {'end': 1492.294, 'text': 'So it makes sense to set lambda equal to NP to try to explore the connection between those.', 'start': 1487.673, 'duration': 4.621}, {'end': 1501.956, 'text': "So this is telling us that p is going to 0 at the same rate as n is going to infinity, cuz we're holding the product constant.", 'start': 1492.894, 'duration': 9.062}], 'summary': 'Exploring the connection between binomial and poisson distributions by setting lambda equal to np.', 'duration': 26.185, 'max_score': 1475.771, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1475771.jpg'}], 'start': 1254.189, 'title': 'Poisson approximation and convergence', 'summary': 'Discusses poisson approximation, lambda calculation, and convergence of binomial to poisson. it explains the approximation of events as poisson, calculation of lambda, and the convergence of binomial distribution to poisson as n gets large and p gets small.', 'chapters': [{'end': 1316.233, 'start': 1254.189, 'title': 'Poisson approximation and lambda calculation', 'summary': "Discusses the poisson approximation, explaining that the number of events occurring is approximately poisson, and lambda is calculated as the sum of the pj's, providing insight into the poisson paradigm.", 'duration': 62.044, 'highlights': ["The number of events that occur is approximately Poisson, and lambda is calculated as the sum of the pj's.", 'Lambda should be the expected number of how many events occur, and it is also referred to as the Poisson approximation and Poisson paradigm.']}, {'end': 1492.294, 'start': 1321.016, 'title': 'Convergence of binomial to poisson', 'summary': 'Explores the convergence of the binomial distribution to the poisson distribution as n gets large and p gets small, showing that even with different probabilities and some dependence, the poisson distribution remains a good approximation.', 'duration': 171.278, 'highlights': ['The number of events that occur in the case of independent events with the same probability is exactly binomial NP. This highlights the relationship between independent events with the same probability and the binomial distribution, providing a fundamental understanding of the concept.', 'The binomial distribution converges to the Poisson distribution as N gets large and P gets small, even with different probabilities and some dependence. This underlines the key concept of convergence of distributions and the robustness of the Poisson distribution as an approximation even with varying probabilities and some dependence.', 'Setting lambda equal to NP explores the connection between the expected values of the binomial and Poisson distributions. This highlights the exploration of the relationship between the expected values of the binomial and Poisson distributions, providing a basis for understanding their connection.']}], 'duration': 238.105, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1254189.jpg', 'highlights': ['Lambda should be the expected number of how many events occur, and it is also referred to as the Poisson approximation and Poisson paradigm.', "The number of events that occur is approximately Poisson, and lambda is calculated as the sum of the pj's.", 'The binomial distribution converges to the Poisson distribution as N gets large and P gets small, even with different probabilities and some dependence.', 'Setting lambda equal to NP explores the connection between the expected values of the binomial and Poisson distributions.', 'The number of events that occur in the case of independent events with the same probability is exactly binomial NP.']}, {'end': 1791.116, 'segs': [{'end': 1616.237, 'src': 'embed', 'start': 1492.894, 'weight': 0, 'content': [{'end': 1501.956, 'text': "So this is telling us that p is going to 0 at the same rate as n is going to infinity, cuz we're holding the product constant.", 'start': 1492.894, 'duration': 9.062}, {'end': 1505.862, 'text': 'All right, so now we wanna show that the PMF converges.', 'start': 1502.881, 'duration': 2.981}, {'end': 1513.363, 'text': 'So find what happens to the PMF.', 'start': 1507.242, 'duration': 6.121}, {'end': 1530.586, 'text': 'What happens to the probability that x equals k, which we keep seeing is n choose k, p to the k, 1- p to the n- k.', 'start': 1517.744, 'duration': 12.842}, {'end': 1532.166, 'text': 'So I wanna know what happens to this thing.', 'start': 1530.586, 'duration': 1.58}, {'end': 1535.487, 'text': "where we're treating k as fixed.", 'start': 1533.505, 'duration': 1.982}, {'end': 1547.562, 'text': "As we're letting n get very large, p get very small, but then we wanna look at one specific value for the PMF, so we're letting k stay constant.", 'start': 1539.092, 'duration': 8.47}, {'end': 1558.674, 'text': "All right, so that's what we wanna do, and then at this point, We just need to do some algebra and a little bit of calculus.", 'start': 1548.584, 'duration': 10.09}, {'end': 1563.218, 'text': 'So right now this is written as a function of p and n.', 'start': 1559.155, 'duration': 4.063}, {'end': 1566.881, 'text': "It's gonna be easier to deal with if we get everything in terms of n.", 'start': 1563.218, 'duration': 3.663}, {'end': 1574.588, 'text': 'And so conveniently, we have p equals lambda over n, so we can write everything in terms of n.', 'start': 1566.881, 'duration': 7.707}, {'end': 1577.03, 'text': 'And let me also write n choose k a different way.', 'start': 1574.588, 'duration': 2.442}, {'end': 1583.989, 'text': "n choose k is the number of ways to choose k people out of n where order doesn't matter.", 'start': 1579.128, 'duration': 4.861}, {'end': 1591.071, 'text': "So that's the same thing as n times n minus 1, blah, blah, blah, n minus k plus 1.", 'start': 1584.389, 'duration': 6.682}, {'end': 1596.793, 'text': 'That would be choosing a committee in a certain order where order matters.', 'start': 1591.071, 'duration': 5.722}, {'end': 1601.694, 'text': "Then we divide by k factorial to reflect the fact that order doesn't matter.", 'start': 1596.833, 'duration': 4.861}, {'end': 1608.136, 'text': "So that's just exactly the same thing as n choose k according to the story, or you can check that easily just using factorials.", 'start': 1601.734, 'duration': 6.402}, {'end': 1616.237, 'text': "I'll plug in p equals lambda to the n, p equals lambda over n.", 'start': 1609.474, 'duration': 6.763}], 'summary': 'The pmf converges as n goes to infinity with p going to 0, showing the probability that x equals k.', 'duration': 123.343, 'max_score': 1492.894, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1492894.jpg'}, {'end': 1686.848, 'src': 'embed', 'start': 1646.654, 'weight': 5, 'content': [{'end': 1648.175, 'text': "Now let's take the limit of this.", 'start': 1646.654, 'duration': 1.521}, {'end': 1658.885, 'text': 'The k is fixed, so k factorial here just stays as k factorial.', 'start': 1654.761, 'duration': 4.124}, {'end': 1665.632, 'text': "Now let's look at, and lambda to the k is also fixed.", 'start': 1662.93, 'duration': 2.702}, {'end': 1673.738, 'text': 'So we have a lambda to the k over k factorial, which is pretty encouraging cuz we want that for our Poisson.', 'start': 1665.792, 'duration': 7.946}, {'end': 1679.923, 'text': "Now, that's lambda to the k over k factorial.", 'start': 1676.06, 'duration': 3.863}, {'end': 1682.765, 'text': "Let's look at this stuff with the n's.", 'start': 1681.324, 'duration': 1.441}, {'end': 1686.848, 'text': 'On top we have k terms.', 'start': 1684.426, 'duration': 2.422}], 'summary': 'Analyzing the limit with k terms and lambda, aiming for poisson distribution.', 'duration': 40.194, 'max_score': 1646.654, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1646654.jpg'}, {'end': 1765.122, 'src': 'embed', 'start': 1737.76, 'weight': 6, 'content': [{'end': 1747.547, 'text': 'That part, I just have to remind you of a general useful limit, that 1 plus x over n to the n goes to e to the x.', 'start': 1737.76, 'duration': 9.787}, {'end': 1752.387, 'text': 'Sometimes this is even taken as the definition of e to the x.', 'start': 1748.519, 'duration': 3.868}, {'end': 1759.077, 'text': 'This is probably the most important limit, for I think of this as the compound interest formula,', 'start': 1752.387, 'duration': 6.69}, {'end': 1765.122, 'text': "because what this is if you have money in a bank that's being compounded a certain number of times per year.", 'start': 1759.077, 'duration': 6.045}], 'summary': 'The limit 1 plus x over n to the n goes to e to the x, important for compound interest.', 'duration': 27.362, 'max_score': 1737.76, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1737760.jpg'}], 'start': 1492.894, 'title': 'Pmf and poisson distribution', 'summary': 'Discusses the convergence of the pmf as n goes to infinity and p goes to 0, emphasizing the calculation of the probability that x equals k. it also explains the poisson distribution limit theorem, with a focus on expressing terms in terms of n and evaluating the limit of the distribution, relevant to compound interest.', 'chapters': [{'end': 1563.218, 'start': 1492.894, 'title': 'Convergence of pmf in probability theory', 'summary': 'Discusses the convergence of the pmf as n goes to infinity and p goes to 0 while keeping the product constant, emphasizing the calculation of the probability that x equals k using n choose k, p to the k, and 1- p to the n- k.', 'duration': 70.324, 'highlights': ['The PMF converges as n goes to infinity and p goes to 0 while keeping the product constant, with a focus on the probability that x equals k using n choose k, p to the k, and 1- p to the n- k.', 'The calculation involves treating k as fixed and letting n get very large and p get very small, while looking at one specific value for the PMF, with k staying constant.', 'Algebra and a little bit of calculus are used to analyze the convergence of the PMF written as a function of p and n.']}, {'end': 1791.116, 'start': 1563.218, 'title': 'Poisson distribution limit theorem', 'summary': 'Explains the poisson distribution limit theorem, with a focus on expressing terms in terms of n, evaluating the limit of the distribution, and applying the general limit 1 + x/n to the power of n goes to e to the x, with relevance to compound interest.', 'duration': 227.898, 'highlights': ['The Poisson distribution is discussed in terms of n and p, with p equals lambda over n, allowing expression of terms in terms of n and simplification of calculations.', "The chapter explains n choose k as a way to choose k people out of n where order doesn't matter, and illustrates it as a committee being chosen in a certain order and then dividing by k factorial to reflect that order doesn't matter.", 'The limit of the Poisson distribution is evaluated, with emphasis on fixed k factorial and fixed lambda to the power of k, leading to the matching of terms and the conclusion that the terms involving n go to 1 as n goes to infinity.', 'The general useful limit 1 plus x over n to the power of n goes to e to the x is discussed, with a connection to compound interest and exponential growth, and the conclusion that the evaluated part goes to e to the minus lambda.']}], 'duration': 298.222, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1492894.jpg', 'highlights': ['The PMF converges as n goes to infinity and p goes to 0 while keeping the product constant, focusing on the probability that x equals k using n choose k, p to the k, and 1- p to the n- k.', 'The calculation involves treating k as fixed and letting n get very large and p get very small, while looking at one specific value for the PMF, with k staying constant.', 'Algebra and a little bit of calculus are used to analyze the convergence of the PMF written as a function of p and n.', 'The Poisson distribution is discussed in terms of n and p, with p equals lambda over n, allowing expression of terms in terms of n and simplification of calculations.', "The chapter explains n choose k as a way to choose k people out of n where order doesn't matter, and illustrates it as a committee being chosen in a certain order and then dividing by k factorial to reflect that order doesn't matter.", 'The limit of the Poisson distribution is evaluated, with emphasis on fixed k factorial and fixed lambda to the power of k, leading to the matching of terms and the conclusion that the terms involving n go to 1 as n goes to infinity.', 'The general useful limit 1 plus x over n to the power of n goes to e to the x is discussed, with a connection to compound interest and exponential growth, and the conclusion that the evaluated part goes to e to the minus lambda.']}, {'end': 2004.502, 'segs': [{'end': 1844.493, 'src': 'embed', 'start': 1796.3, 'weight': 0, 'content': [{'end': 1798.201, 'text': "And that's just the Poisson PMF.", 'start': 1796.3, 'duration': 1.901}, {'end': 1812.733, 'text': 'At evaluated at k, okay? So that shows that binomials converge to Poisson when we do it in this way.', 'start': 1803.165, 'duration': 9.568}, {'end': 1827.35, 'text': 'Another example that I like to think of, just as more intuition on this, is counting the number of raindrops that fall in some region.', 'start': 1818.784, 'duration': 8.566}, {'end': 1833.895, 'text': "I don't know why I like this example, but I just find it very, very intuitive for understanding the connection between binomial and Poisson.", 'start': 1827.871, 'duration': 6.024}, {'end': 1835.436, 'text': "So I'm just gonna draw a quick picture.", 'start': 1833.915, 'duration': 1.521}, {'end': 1837.678, 'text': 'So this is the raindrop example.', 'start': 1835.456, 'duration': 2.222}, {'end': 1844.493, 'text': "So imagine we have a piece of paper, which I'll represent as just this rectangle.", 'start': 1840.25, 'duration': 4.243}], 'summary': 'Binomials converge to poisson, illustrated with raindrop example.', 'duration': 48.193, 'max_score': 1796.3, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1796300.jpg'}, {'end': 1931.54, 'src': 'embed', 'start': 1868.834, 'weight': 2, 'content': [{'end': 1873.737, 'text': "So draw as many as you want, but I'm just gonna draw a grid.", 'start': 1868.834, 'duration': 4.903}, {'end': 1881.822, 'text': "And imagine that we've broken it up into millions and millions of tiny little squares.", 'start': 1874.998, 'duration': 6.824}, {'end': 1886.165, 'text': 'okay?. Now for each individual, and I wanna count the number of raindrops.', 'start': 1881.822, 'duration': 4.343}, {'end': 1892.767, 'text': 'in some time interval for each individual square.', 'start': 1887.904, 'duration': 4.863}, {'end': 1906.255, 'text': 'if we break this up into tiny enough pieces, each individual square is unlikely to get a raindrop hitting exactly in that.', 'start': 1892.767, 'duration': 13.488}, {'end': 1907.856, 'text': 'if we make them small enough, right?', 'start': 1906.255, 'duration': 1.601}, {'end': 1910.218, 'text': 'So each one is kind of unlikely.', 'start': 1908.817, 'duration': 1.401}, {'end': 1912.999, 'text': 'but we have a huge number of little squares, okay?', 'start': 1910.218, 'duration': 2.781}, {'end': 1917.542, 'text': 'So we are gonna get some raindrops hitting this piece of paper probably.', 'start': 1913.039, 'duration': 4.503}, {'end': 1926.078, 'text': 'And lambda is gonna be a measure of the intensity of how hard the rain is coming.', 'start': 1919.235, 'duration': 6.843}, {'end': 1931.54, 'text': "And so let's think should we use a binomial distribution for this?", 'start': 1926.778, 'duration': 4.762}], 'summary': 'Using grid to count raindrops, considering binomial distribution.', 'duration': 62.706, 'max_score': 1868.834, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1868834.jpg'}, {'end': 2004.502, 'src': 'embed', 'start': 1978.409, 'weight': 4, 'content': [{'end': 1989.695, 'text': "but even if it were exactly binomial, If we had binomial of like a trillion and then some tiny number, that's very, very hard to work with,", 'start': 1978.409, 'duration': 11.286}, {'end': 1991.316, 'text': 'certainly very hard to work with by hand.', 'start': 1989.695, 'duration': 1.621}, {'end': 2001.401, 'text': 'But even on a computer computers are gonna have a lot of trouble handling, even 1, 000 factorial,', 'start': 1991.676, 'duration': 9.725}, {'end': 2004.502, 'text': "or you're gonna run into some computational difficulties.", 'start': 2001.401, 'duration': 3.101}], 'summary': 'Handling a trillion binomial computations is very difficult, even for computers.', 'duration': 26.093, 'max_score': 1978.409, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1978409.jpg'}], 'start': 1796.3, 'title': 'Connection between binomial and poisson distributions', 'summary': 'Explores the relationship between binomial and poisson distributions, demonstrating convergence and intuition through examples like poisson pmf and counting raindrops. it also discusses modeling rainfall with binomial distribution, addressing challenges in handling large factorial computations.', 'chapters': [{'end': 1844.493, 'start': 1796.3, 'title': 'Connection between binomial and poisson', 'summary': 'Explains the connection between binomial and poisson distributions through examples like the poisson pmf and counting the number of raindrops, illustrating convergence and intuition.', 'duration': 48.193, 'highlights': ['The connection between binomial and Poisson distributions is illustrated through the example of the Poisson PMF, showing convergence at evaluated at k.', 'Counting the number of raindrops falling in a region is used as an intuitive example to understand the connection between binomial and Poisson distributions.']}, {'end': 2004.502, 'start': 1845.154, 'title': 'Modeling rainfall with binomial distribution', 'summary': 'Discusses modeling rainfall using a grid of tiny squares, applying binomial distribution to count raindrops, and the challenges of handling large factorial computations in this context.', 'duration': 159.348, 'highlights': ['The process of modeling rainfall using a grid of tiny squares and applying binomial distribution to count raindrops.', 'The challenge of handling large factorial computations, such as 1,000 factorial, in the context of modeling rainfall.', "The consideration of using binomial distribution to measure the intensity of rain, with lambda as a measure of the rain's intensity."]}], 'duration': 208.202, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY1796300.jpg', 'highlights': ['The connection between binomial and Poisson distributions is illustrated through the example of the Poisson PMF, showing convergence at evaluated at k.', 'Counting the number of raindrops falling in a region is used as an intuitive example to understand the connection between binomial and Poisson distributions.', 'The process of modeling rainfall using a grid of tiny squares and applying binomial distribution to count raindrops.', "The consideration of using binomial distribution to measure the intensity of rain, with lambda as a measure of the rain's intensity.", 'The challenge of handling large factorial computations, such as 1,000 factorial, in the context of modeling rainfall.']}, {'end': 2564.768, 'segs': [{'end': 2036.586, 'src': 'embed', 'start': 2005.203, 'weight': 4, 'content': [{'end': 2009.485, 'text': 'Binomial has a lot of complications to it, where the Poisson is much simpler to deal with.', 'start': 2005.203, 'duration': 4.282}, {'end': 2015.199, 'text': 'A Poisson approximation seems reasonable here cuz we have a huge number of little squares, each one is very unlikely.', 'start': 2010.478, 'duration': 4.721}, {'end': 2023.161, 'text': "All right, so let's do one example like with the birthday problem type of setup.", 'start': 2016.359, 'duration': 6.802}, {'end': 2030.782, 'text': 'And you have one on the next homework, homework five, related to this too.', 'start': 2026.481, 'duration': 4.301}, {'end': 2036.586, 'text': "Let's do the problem of, triple birthday matches.", 'start': 2031.342, 'duration': 5.244}], 'summary': 'Comparing binomial and poisson distributions, applying poisson approximation for a large number of unlikely events, discussing a birthday problem example, and mentioning a related homework assignment.', 'duration': 31.383, 'max_score': 2005.203, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2005203.jpg'}, {'end': 2131.221, 'src': 'embed', 'start': 2099.728, 'weight': 0, 'content': [{'end': 2104.429, 'text': "But you can find some group of three people who all have the same birthday, okay? That's the problem.", 'start': 2099.728, 'duration': 4.701}, {'end': 2114.532, 'text': "So if you try to do this in a way, that's analogous to how we solved the original basic birthday problem.", 'start': 2108.01, 'duration': 6.522}, {'end': 2118.773, 'text': 'this is actually pretty difficult, very difficult.', 'start': 2114.532, 'duration': 4.241}, {'end': 2120.554, 'text': 'You can try it.', 'start': 2119.974, 'duration': 0.58}, {'end': 2131.221, 'text': "But it's not gonna be nice, okay? But with a Poisson approximation, this actually should be pretty easy.", 'start': 2121.448, 'duration': 9.773}], 'summary': 'Finding a group of three people with the same birthday is difficult, but a poisson approximation makes it easier.', 'duration': 31.493, 'max_score': 2099.728, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2099728.jpg'}, {'end': 2218.867, 'src': 'embed', 'start': 2191.362, 'weight': 1, 'content': [{'end': 2199.044, 'text': '253 is reasonably large, right? So the more relevant quantity is n choose 3.', 'start': 2191.362, 'duration': 7.682}, {'end': 2205.785, 'text': 'So even if n is, I mean, n should at least be 10 or 20 or something like that.', 'start': 2199.044, 'duration': 6.741}, {'end': 2209.025, 'text': 'n does not have to be in the hundreds for this to work well.', 'start': 2206.105, 'duration': 2.92}, {'end': 2218.867, 'text': "Because even with some double digit n, when you do n choose 3, it's gonna be pretty large, right? Okay, so there are n choose 3 triplets of people.", 'start': 2209.045, 'duration': 9.822}], 'summary': 'Even with double-digit n, n choose 3 yields a reasonably large number of triplets.', 'duration': 27.505, 'max_score': 2191.362, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2191362.jpg'}, {'end': 2340.746, 'src': 'heatmap', 'start': 2275.912, 'weight': 2, 'content': [{'end': 2277.534, 'text': 'We know the expected value exactly.', 'start': 2275.912, 'duration': 1.622}, {'end': 2282.233, 'text': 'expected value of the number of triple matches.', 'start': 2278.351, 'duration': 3.882}, {'end': 2290.376, 'text': 'We get that immediately by indicator random variables, linearity, and symmetry.', 'start': 2286.234, 'duration': 4.142}, {'end': 2292.997, 'text': "That is, it's n choose 3 of these indicators.", 'start': 2290.796, 'duration': 2.201}, {'end': 2300.581, 'text': "And for each one, now I'm imagining that I have the first three people, just for concreteness.", 'start': 2293.458, 'duration': 7.123}, {'end': 2305.143, 'text': 'First person can have whatever birthday, and the second person has to match the first person.', 'start': 2301.021, 'duration': 4.122}, {'end': 2312.699, 'text': "1 over 365, and then the third person also has to match, so it's 1 over 365 squared.", 'start': 2306.796, 'duration': 5.903}, {'end': 2314.259, 'text': "That's exact.", 'start': 2313.439, 'duration': 0.82}, {'end': 2318.241, 'text': "So that's the exact number of triple matches.", 'start': 2316.2, 'duration': 2.041}, {'end': 2326.765, 'text': "Notice though that if there's a group of four people who have the same birthday, we are counting that for each set of three.", 'start': 2318.821, 'duration': 7.944}, {'end': 2332.007, 'text': "So there are four choices of three people out of the four, and we're counting that as four matches.", 'start': 2326.865, 'duration': 5.142}, {'end': 2336.664, 'text': "Okay, so that's the exact answer for the expected value.", 'start': 2334.022, 'duration': 2.642}, {'end': 2340.746, 'text': 'But I said find the approximate probability, not the.', 'start': 2337.144, 'duration': 3.602}], 'summary': 'Exact expected value of triple matches is calculated using n choose 3 and 1/365^2, accounting for groups of four.', 'duration': 64.834, 'max_score': 2275.912, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2275912.jpg'}, {'end': 2354.154, 'src': 'embed', 'start': 2326.865, 'weight': 3, 'content': [{'end': 2332.007, 'text': "So there are four choices of three people out of the four, and we're counting that as four matches.", 'start': 2326.865, 'duration': 5.142}, {'end': 2336.664, 'text': "Okay, so that's the exact answer for the expected value.", 'start': 2334.022, 'duration': 2.642}, {'end': 2340.746, 'text': 'But I said find the approximate probability, not the.', 'start': 2337.144, 'duration': 3.602}, {'end': 2343.948, 'text': "I didn't say find the exact expected value, okay?", 'start': 2340.746, 'duration': 3.202}, {'end': 2347.57, 'text': "So now we're gonna use the Poisson approximation.", 'start': 2345.409, 'duration': 2.161}, {'end': 2354.154, 'text': "So let's let X equal number of triple matches.", 'start': 2350.052, 'duration': 4.102}], 'summary': 'Using poisson approximation to find the approximate probability of triple matches.', 'duration': 27.289, 'max_score': 2326.865, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2326865.jpg'}, {'end': 2547.414, 'src': 'embed', 'start': 2521.812, 'weight': 5, 'content': [{'end': 2531.28, 'text': "And as an approximation, that's gonna be 1 minus e to the minus lambda lambda to the 0 over 0 factorial,", 'start': 2521.812, 'duration': 9.468}, {'end': 2534.883, 'text': 'which of course is just 1 minus e to the minus lambda.', 'start': 2531.28, 'duration': 3.603}, {'end': 2540.208, 'text': "So that's 1 minus e to the minus lambda, where lambda is this.", 'start': 2535.524, 'duration': 4.684}, {'end': 2547.414, 'text': 'So notice that this is something that you could calculate easily using a calculator or a computer.', 'start': 2540.908, 'duration': 6.506}], 'summary': 'Approximation formula: 1 - e^-λ, λ can be calculated easily.', 'duration': 25.602, 'max_score': 2521.812, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2521812.jpg'}], 'start': 2005.203, 'title': 'Poisson approximation for triple birthday matches', 'summary': 'Discusses the poisson approximation to calculate the probability of three people sharing the same birthday, highlighting the relevance of n choose 3 and the indicator random variables. it also emphasizes the validity of the approximation based on the large number of possible matches, small probability of success for each trial, and weak dependence between trials.', 'chapters': [{'end': 2326.765, 'start': 2005.203, 'title': 'Poisson approximation for triple birthday matches', 'summary': 'Discusses the poisson approximation to calculate the probability of three people sharing the same birthday, highlighting the relevance of n choose 3 and the indicator random variables, providing a simpler alternative to the exact calculation.', 'duration': 321.562, 'highlights': ['The Poisson approximation is introduced to calculate the probability of three people sharing the same birthday, providing a simpler alternative to the exact calculation, which is particularly useful for larger values of n.', 'The relevance of n choose 3 is emphasized, indicating that n does not have to be very large for the Poisson paradigm to be appropriate, as even with some double-digit n, n choose 3 yields a reasonably large quantity.', 'The concept of indicator random variables for each triplet of people is explained, where the expected value of the number of triple matches can be determined using linearity and symmetry, providing insight into the calculation process.']}, {'end': 2564.768, 'start': 2326.865, 'title': 'Approximating triple birthday matches', 'summary': 'Discusses the use of poisson approximation to calculate the approximate probability of at least one triple birthday match, emphasizing the validity of the approximation based on the large number of possible matches, small probability of success for each trial, and weak dependence between trials.', 'duration': 237.903, 'highlights': ['The Poisson approximation is used to calculate the probability of at least one triple birthday match. The chapter focuses on using the Poisson approximation to determine the probability of at least one triple birthday match instead of the exact expected value.', 'Validity of the Poisson approximation is discussed based on the large number of possible matches, small probability of success for each trial, and weak dependence between trials. The validity of the Poisson approximation is emphasized by considering the large number of possible matches, small probability of success for each trial, and weak dependence between trials, justifying the use of the approximation.', 'Calculation of the approximate probability is simplified using 1 minus e to the power of minus lambda, where lambda represents the expected value. The calculation of the approximate probability is simplified using 1 minus e to the power of minus lambda, providing a convenient method for quick approximations without complex calculations.']}], 'duration': 559.565, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TD1N4hxqMzY/pics/TD1N4hxqMzY2005203.jpg', 'highlights': ['The Poisson approximation simplifies the calculation of the probability of three people sharing the same birthday, particularly useful for larger values of n.', 'The relevance of n choose 3 is emphasized, indicating that n does not have to be very large for the Poisson paradigm to be appropriate.', 'The concept of indicator random variables for each triplet of people is explained, providing insight into the calculation process.', 'The chapter focuses on using the Poisson approximation to determine the probability of at least one triple birthday match instead of the exact expected value.', 'Validity of the Poisson approximation is discussed based on the large number of possible matches, small probability of success for each trial, and weak dependence between trials.', 'Calculation of the approximate probability is simplified using 1 minus e to the power of minus lambda, providing a convenient method for quick approximations without complex calculations.']}], 'highlights': ['The Poisson distribution is highlighted as the most widely used distribution for modeling discrete data in real-world applications, particularly in scenarios where the count of occurrences with small probabilities, such as the number of emails received in an hour, needs to be modeled.', 'The Poisson distribution is validated as a probability mass function by ensuring that its non-negative values sum to 1, which is proven using the Taylor series for e to the lambda and the calculation of the expected value.', 'The Poisson paradigm is an approximation used in various scenarios, such as the number of chocolate chips in a cookie and the number of earthquakes in a year, where events are unlikely but can occur in numerous possible locations, making it a useful approximation in many problems.', 'The PMF converges as n goes to infinity and p goes to 0 while keeping the product constant, focusing on the probability that x equals k using n choose k, p to the k, and 1- p to the n- k.', 'The connection between binomial and Poisson distributions is illustrated through the example of the Poisson PMF, showing convergence at evaluated at k.', 'The Poisson approximation simplifies the calculation of the probability of three people sharing the same birthday, particularly useful for larger values of n.']}