title

Lecture 9: Expectation, Indicator Random Variables, Linearity | Statistics 110

description

We discuss expected values and the meaning of means, and introduce some very useful tools for finding expected values: indicator r.v.s, linearity, and symmetry. The fundamental bridge connects probability and expectation. We also introduce the Geometric distribution.

detail

{'title': 'Lecture 9: Expectation, Indicator Random Variables, Linearity | Statistics 110', 'heatmap': [{'end': 2124.745, 'start': 2083.244, 'weight': 0.803}, {'end': 2360.369, 'start': 2324.841, 'weight': 0.76}, {'end': 2726.452, 'start': 2690.072, 'weight': 0.707}, {'end': 3020.375, 'start': 2924.892, 'weight': 1}], 'summary': 'Covers random variable distributions, cdf, averages, and properties of random variables and independence, emphasizing the application of linearity in expectation for random variables and probability properties including indicator random variables with an example resulting in a probability of 5/13.', 'chapters': [{'end': 110.065, 'segs': [{'end': 28.865, 'src': 'embed', 'start': 0.009, 'weight': 0, 'content': [{'end': 2.51, 'text': "We've been talking about random variables and their distributions.", 'start': 0.009, 'duration': 2.501}, {'end': 6.473, 'text': 'And the main topic for today is averages.', 'start': 3.591, 'duration': 2.882}, {'end': 8.414, 'text': 'That is, how do you compute the average?', 'start': 7.073, 'duration': 1.341}, {'end': 13.237, 'text': 'First of all, what does an average mean and how do you compute the average of a random variable?', 'start': 8.534, 'duration': 4.703}, {'end': 18.52, 'text': 'Okay, but before we do that, I want to say a little bit more about CDF.', 'start': 14.557, 'duration': 3.963}, {'end': 24.803, 'text': 'I think it seems a little mysterious the first time you see it.', 'start': 21.121, 'duration': 3.682}, {'end': 28.865, 'text': 'You have this random variable and then you assign it a function.', 'start': 25.083, 'duration': 3.782}], 'summary': 'Discussing random variables, distributions, and average computation in the context of statistics.', 'duration': 28.856, 'max_score': 0.009, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU9.jpg'}, {'end': 110.065, 'src': 'embed', 'start': 53.824, 'weight': 1, 'content': [{'end': 54.985, 'text': "It doesn't have to be discrete.", 'start': 53.824, 'duration': 1.161}, {'end': 66.532, 'text': 'That is, we have a random variable capital, X, and we let f of x equal to probably x, less than or equal to little x as a function of little x,', 'start': 55.845, 'duration': 10.687}, {'end': 67.092, 'text': 'which is real.', 'start': 66.532, 'duration': 0.56}, {'end': 78.221, 'text': "Even if x only takes integer values, this is a function that's well defined for all real numbers x.", 'start': 71.172, 'duration': 7.049}, {'end': 80.283, 'text': "Okay, so that's the function, that's called the CDF.", 'start': 78.221, 'duration': 2.062}, {'end': 85.369, 'text': 'But I wanna talk more about what are the properties and how do you use this thing.', 'start': 80.784, 'duration': 4.585}, {'end': 87.232, 'text': 'So a picture.', 'start': 86.651, 'duration': 0.581}, {'end': 90.751, 'text': 'a discrete example.', 'start': 89.629, 'duration': 1.122}, {'end': 97.841, 'text': 'By the way, make sure you spell discrete, D-I-S-C-R-E-T-E.', 'start': 92.353, 'duration': 5.488}, {'end': 101.526, 'text': "It's a different word from R-E-E-T.", 'start': 98.502, 'duration': 3.024}, {'end': 106.922, 'text': "Don't confuse the two senses of discrete, otherwise it could be a little embarrassing.", 'start': 103.139, 'duration': 3.783}, {'end': 109.024, 'text': 'So this is a discrete random variable.', 'start': 107.362, 'duration': 1.662}, {'end': 110.065, 'text': "I'm gonna draw a CDF.", 'start': 109.084, 'duration': 0.981}], 'summary': 'Discussion of cdf for a random variable x and a discrete example.', 'duration': 56.241, 'max_score': 53.824, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU53824.jpg'}], 'start': 0.009, 'title': 'Random variable distributions and averages', 'summary': 'Delves into random variable distributions, cdf, and computation of averages, highlighting the significance of cdf in understanding distributions and emphasizing its practical application.', 'chapters': [{'end': 110.065, 'start': 0.009, 'title': 'Understanding random variable distributions and averages', 'summary': 'Discusses random variable distributions, cdf, and the computation of averages, emphasizing the importance and application of cdf in understanding random variables and their distributions.', 'duration': 110.056, 'highlights': ['Understanding the concept of CDF and its function associated with a random variable, explaining its properties and application in computing averages.', 'Explaining the relevance of CDF for both discrete and non-discrete random variables, emphasizing its well-defined function for all real numbers.', "Highlighting the importance of correctly spelling 'discrete' to avoid confusion, and illustrating the drawing of a CDF for a discrete random variable."]}], 'duration': 110.056, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU9.jpg', 'highlights': ['Understanding the concept of CDF and its function associated with a random variable, explaining its properties and application in computing averages.', 'Explaining the relevance of CDF for both discrete and non-discrete random variables, emphasizing its well-defined function for all real numbers.', "Highlighting the importance of correctly spelling 'discrete' to avoid confusion, and illustrating the drawing of a CDF for a discrete random variable."]}, {'end': 586, 'segs': [{'end': 140.081, 'src': 'embed', 'start': 110.265, 'weight': 2, 'content': [{'end': 111.846, 'text': "It's gonna look like a step function.", 'start': 110.265, 'duration': 1.581}, {'end': 114.388, 'text': "So it's gonna jump.", 'start': 113.348, 'duration': 1.04}, {'end': 121.536, 'text': "So let's draw one whose possible values are 0, 1, 2, or 3.", 'start': 114.669, 'duration': 6.867}, {'end': 127.918, 'text': 'We could have a discrete random variable where it could take on any non-negative integer value, so it could go off to infinity.', 'start': 121.536, 'duration': 6.382}, {'end': 131.699, 'text': "But right now I'm just looking at one that's 0, 1, 2, or 3.", 'start': 128.018, 'duration': 3.681}, {'end': 133.199, 'text': 'So it has four possible values.', 'start': 131.699, 'duration': 1.5}, {'end': 140.081, 'text': "So on the negative side, it's just 0 because it can't be negative.", 'start': 135.179, 'duration': 4.902}], 'summary': 'Describes a discrete random variable with 4 possible values.', 'duration': 29.816, 'max_score': 110.265, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU110265.jpg'}, {'end': 312.99, 'src': 'embed', 'start': 257.865, 'weight': 0, 'content': [{'end': 263.107, 'text': 'So the jump sizes are the PMF in this picture.', 'start': 257.865, 'duration': 5.242}, {'end': 265.778, 'text': 'What did we say about PMFs?', 'start': 264.417, 'duration': 1.361}, {'end': 268.981, 'text': 'We said they have to be non-negative and they add up to 1, right?', 'start': 265.858, 'duration': 3.123}, {'end': 274.367, 'text': "And if you add up all these jumps, that's just saying we're going from 0 up to 1..", 'start': 269.322, 'duration': 5.045}, {'end': 277.129, 'text': 'So from the CDF, we could recover the PMF.', 'start': 274.367, 'duration': 2.762}, {'end': 281.373, 'text': 'From the PMF, we could recover the CDF just by summing things up.', 'start': 277.59, 'duration': 3.783}, {'end': 283.095, 'text': "I'll talk a little bit more about that.", 'start': 281.774, 'duration': 1.321}, {'end': 285.317, 'text': 'But just for an example.', 'start': 284.016, 'duration': 1.301}, {'end': 301.888, 'text': "if we wanted to know, For example, if we took the probability that X is, suppose we wanna know what's the probability that X is between 1 and 3..", 'start': 285.317, 'duration': 16.571}, {'end': 306.288, 'text': 'I just made up numbers, just for example, but you can make this any A and B, okay?', 'start': 301.888, 'duration': 4.4}, {'end': 312.99, 'text': 'Well, one way to think about that would be to say that so suppose we wanna find this in terms of the CDF.', 'start': 306.609, 'duration': 6.381}], 'summary': 'Pmfs are non-negative, add up to 1. cdf helps recover pmf and vice versa. probabilities can be calculated using cdf.', 'duration': 55.125, 'max_score': 257.865, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU257865.jpg'}, {'end': 480.863, 'src': 'embed', 'start': 445.352, 'weight': 1, 'content': [{'end': 450.496, 'text': 'Okay, now, so in general, CDFs have three important properties.', 'start': 445.352, 'duration': 5.144}, {'end': 455.179, 'text': 'And we can see all three from the picture, basically.', 'start': 452.637, 'duration': 2.542}, {'end': 467.199, 'text': "First of all, it's increasing.", 'start': 464.958, 'duration': 2.241}, {'end': 475.941, 'text': "And I don't mean strictly increasing, so it's allowed to be flat like that, okay? But it can only go up, not down.", 'start': 470.2, 'duration': 5.741}, {'end': 480.863, 'text': "And to prove that that's true, all you have to do is look at the definition.", 'start': 476.862, 'duration': 4.001}], 'summary': 'Cdfs have three important properties: increasing, not strictly, and allowed to be flat.', 'duration': 35.511, 'max_score': 445.352, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU445352.jpg'}], 'start': 110.265, 'title': 'Discrete random variables and cdfs', 'summary': 'Illustrates a step function example of a discrete random variable with four possible values and discusses the properties of cumulative distribution functions (cdfs), emphasizing its three important properties: increasing, right continuous, and limits as x goes to infinity or minus infinity.', 'chapters': [{'end': 283.095, 'start': 110.265, 'title': 'Step function example - discrete random variable', 'summary': 'Illustrates a step function example of a discrete random variable with four possible values 0, 1, 2, or 3. it demonstrates how the jumps in the function represent the probability mass function (pmf) and the cumulative distribution function (cdf) and explains the relationship between them.', 'duration': 172.83, 'highlights': ['The function represents a discrete random variable with four possible values 0, 1, 2, or 3. The example showcases a discrete random variable with specific possible values, demonstrating its bounded nature.', 'The jumps in the function correspond to the probability mass function (PMF), where each jump represents the probability of the variable taking a specific value. The jumps in the function directly relate to the PMF, illustrating the probability of the variable taking on specific values.', 'The cumulative distribution function (CDF) can be derived from the PMF by summing up the jump sizes, and vice versa. The relationship between CDF and PMF is explained, highlighting how the CDF can be obtained from the PMF by summing the jump sizes, and vice versa.']}, {'end': 586, 'start': 284.016, 'title': 'Properties of cdfs', 'summary': 'Discusses the properties of cumulative distribution functions (cdfs) and demonstrates how the cdf can be used to compute the probability of an interval, emphasizing its three important properties: increasing, right continuous, and limits as x goes to infinity or minus infinity.', 'duration': 301.984, 'highlights': ['The CDF allows computation of the probability of an interval like f(b) - f(a), showcasing its versatility for discrete, continuous, or any type of random variable. The CDF allows computation of the probability of an interval like f(b) - f(a), showcasing its versatility for discrete, continuous, or any type of random variable.', 'The three important properties of CDFs include being increasing, right continuous, and having limits as x goes to infinity or minus infinity. The three important properties of CDFs include being increasing, right continuous, and having limits as x goes to infinity or minus infinity.', 'The chapter emphasizes that the CDF is a powerful tool for computing any probability for x and demonstrates its application to find the probability of x being between 1 and 3. The chapter emphasizes that the CDF is a powerful tool for computing any probability for x and demonstrates its application to find the probability of x being between 1 and 3.']}], 'duration': 475.735, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU110265.jpg', 'highlights': ['The CDF allows computation of the probability of an interval like f(b) - f(a), showcasing its versatility for discrete, continuous, or any type of random variable.', 'The three important properties of CDFs include being increasing, right continuous, and having limits as x goes to infinity or minus infinity.', 'The function represents a discrete random variable with four possible values 0, 1, 2, or 3. The example showcases a discrete random variable with specific possible values, demonstrating its bounded nature.', 'The cumulative distribution function (CDF) can be derived from the PMF by summing up the jump sizes, and vice versa. The relationship between CDF and PMF is explained, highlighting how the CDF can be obtained from the PMF by summing the jump sizes, and vice versa.']}, {'end': 976.109, 'segs': [{'end': 675.864, 'src': 'embed', 'start': 620.741, 'weight': 0, 'content': [{'end': 623.703, 'text': "Right now we're talking about the other direction, that we started with a random variable.", 'start': 620.741, 'duration': 2.962}, {'end': 629.926, 'text': 'And we said, what does the CDF looks like? Well, in the discrete case, it could look something like that.', 'start': 624.363, 'duration': 5.563}, {'end': 633.028, 'text': 'And in a continuous case, it would be a continuous curve.', 'start': 629.986, 'duration': 3.042}, {'end': 640.352, 'text': "But in any case, it's gonna satisfy these three things, okay? So, all right, those are properties of CDFs.", 'start': 633.088, 'duration': 7.264}, {'end': 645.414, 'text': 'One more definition is independence.', 'start': 643.173, 'duration': 2.241}, {'end': 655.138, 'text': "So we already know what independence of events means and we've been talking a little bit about independence of random variables, kind of intuitively.", 'start': 648.155, 'duration': 6.983}, {'end': 660.36, 'text': "but I wanna say what's the precise definition of independence of random variables?", 'start': 655.138, 'duration': 5.222}, {'end': 665.386, 'text': 'So we say X and Y are.', 'start': 663.883, 'duration': 1.503}, {'end': 667.71, 'text': 'this is just a definition.', 'start': 665.386, 'duration': 2.324}, {'end': 675.864, 'text': 'X and Y, which are both random variables, are independent if.', 'start': 667.71, 'duration': 8.154}], 'summary': 'Discussing properties of cdfs and precise definition of independence of random variables.', 'duration': 55.123, 'max_score': 620.741, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU620741.jpg'}, {'end': 959.24, 'src': 'embed', 'start': 933.469, 'weight': 4, 'content': [{'end': 938.773, 'text': "But beforehand, we may wanna make some predictions, right? We may wanna say on average what's gonna happen.", 'start': 933.469, 'duration': 5.304}, {'end': 943.288, 'text': "Okay, so that's one reason, average is a very, very familiar concept.", 'start': 939.485, 'duration': 3.803}, {'end': 950.233, 'text': 'But actually the importance goes beyond that too, because you could just say well,', 'start': 945.229, 'duration': 5.004}, {'end': 955.998, 'text': 'the average is only just telling you a one number summary of the center of the distribution in some sense.', 'start': 950.233, 'duration': 5.765}, {'end': 959.24, 'text': "And that's important, but still, that's just a one number summary.", 'start': 956.318, 'duration': 2.922}], 'summary': 'Average is a familiar concept but provides only a one number summary of the distribution center.', 'duration': 25.771, 'max_score': 933.469, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU933469.jpg'}], 'start': 591.142, 'title': 'Properties and independence', 'summary': 'Discusses the properties of cumulative distribution functions, emphasizing their significance in determining a valid cdf and visual representation, and also introduces the concept of independence of random variables and averages, defining their importance in predicting outcomes.', 'chapters': [{'end': 640.352, 'start': 591.142, 'title': 'Properties of cumulative distribution functions', 'summary': 'Discusses the properties of cumulative distribution functions, highlighting the three key characteristics and emphasizing their significance in determining a valid cdf and the visual representation in discrete and continuous cases.', 'duration': 49.21, 'highlights': ['The three key characteristics of a valid Cumulative Distribution Function (CDF) are discussed, emphasizing their importance in determining a valid CDF.', 'The visual representation of a CDF in the discrete case is described as a potential step function, while in the continuous case, it is illustrated as a continuous curve.', 'The discussion emphasizes the significance of the three key characteristics in determining a valid CDF, providing clarity on its properties and functions.']}, {'end': 976.109, 'start': 643.173, 'title': 'Independence of random variables & averages', 'summary': 'Introduces the concept of independence of random variables, defining it as the joint probability of two random variables being equal to the product of their individual probabilities, and also covers the concept of averages and their importance in summarizing the center of a distribution and predicting outcomes.', 'duration': 332.936, 'highlights': ['Independence of random variables is defined as the joint probability of two random variables being equal to the product of their individual probabilities, with the equation being easier to work with in the discrete case. This definition explains that for two random variables X and Y to be independent, the joint probability of X and Y being less than or equal to specific values is equal to the product of their individual probabilities. In the discrete case, this equation is easier to work with and can be expressed using the joint and individual probability mass functions (PMFs).', 'The concept of averages is introduced, emphasizing its importance in summarizing the center of a distribution and predicting outcomes. The importance of averages extends to making predictions about the expected outcome of a random variable and providing a one-number summary of the center of a distribution. Additionally, averages are essential for measuring the spread of a distribution, leading to discussions about variance and standard deviation.']}], 'duration': 384.967, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU591142.jpg', 'highlights': ['The three key characteristics of a valid Cumulative Distribution Function (CDF) are discussed, emphasizing their importance in determining a valid CDF.', 'The visual representation of a CDF in the discrete case is described as a potential step function, while in the continuous case, it is illustrated as a continuous curve.', 'The discussion emphasizes the significance of the three key characteristics in determining a valid CDF, providing clarity on its properties and functions.', 'Independence of random variables is defined as the joint probability of two random variables being equal to the product of their individual probabilities, with the equation being easier to work with in the discrete case.', 'The concept of averages is introduced, emphasizing its importance in summarizing the center of a distribution and predicting outcomes. The importance of averages extends to making predictions about the expected outcome of a random variable and providing a one-number summary of the center of a distribution.']}, {'end': 2019.398, 'segs': [{'end': 1032.943, 'src': 'embed', 'start': 1000.119, 'weight': 3, 'content': [{'end': 1007.144, 'text': '1 plus 2 plus 3 plus 4 plus 5 plus 6 divided by 6 equals 3 and a half.', 'start': 1000.119, 'duration': 7.025}, {'end': 1015.11, 'text': "Okay, well, that's the starting point of averages, very, very simple.", 'start': 1011.327, 'duration': 3.783}, {'end': 1021.079, 'text': "And by the way, does everyone know that you can, if you wanna do this, you don't actually have to add up all the numbers.", 'start': 1016.278, 'duration': 4.801}, {'end': 1029.662, 'text': 'You can just add the first one and the last one and average those.', 'start': 1021.099, 'duration': 8.563}, {'end': 1032.943, 'text': 'That is the average of one and six is three and a half.', 'start': 1030.021, 'duration': 2.922}], 'summary': 'Average of 1 to 6 can be calculated using first and last numbers, yielding 3.5.', 'duration': 32.824, 'max_score': 1000.119, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU1000119.jpg'}, {'end': 1127.094, 'src': 'embed', 'start': 1073.214, 'weight': 0, 'content': [{'end': 1082.777, 'text': 'And then so Gauss saw these numbers and he said, well, Gauss immediately saw, well, you can pair the 1 and the 100 and the 2 and the 99.', 'start': 1073.214, 'duration': 9.563}, {'end': 1084.097, 'text': '1 and 100 is 101, 2 and 99 is 101, 3 and 98 is 101.', 'start': 1082.777, 'duration': 1.32}, {'end': 1088.119, 'text': "So all the pairs add up to 101, so he just immediately wrote down 101 and there's 50 pairs equals 50, 50.", 'start': 1084.097, 'duration': 4.022}, {'end': 1089.58, 'text': 'And just immediately wrote down the answer.', 'start': 1088.119, 'duration': 1.461}, {'end': 1107.368, 'text': "So that's just an example of of what I was just saying about an arithmetic series.", 'start': 1089.62, 'duration': 17.748}, {'end': 1112.513, 'text': "So it's useful in this class, but it's just useful in general.", 'start': 1107.808, 'duration': 4.705}, {'end': 1123.451, 'text': "Everyone should know that if you want to add up the numbers from 1 to n, If I wanna average the numbers from 1 to n, that's n plus 1 over 2,", 'start': 1113.134, 'duration': 10.317}, {'end': 1124.392, 'text': 'which is what I just said.', 'start': 1123.451, 'duration': 0.941}, {'end': 1127.094, 'text': "It's the same thing as averaging the first number and the last number.", 'start': 1124.432, 'duration': 2.662}], 'summary': 'Gauss paired numbers to sum to 101, 50 pairs equals 50, 50, demonstrating an arithmetic series.', 'duration': 53.88, 'max_score': 1073.214, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU1073214.jpg'}, {'end': 1276.703, 'src': 'embed', 'start': 1243.673, 'weight': 6, 'content': [{'end': 1252.101, 'text': "So I would do a weighted average, which would be, I'm giving this term weight 5 eighths, cuz there's five ones.", 'start': 1243.673, 'duration': 8.428}, {'end': 1261.146, 'text': "And I'm giving the 3s weight 2 eighths, and I'm giving the 5 weight 1 eighth.", 'start': 1254.983, 'duration': 6.163}, {'end': 1266.088, 'text': 'And that would be the same right?', 'start': 1265.027, 'duration': 1.061}, {'end': 1268.921, 'text': "It's just simple arithmetic,", 'start': 1267.381, 'duration': 1.54}, {'end': 1276.703, 'text': 'but the key point is that I can take the ungrouped average or I can group them according to common values and then average those.', 'start': 1268.921, 'duration': 7.782}], 'summary': 'Weighted average calculated with term weights: 5/8 for 1s, 2/8 for 3s, and 1/8 for 5s, illustrating grouping and averaging.', 'duration': 33.03, 'max_score': 1243.673, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU1243673.jpg'}, {'end': 1374.6, 'src': 'embed', 'start': 1349.378, 'weight': 1, 'content': [{'end': 1354.517, 'text': 'Standard notation in statistics is to write E E means expected value.', 'start': 1349.378, 'duration': 5.139}, {'end': 1365.856, 'text': "E of x equals, and now I'm just gonna use this same idea that we're summing up values times weights.", 'start': 1356.753, 'duration': 9.103}, {'end': 1369.918, 'text': 'And the obvious weight to use would be the probabilities right?', 'start': 1366.337, 'duration': 3.581}, {'end': 1374.6, 'text': 'So, if I wanna know, the average of x and x has different possible values.', 'start': 1370.338, 'duration': 4.262}], 'summary': 'Standard notation in statistics: e is the expected value, calculated as the sum of values times probabilities.', 'duration': 25.222, 'max_score': 1349.378, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU1349378.jpg'}, {'end': 1500.96, 'src': 'embed', 'start': 1473.593, 'weight': 7, 'content': [{'end': 1477.074, 'text': 'Bernoulli p means it can only equal 0 or 1.', 'start': 1473.593, 'duration': 3.481}, {'end': 1478.574, 'text': "So that's very, very easy to deal with.", 'start': 1477.074, 'duration': 1.5}, {'end': 1480.335, 'text': 'So we can do this one very quickly.', 'start': 1478.614, 'duration': 1.721}, {'end': 1485.385, 'text': "Well, let's just write it down.", 'start': 1483.523, 'duration': 1.862}, {'end': 1496.255, 'text': 'Expected value of X equals 1 times the probability that X equals 1, plus 0 times the probability that X equals 0, but that term is 0 anyway.', 'start': 1485.745, 'duration': 10.51}, {'end': 1500.96, 'text': "And that's just equal to p by definition.", 'start': 1499.258, 'duration': 1.702}], 'summary': 'Bernoulli p equals the expected value of x, making calculations quick and simple.', 'duration': 27.367, 'max_score': 1473.593, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU1473593.jpg'}, {'end': 1992.399, 'src': 'embed', 'start': 1963.011, 'weight': 2, 'content': [{'end': 1966.392, 'text': "So we're gonna use linearity over and over and over again.", 'start': 1963.011, 'duration': 3.381}, {'end': 1970.914, 'text': 'The single most useful thing about expectation.', 'start': 1968.313, 'duration': 2.601}, {'end': 1979.917, 'text': 'Linearity says that the expected value of x plus y, you can just add them up.', 'start': 1972.454, 'duration': 7.463}, {'end': 1986.716, 'text': 'And this is always true, even if X and Y are dependent.', 'start': 1982.113, 'duration': 4.603}, {'end': 1992.399, 'text': 'I think this is kind of intuitively seems fairly obvious if X and Y are independent.', 'start': 1987.356, 'duration': 5.043}], 'summary': 'Using linearity multiple times, the expected value can be easily added even when x and y are dependent.', 'duration': 29.388, 'max_score': 1963.011, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU1963011.jpg'}], 'start': 976.249, 'title': 'Finding averages, arithmetic series, and random variables', 'summary': 'Covers finding averages of numbers, introducing arithmetic series and weighted averages, and computing the average of discrete random variables. it demonstrates averaging methods and properties, including the calculation of the average of a set of numbers as 3.5 and the application of linearity in expectation for random variables.', 'chapters': [{'end': 1032.943, 'start': 976.249, 'title': 'Finding averages of numbers', 'summary': 'Introduces the concept of finding averages of numbers and demonstrates the calculation of the average of a set of numbers as 3.5 by adding them up and dividing by the count. additionally, it highlights the shortcut method of averaging by adding the first and last numbers and dividing by 2.', 'duration': 56.694, 'highlights': ['The average of a set of numbers 1, 2, 3, 4, 5, 6 is calculated as 3.5 by adding them up and dividing by 6.', 'An alternative method for finding the average is demonstrated by adding the first and last numbers (1 and 6) and dividing by 2 to yield the average as 3.5.']}, {'end': 1320.988, 'start': 1033.742, 'title': 'Arithmetic series and weighted averages', 'summary': 'Explains the concept of arithmetic series using the story of gauss, the formula for summing numbers from 1 to n, and introduces the concept of weighted averages with examples demonstrating the application of the concept.', 'duration': 287.246, 'highlights': ["The story of Gauss at age ten and the numbers from one to 100, where Gauss immediately realized that the sum of numbers from 1 to 100 is 5050 by pairing the numbers and applying the concept of arithmetic series. Gauss's method of quickly summing numbers from 1 to 100 by pairing the numbers and recognizing the pattern, leading to the sum of 5050.", 'The formula for summing numbers from 1 to n, n + 1 over 2, is introduced and explained using the concept of averaging the first and last numbers in the sequence. Introduction and explanation of the formula for summing numbers from 1 to n, n + 1 over 2, by relating it to averaging the first and last numbers in the sequence.', 'Explaining the concept of weighted averages using an example of averaging numbers with repetitions, where the weights are assigned based on the frequency of each number in the sequence. Introduction of the concept of weighted averages using an example of averaging numbers with repetitions and assigning weights based on the frequency of each number in the sequence.']}, {'end': 2019.398, 'start': 1321.328, 'title': 'Average of discrete random variable', 'summary': 'Discusses the computation of expected value for discrete random variables, using the sum of values multiplied by their respective probabilities. it elaborates on the calculation for bernoulli and binomial random variables and emphasizes the application of linearity in expectation as the most significant property for computation.', 'duration': 698.07, 'highlights': ['The expected value of a discrete random variable x is calculated using the sum of values multiplied by their respective probabilities, denoted as E(X) = ∑x * P(X=x), with a focus on assigning higher weights to more likely values and lower weights to unlikely values.', 'The expected value of a Bernoulli random variable with probability p is given by E(X) = 1 * p + 0 * (1 - p) = p, representing a fundamental equation that bridges between expected values and probabilities, enabling the reinterpretation of probabilities as expected values of indicator random variables.', 'The application of linearity in expectation, which allows for the addition of expected values of dependent variables, is emphasized as the single most important property for computation, providing a more efficient approach for calculating the expected value of binomial random variables.']}], 'duration': 1043.149, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU976249.jpg', 'highlights': ['The formula for summing numbers from 1 to n, n + 1 over 2, is introduced and explained using the concept of averaging the first and last numbers in the sequence.', 'The expected value of a discrete random variable x is calculated using the sum of values multiplied by their respective probabilities, denoted as E(X) = ∑x * P(X=x), with a focus on assigning higher weights to more likely values and lower weights to unlikely values.', 'The application of linearity in expectation, which allows for the addition of expected values of dependent variables, is emphasized as the single most important property for computation, providing a more efficient approach for calculating the expected value of binomial random variables.', 'The average of a set of numbers 1, 2, 3, 4, 5, 6 is calculated as 3.5 by adding them up and dividing by 6.', 'An alternative method for finding the average is demonstrated by adding the first and last numbers (1 and 6) and dividing by 2 to yield the average as 3.5.', "The story of Gauss at age ten and the numbers from one to 100, where Gauss immediately realized that the sum of numbers from 1 to 100 is 5050 by pairing the numbers and applying the concept of arithmetic series. Gauss's method of quickly summing numbers from 1 to 100 by pairing the numbers and recognizing the pattern, leading to the sum of 5050.", 'Introduction of the concept of weighted averages using an example of averaging numbers with repetitions and assigning weights based on the frequency of each number in the sequence.', 'The expected value of a Bernoulli random variable with probability p is given by E(X) = 1 * p + 0 * (1 - p) = p, representing a fundamental equation that bridges between expected values and probabilities, enabling the reinterpretation of probabilities as expected values of indicator random variables.']}, {'end': 2397.373, 'segs': [{'end': 2045.714, 'src': 'embed', 'start': 2019.458, 'weight': 0, 'content': [{'end': 2023.579, 'text': 'One other thing about linearity, linearity is normally stated as two things.', 'start': 2019.458, 'duration': 4.121}, {'end': 2026.9, 'text': 'One is this, and the other one is that we can take out constants.', 'start': 2023.739, 'duration': 3.161}, {'end': 2029.62, 'text': 'And this one is kind of more obvious, but also useful.', 'start': 2027.34, 'duration': 2.28}, {'end': 2039.813, 'text': 'So if C is a constant, If C is a constant, we can just take it out, okay? But this is the most interesting one.', 'start': 2030.101, 'duration': 9.712}, {'end': 2042.473, 'text': 'Expected value of the sum is the sum of the expected values.', 'start': 2039.933, 'duration': 2.54}, {'end': 2045.714, 'text': "All right, so now let's redo this binomial thing.", 'start': 2043.253, 'duration': 2.461}], 'summary': 'Linearity in statistics allows for constant extraction and sums of expected values.', 'duration': 26.256, 'max_score': 2019.458, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2019458.jpg'}, {'end': 2124.745, 'src': 'heatmap', 'start': 2068.476, 'weight': 1, 'content': [{'end': 2070.038, 'text': "Let's redo that a better way.", 'start': 2068.476, 'duration': 1.562}, {'end': 2072.36, 'text': 'Redo the binomial.', 'start': 2071.279, 'duration': 1.081}, {'end': 2082.503, 'text': "Well, I kind of didn't leave much space here, but we don't actually need much space, because all we have to do is think about linearity.", 'start': 2074.639, 'duration': 7.864}, {'end': 2091.348, 'text': "Remember the binomial, np, we can think of it as the sum of n iid Bernoulli p's.", 'start': 2083.244, 'duration': 8.104}, {'end': 2103.255, 'text': "Each of those Bernoulli p's has expected value p, and there's n of them, so it's n times p, by linearity.", 'start': 2092.049, 'duration': 11.206}, {'end': 2110.677, 'text': "So I don't know about you, but I prefer this method to this method.", 'start': 2105.794, 'duration': 4.883}, {'end': 2114.999, 'text': "So that's a calculation you can all do in your head.", 'start': 2111.938, 'duration': 3.061}, {'end': 2121.263, 'text': "Each Bernoulli has expected value p, and there's n of them, so it's np.", 'start': 2115.78, 'duration': 5.483}, {'end': 2124.745, 'text': "I'll write that out as well.", 'start': 2122.844, 'duration': 1.901}], 'summary': "Binomial np can be calculated as the sum of n iid bernoulli p's, with expected value n*p.", 'duration': 34.779, 'max_score': 2068.476, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2068476.jpg'}, {'end': 2217.968, 'src': 'embed', 'start': 2189.244, 'weight': 2, 'content': [{'end': 2197.771, 'text': 'Well, the hypergeometric, remember it had that PMF involving these binomial coefficients and this complicated looking thing.', 'start': 2189.244, 'duration': 8.527}, {'end': 2200.814, 'text': 'It would be kind of a pain to try to do that.', 'start': 2197.791, 'duration': 3.023}, {'end': 2202.815, 'text': 'A hypergeometric looks pretty complicated.', 'start': 2200.854, 'duration': 1.961}, {'end': 2208.8, 'text': "But instead, let's just think in terms of indicator random variables.", 'start': 2203.616, 'duration': 5.184}, {'end': 2217.968, 'text': "So let's let Xj be indicator of jth card being an ace.", 'start': 2209.461, 'duration': 8.507}], 'summary': 'Hypergeometric pmf involving binomial coefficients. using indicator random variables to simplify.', 'duration': 28.724, 'max_score': 2189.244, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2189244.jpg'}, {'end': 2305.543, 'src': 'embed', 'start': 2270.583, 'weight': 4, 'content': [{'end': 2273.824, 'text': "That's just how we count, right? We want to count to three, you go one, two, three.", 'start': 2270.583, 'duration': 3.241}, {'end': 2276.745, 'text': 'You just add one each time you want to increment.', 'start': 2273.844, 'duration': 2.901}, {'end': 2282.526, 'text': "By linearity, that's just the sum of these.", 'start': 2277.765, 'duration': 4.761}, {'end': 2288.608, 'text': "That's linearity.", 'start': 2282.547, 'duration': 6.061}, {'end': 2291.709, 'text': 'By symmetry, so this is linearity.', 'start': 2289.388, 'duration': 2.321}, {'end': 2296.776, 'text': 'This step is indicator random variables.', 'start': 2294.554, 'duration': 2.222}, {'end': 2305.543, 'text': "And then by symmetry, that's really just the same thing five times, symmetry.", 'start': 2300.038, 'duration': 5.505}], 'summary': 'Counting method explained using linearity and symmetry, incrementing by one each time.', 'duration': 34.96, 'max_score': 2270.583, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2270583.jpg'}, {'end': 2371.433, 'src': 'heatmap', 'start': 2324.841, 'weight': 3, 'content': [{'end': 2329.887, 'text': 'E equals the same as the probability that that card is an ace.', 'start': 2324.841, 'duration': 5.046}, {'end': 2335.473, 'text': 'So this is equal to five times the probability that the first card is an ace.', 'start': 2330.608, 'duration': 4.865}, {'end': 2339.658, 'text': "That's the fundamental bridge.", 'start': 2338.476, 'duration': 1.182}, {'end': 2347.502, 'text': 'Probably that the first card is an ace, well, 4 over 52 or 1 13th, so this is just 5 13ths.', 'start': 2343.891, 'duration': 3.611}, {'end': 2357.528, 'text': 'Okay, so I wrote this out in a lot of detail just so you could see exactly what steps were being used.', 'start': 2353.046, 'duration': 4.482}, {'end': 2360.369, 'text': 'But this is also one that you could do in your head right?', 'start': 2357.748, 'duration': 2.621}, {'end': 2363.09, 'text': "It's just saying I have five indicator random variables.", 'start': 2360.389, 'duration': 2.701}, {'end': 2365.411, 'text': "each one is 1, 13th, so it's 5, 13ths.", 'start': 2363.09, 'duration': 2.321}, {'end': 2371.433, 'text': "So it's actually completely analogous to this, where it's just something you can immediately do in your head.", 'start': 2365.951, 'duration': 5.482}], 'summary': 'Probability of drawing an ace is 5/13th, using indicator random variables.', 'duration': 46.592, 'max_score': 2324.841, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2324841.jpg'}], 'start': 2019.458, 'title': 'Probability properties', 'summary': 'Discusses the linearity property in probability, emphasizing how constants can be taken out and the expected value of the sum is the sum of the expected values. it also demonstrates the application in simplifying calculations for binomial and hypergeometric distributions. additionally, it explains the concept of indicator random variables and their application in calculating the probability of drawing a specific card from a shuffled deck, resulting in a probability of 5/13.', 'chapters': [{'end': 2217.968, 'start': 2019.458, 'title': 'Linearity property in probability', 'summary': 'Discusses the linearity property in probability, highlighting how constants can be taken out and the expected value of the sum is the sum of the expected values. it also demonstrates the application of linearity in simplifying calculations for binomial and hypergeometric distributions.', 'duration': 198.51, 'highlights': ['The expected value of the sum is the sum of the expected values The chapter emphasizes the property that the expected value of the sum of random variables equals the sum of their individual expected values, showcasing the application of linearity in probability calculations.', "Application of linearity in simplifying binomial distribution calculations It demonstrates how linearity simplifies the calculation for the binomial distribution by expressing it as the sum of n iid Bernoulli p's, showcasing the efficiency of applying the linearity property.", 'Simplified approach using indicator random variables for hypergeometric distribution The chapter illustrates a simplified approach for calculating the expected value of a hypergeometric distribution by using indicator random variables, offering a more efficient method compared to the traditional PMF involving binomial coefficients.']}, {'end': 2397.373, 'start': 2220.137, 'title': 'Card counting probability', 'summary': 'Explains the concept of indicator random variables and their application in calculating the probability of drawing a specific card from a shuffled deck, with the final step resulting in a probability of 5/13.', 'duration': 177.236, 'highlights': ['The concept of indicator random variables is applied to calculate the probability of drawing a specific card from a shuffled deck, resulting in a probability of 5/13.', 'The explanation emphasizes the symmetry and linearity of the situation, enabling a straightforward calculation of the probability.', 'The use of indicator random variables and linearity is highlighted as a powerful strategy in probability calculations.']}], 'duration': 377.915, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2019458.jpg', 'highlights': ['The expected value of the sum is the sum of the expected values. It emphasizes the property that the expected value of the sum of random variables equals the sum of their individual expected values, showcasing the application of linearity in probability calculations.', "Application of linearity in simplifying binomial distribution calculations. It demonstrates how linearity simplifies the calculation for the binomial distribution by expressing it as the sum of n iid Bernoulli p's, showcasing the efficiency of applying the linearity property.", 'Simplified approach using indicator random variables for hypergeometric distribution. The chapter illustrates a simplified approach for calculating the expected value of a hypergeometric distribution by using indicator random variables, offering a more efficient method compared to the traditional PMF involving binomial coefficients.', 'The concept of indicator random variables is applied to calculate the probability of drawing a specific card from a shuffled deck, resulting in a probability of 5/13. The explanation emphasizes the symmetry and linearity of the situation, enabling a straightforward calculation of the probability.', 'The use of indicator random variables and linearity is highlighted as a powerful strategy in probability calculations.']}, {'end': 3021.156, 'segs': [{'end': 2454.642, 'src': 'embed', 'start': 2428.339, 'weight': 0, 'content': [{'end': 2436.595, 'text': 'And even though in the hypergeometric the trials are dependent, What this says is that for the expected value, it still looks as if it were binomial,', 'start': 2428.339, 'duration': 8.256}, {'end': 2438.556, 'text': "even though it's not for the expected value.", 'start': 2436.595, 'duration': 1.961}, {'end': 2441.197, 'text': "For other things we'll do later, we'll see differences with hypergeometric.", 'start': 2438.596, 'duration': 2.601}, {'end': 2447.499, 'text': "But just for the expected value, it's just n times the probability that an individual one has whatever property.", 'start': 2441.557, 'duration': 5.942}, {'end': 2454.642, 'text': "Okay, so we'll do more examples with indicators and stuff next time.", 'start': 2449.06, 'duration': 5.582}], 'summary': 'In hypergeometric trials, the expected value appears binomial, despite being dependent, n times the individual probability.', 'duration': 26.303, 'max_score': 2428.339, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2428339.jpg'}, {'end': 2512.261, 'src': 'embed', 'start': 2481.735, 'weight': 1, 'content': [{'end': 2487.897, 'text': "As with the binomial, we're assuming that we have Bernoulli trials, independent Bernoulli trials.", 'start': 2481.735, 'duration': 6.162}, {'end': 2496.577, 'text': "for example, flipping a coin, but just anything where you're repeating the same experiment over and over again with independent Bernoulli trials.", 'start': 2489.53, 'duration': 7.047}, {'end': 2499.7, 'text': 'Each trial has the same probability p of success.', 'start': 2497.298, 'duration': 2.402}, {'end': 2512.261, 'text': "And the story of the geometric is that it's the first, It's how many failures before the first success.", 'start': 2501.402, 'duration': 10.859}], 'summary': 'Geometric distribution models number of failures before first success in independent bernoulli trials.', 'duration': 30.526, 'max_score': 2481.735, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2481735.jpg'}, {'end': 2726.452, 'src': 'heatmap', 'start': 2690.072, 'weight': 0.707, 'content': [{'end': 2693.555, 'text': "1 minus q is p, so that's just 1, and that's what we wanted.", 'start': 2690.072, 'duration': 3.483}, {'end': 2696.918, 'text': "So that's why this is called the geometric distribution.", 'start': 2694.616, 'duration': 2.302}, {'end': 2699.5, 'text': "It's because this thing is a geometric series.", 'start': 2696.998, 'duration': 2.502}, {'end': 2705.766, 'text': 'Just like for the binomial distribution, you get the binomial theorem that shows that the PMF is valid.', 'start': 2700.241, 'duration': 5.525}, {'end': 2707.988, 'text': "All right, so now let's do the expected value.", 'start': 2706.406, 'duration': 1.582}, {'end': 2716.107, 'text': "So again, there's more than one way to do this.", 'start': 2713.266, 'duration': 2.841}, {'end': 2726.452, 'text': "So we're letting x be geometric p, and we're gonna try to compute the expected value of x, first of all by the definition.", 'start': 2717.888, 'duration': 8.564}], 'summary': 'The geometric distribution is characterized by a geometric series and has valid pmf; the expected value can be computed in multiple ways.', 'duration': 36.38, 'max_score': 2690.072, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2690072.jpg'}, {'end': 2726.452, 'src': 'embed', 'start': 2696.998, 'weight': 2, 'content': [{'end': 2699.5, 'text': "It's because this thing is a geometric series.", 'start': 2696.998, 'duration': 2.502}, {'end': 2705.766, 'text': 'Just like for the binomial distribution, you get the binomial theorem that shows that the PMF is valid.', 'start': 2700.241, 'duration': 5.525}, {'end': 2707.988, 'text': "All right, so now let's do the expected value.", 'start': 2706.406, 'duration': 1.582}, {'end': 2716.107, 'text': "So again, there's more than one way to do this.", 'start': 2713.266, 'duration': 2.841}, {'end': 2726.452, 'text': "So we're letting x be geometric p, and we're gonna try to compute the expected value of x, first of all by the definition.", 'start': 2717.888, 'duration': 8.564}], 'summary': 'The lecture covers geometric series, binomial distribution, and expected value computation.', 'duration': 29.454, 'max_score': 2696.998, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2696998.jpg'}, {'end': 3020.375, 'src': 'heatmap', 'start': 2924.892, 'weight': 1, 'content': [{'end': 2933.479, 'text': "Well, let's just think about it in terms of flipping a coin with probability p of heads over and over again until the coin lands heads for the first time.", 'start': 2924.892, 'duration': 8.587}, {'end': 2934.64, 'text': 'Count the number of failures.', 'start': 2933.619, 'duration': 1.021}, {'end': 2936.421, 'text': 'So you want to solve for c.', 'start': 2935.06, 'duration': 1.361}, {'end': 2937.382, 'text': "Well, there's two cases.", 'start': 2936.421, 'duration': 0.961}, {'end': 2941.525, 'text': "This is very similar to the first step analysis, like the gambler's ruin.", 'start': 2938.023, 'duration': 3.502}, {'end': 2945.348, 'text': 'Either the first coin flip is heads.', 'start': 2942.146, 'duration': 3.202}, {'end': 2954.784, 'text': "In that case, x is 0 because we had no failures, right? So in that case, it's 0, and that case happens with probability p.", 'start': 2946.129, 'duration': 8.655}, {'end': 2956.265, 'text': 'That means success the first time.', 'start': 2954.784, 'duration': 1.481}, {'end': 2968.031, 'text': 'The other case is failure the first time, okay? Which happens with probability q.', 'start': 2956.845, 'duration': 11.186}, {'end': 2977.496, 'text': "Now, if it's failure the first time We have that one failure, but then notice it's the same problem again.", 'start': 2968.031, 'duration': 9.465}, {'end': 2981.42, 'text': "right?. The coin is memoryless, the coin is not out to get you, the coin's not trying to help you.", 'start': 2977.496, 'duration': 3.924}, {'end': 2987.146, 'text': "It's just a coin, one failure, then the problem restarted.", 'start': 2981.44, 'duration': 5.706}, {'end': 2993.933, 'text': "So it's the same problem again, so it's plus C again.", 'start': 2988.047, 'duration': 5.886}, {'end': 3003.643, 'text': "Okay, so what's that? That's just q plus cq, solve for c, 1 minus q is p.", 'start': 2994.696, 'duration': 8.947}, {'end': 3007.165, 'text': 'So if we solve that equation, we get q over p.', 'start': 3003.643, 'duration': 3.522}, {'end': 3010.007, 'text': "So this way, I like more because we don't need to use any calculus.", 'start': 3007.165, 'duration': 2.842}, {'end': 3013.95, 'text': 'We just write down what does it mean to be geometric in the story.', 'start': 3010.027, 'duration': 3.923}, {'end': 3018.414, 'text': 'This way we use a little calculus, but either way we get q over p.', 'start': 3014.111, 'duration': 4.303}, {'end': 3020.375, 'text': "Okay, so that's all for today.", 'start': 3018.414, 'duration': 1.961}], 'summary': 'In a coin flip scenario, the average number of failures before the first success is q over p.', 'duration': 95.483, 'max_score': 2924.892, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2924892.jpg'}], 'start': 2397.433, 'title': 'Geometric distribution', 'summary': 'Discusses the expected value of a hypergeometric distribution and its resemblance to a binomial distribution, introduces the geometric distribution with parameter p, explains the geometric distribution as the number of failures before the first success, and derives its expected value resulting in an expected value of q/p.', 'chapters': [{'end': 2481.735, 'start': 2397.433, 'title': 'Expected value and geometric distribution', 'summary': 'Discusses the expected value of a hypergeometric distribution, highlighting that it resembles a binomial distribution for the expected value, and introduces the geometric distribution with parameter p.', 'duration': 84.302, 'highlights': ['The expected value of a hypergeometric distribution resembles a binomial distribution despite the trials being dependent, calculated as n times the probability that an individual one has a specific property.', 'Introduction of the geometric distribution with parameter p, distinct from the hypergeometric distribution.']}, {'end': 3021.156, 'start': 2481.735, 'title': 'Geometric distribution', 'summary': 'Explains the geometric distribution, defining it as the number of failures before the first success in a series of independent bernoulli trials with the same probability of success, illustrating its probability mass function (pmf), and deriving its expected value using both a calculus-based method and a story-based method, ultimately resulting in an expected value of q/p.', 'duration': 539.421, 'highlights': ['The geometric distribution is defined as the number of failures before the first success in a series of independent Bernoulli trials with the same probability of success.', 'The probability mass function (PMF) of the geometric distribution is given by pq^(k), where k goes from 0 to infinity, and the sum of the PMF equals 1, validating it as a valid PMF.', 'The expected value of the geometric distribution, denoted as E(X), is derived as q/p using both a calculus-based method and a story-based method, demonstrating the versatility of approaches in solving for the expected value.']}], 'duration': 623.723, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/LX2q356N2rU/pics/LX2q356N2rU2397433.jpg', 'highlights': ['The expected value of a hypergeometric distribution resembles a binomial distribution despite the trials being dependent, calculated as n times the probability that an individual one has a specific property.', 'The geometric distribution is defined as the number of failures before the first success in a series of independent Bernoulli trials with the same probability of success.', 'The expected value of the geometric distribution, denoted as E(X), is derived as q/p using both a calculus-based method and a story-based method, demonstrating the versatility of approaches in solving for the expected value.']}], 'highlights': ['The CDF allows computation of the probability of an interval like f(b) - f(a), showcasing its versatility for discrete, continuous, or any type of random variable.', 'The three important properties of CDFs include being increasing, right continuous, and having limits as x goes to infinity or minus infinity.', 'The function represents a discrete random variable with four possible values 0, 1, 2, or 3. The example showcases a discrete random variable with specific possible values, demonstrating its bounded nature.', 'The cumulative distribution function (CDF) can be derived from the PMF by summing up the jump sizes, and vice versa. The relationship between CDF and PMF is explained, highlighting how the CDF can be obtained from the PMF by summing the jump sizes, and vice versa.', 'Independence of random variables is defined as the joint probability of two random variables being equal to the product of their individual probabilities, with the equation being easier to work with in the discrete case.', 'The concept of averages is introduced, emphasizing its importance in summarizing the center of a distribution and predicting outcomes. The importance of averages extends to making predictions about the expected outcome of a random variable and providing a one-number summary of the center of a distribution.', 'The expected value of a discrete random variable x is calculated using the sum of values multiplied by their respective probabilities, denoted as E(X) = ∑x * P(X=x), with a focus on assigning higher weights to more likely values and lower weights to unlikely values.', 'The application of linearity in expectation, which allows for the addition of expected values of dependent variables, is emphasized as the single most important property for computation, providing a more efficient approach for calculating the expected value of binomial random variables.', 'The expected value of the sum is the sum of the expected values. It emphasizes the property that the expected value of the sum of random variables equals the sum of their individual expected values, showcasing the application of linearity in probability calculations.', "Application of linearity in simplifying binomial distribution calculations. It demonstrates how linearity simplifies the calculation for the binomial distribution by expressing it as the sum of n iid Bernoulli p's, showcasing the efficiency of applying the linearity property.", 'Simplified approach using indicator random variables for hypergeometric distribution. The chapter illustrates a simplified approach for calculating the expected value of a hypergeometric distribution by using indicator random variables, offering a more efficient method compared to the traditional PMF involving binomial coefficients.', 'The concept of indicator random variables is applied to calculate the probability of drawing a specific card from a shuffled deck, resulting in a probability of 5/13. The explanation emphasizes the symmetry and linearity of the situation, enabling a straightforward calculation of the probability.', 'The expected value of a hypergeometric distribution resembles a binomial distribution despite the trials being dependent, calculated as n times the probability that an individual one has a specific property.', 'The geometric distribution is defined as the number of failures before the first success in a series of independent Bernoulli trials with the same probability of success.', 'The expected value of the geometric distribution, denoted as E(X), is derived as q/p using both a calculus-based method and a story-based method, demonstrating the versatility of approaches in solving for the expected value.']}