title

Lecture 19: Joint, Conditional, and Marginal Distributions | Statistics 110

description

We discuss joint, conditional, and marginal distributions (continuing from Lecture 18), the 2-D LOTUS, the fact that E(XY)=E(X)E(Y) if X and Y are independent, the expected distance between 2 random points, and the chicken-egg problem.

detail

{'title': 'Lecture 19: Joint, Conditional, and Marginal Distributions | Statistics 110', 'heatmap': [{'end': 1475.152, 'start': 1442.196, 'weight': 0.781}, {'end': 2108.264, 'start': 2074.516, 'weight': 0.797}, {'end': 2979.166, 'start': 2947.571, 'weight': 1}], 'summary': "The lecture series covers topics such as joint, conditional, and marginal distributions, probability, double integrals, conditional probability density functions, bayes' rule, expected values, poisson distribution, and more, providing insights and calculations to understand probability and distributions effectively.", 'chapters': [{'end': 210.113, 'segs': [{'end': 38.347, 'src': 'embed', 'start': 0.659, 'weight': 1, 'content': [{'end': 3.903, 'text': 'Okay, so last time we were talking about joint distributions.', 'start': 0.659, 'duration': 3.244}, {'end': 11.75, 'text': 'And just to kind of quickly remind everyone, the big theme right now is joint, conditional, and marginal distributions.', 'start': 4.283, 'duration': 7.467}, {'end': 16.315, 'text': 'And everyone needs to get comfortable at how all those concepts relate.', 'start': 11.83, 'duration': 4.485}, {'end': 20.579, 'text': "So there's three different types of things, joint, conditional, and marginal.", 'start': 16.395, 'duration': 4.184}, {'end': 27.617, 'text': 'And we were talking about joint and marginal distributions last time.', 'start': 24.174, 'duration': 3.443}, {'end': 33.322, 'text': "Not so much about conditional distributions, but it's analogous to stuff we've already seen about conditioning.", 'start': 27.657, 'duration': 5.665}, {'end': 38.347, 'text': 'So those are the three key words, joint, conditional, and marginal distributions.', 'start': 34.043, 'duration': 4.304}], 'summary': 'Focus on joint, conditional, and marginal distributions for understanding.', 'duration': 37.688, 'max_score': 0.659, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ659.jpg'}, {'end': 161.139, 'src': 'embed', 'start': 109.917, 'weight': 0, 'content': [{'end': 116.099, 'text': "So this extends to as many as you want, but it's just easier to write it down and think about for two of them.", 'start': 109.917, 'duration': 6.182}, {'end': 121.621, 'text': "But it's more general than this, okay? Without the joint CDF, that always makes sense.", 'start': 116.759, 'duration': 4.862}, {'end': 126.763, 'text': 'They can be discrete, continuous, mixtures of discrete and continuous or anything.', 'start': 122.941, 'duration': 3.822}, {'end': 134.967, 'text': 'In the continuous case, then we have a joint PDF, which I talked a little bit about.', 'start': 127.463, 'duration': 7.504}, {'end': 139.009, 'text': "But I don't think I wrote down how to get from the joint CDF to the joint PDF.", 'start': 135.007, 'duration': 4.002}, {'end': 141.45, 'text': 'So then we have a joint PDF.', 'start': 139.83, 'duration': 1.62}, {'end': 148.494, 'text': "And it's analogous to the one-dimensional case.", 'start': 145.973, 'duration': 2.521}, {'end': 154.854, 'text': 'Where in the one dimensional case, we take the derivative of the CDF to get the PDF.', 'start': 150.871, 'duration': 3.983}, {'end': 161.139, 'text': "In this case, we take the derivative except that it's a function of two variables, so we're gonna take two partial derivatives.", 'start': 154.894, 'duration': 6.245}], 'summary': 'Discussion about joint cdf and joint pdf in statistics.', 'duration': 51.222, 'max_score': 109.917, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ109917.jpg'}], 'start': 0.659, 'title': 'Understanding distributions and working with multiple variables', 'summary': 'Emphasizes joint, conditional, and marginal distributions, and extends the discussion to working with multiple random variables, highlighting the importance of understanding joint cdf and pdf.', 'chapters': [{'end': 38.347, 'start': 0.659, 'title': 'Understanding joint, conditional, and marginal distributions', 'summary': 'Emphasizes the concepts of joint, conditional, and marginal distributions, highlighting the importance of understanding how these concepts relate, with a focus on joint and marginal distributions discussed in the previous session.', 'duration': 37.688, 'highlights': ['The chapter emphasizes the importance of understanding joint, conditional, and marginal distributions and how these concepts relate.', 'Last time, the focus was on joint and marginal distributions, with less attention given to conditional distributions.', 'The big theme revolves around joint, conditional, and marginal distributions, and it is essential for everyone to grasp how these concepts interrelate.']}, {'end': 210.113, 'start': 41.32, 'title': 'Working with multiple random variables', 'summary': 'Discusses the extension of working with one random variable to multiple random variables, emphasizing the importance of understanding joint cdf and pdf, and the process of obtaining the joint pdf from the joint cdf through partial derivatives.', 'duration': 168.793, 'highlights': ['Understanding joint CDF and PDF is crucial for working with multiple random variables, as it extends to as many variables as needed and can be discrete, continuous, or mixtures of both. ', 'The process of obtaining the joint PDF from the joint CDF involves taking two partial derivatives and is analogous to the one-dimensional case, where the derivative of the CDF gives the PDF. ', "Emphasizing the cumulative nature of the study, the instructor clarifies that if there's trouble with one random variable's CDF, understanding two of them simultaneously becomes challenging. "]}], 'duration': 209.454, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ659.jpg', 'highlights': ['Understanding joint CDF and PDF is crucial for working with multiple random variables, as it extends to as many variables as needed and can be discrete, continuous, or mixtures of both.', 'The chapter emphasizes the importance of understanding joint, conditional, and marginal distributions and how these concepts relate.', 'The big theme revolves around joint, conditional, and marginal distributions, and it is essential for everyone to grasp how these concepts interrelate.', 'The process of obtaining the joint PDF from the joint CDF involves taking two partial derivatives and is analogous to the one-dimensional case, where the derivative of the CDF gives the PDF.']}, {'end': 504.214, 'segs': [{'end': 352.014, 'src': 'embed', 'start': 324.492, 'weight': 2, 'content': [{'end': 329.215, 'text': "that's uniform over a square, or uniform over a circle, that kind of thing.", 'start': 324.492, 'duration': 4.723}, {'end': 337.861, 'text': 'And in that uniform case, we can interpret probability as proportional to area, okay? So in the uniform case, probability is proportional to area.', 'start': 330.115, 'duration': 7.746}, {'end': 341.824, 'text': "And then I could say, well, I'm just gonna do something proportional to the area of the blob.", 'start': 337.881, 'duration': 3.943}, {'end': 345.987, 'text': 'And at least I can think more geometrically, okay?', 'start': 342.204, 'duration': 3.783}, {'end': 352.014, 'text': "But anyway, conceptually It's analogous.", 'start': 347.528, 'duration': 4.486}], 'summary': 'Probability in uniform cases is proportional to area, aiding geometric interpretation.', 'duration': 27.522, 'max_score': 324.492, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ324492.jpg'}, {'end': 405.001, 'src': 'embed', 'start': 374.437, 'weight': 1, 'content': [{'end': 378.961, 'text': 'Those are joint distributions and I talked a little bit last time about how to get the marginal.', 'start': 374.437, 'duration': 4.524}, {'end': 381.203, 'text': "And it's very straightforward.", 'start': 380.102, 'duration': 1.101}, {'end': 385.467, 'text': 'Marginal PDF of X.', 'start': 382.344, 'duration': 3.123}, {'end': 390.471, 'text': 'To get the marginal PDF of X, we just integrate out the Y.', 'start': 385.467, 'duration': 5.004}, {'end': 394.895, 'text': 'So we just integrate minus infinity to infinity, f of x, y, dy.', 'start': 390.471, 'duration': 4.424}, {'end': 403.38, 'text': "Notice that by doing this, we'll get something that's now a function of x.", 'start': 397.839, 'duration': 5.541}, {'end': 405.001, 'text': 'X is just treated as a constant here.', 'start': 403.38, 'duration': 1.621}], 'summary': 'Obtaining marginal pdf of x involves integrating out y, resulting in a function of x.', 'duration': 30.564, 'max_score': 374.437, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ374437.jpg'}, {'end': 479.696, 'src': 'embed', 'start': 456.597, 'weight': 0, 'content': [{'end': 463.603, 'text': "one way to think of it is to say if we let A be the entire plane like everything, we'd better get 1, right?", 'start': 456.597, 'duration': 7.006}, {'end': 464.864, 'text': "Otherwise it wouldn't make any sense.", 'start': 463.623, 'duration': 1.241}, {'end': 471.75, 'text': 'The other way to think of it is, this is supposed to be the density of X just viewed as X in its own right.', 'start': 465.604, 'duration': 6.146}, {'end': 474.832, 'text': 'So if we integrate this dx, we have to get 1.', 'start': 472.05, 'duration': 2.782}, {'end': 477.214, 'text': 'Otherwise, we did not find a valid marginal PDF.', 'start': 474.832, 'duration': 2.382}, {'end': 479.696, 'text': 'So that has to integrate to 1.', 'start': 477.754, 'duration': 1.942}], 'summary': 'In probability, integrating the density function must yield 1 for a valid marginal pdf.', 'duration': 23.099, 'max_score': 456.597, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ456597.jpg'}], 'start': 210.113, 'title': 'Probability and joint distributions', 'summary': 'Covers double integrals, joint and marginal distributions, and emphasizes the interpretation of probability in uniform distribution and the process of obtaining marginal pdf through integration. it also ensures that the double integral of a joint pdf equals 1.', 'chapters': [{'end': 345.987, 'start': 210.113, 'title': 'Double integrals and probability', 'summary': 'Explains the concept of double integrals, joint pdf, and the interpretation of probability in the context of uniform distribution, emphasizing that probability is proportional to area.', 'duration': 135.874, 'highlights': ['The joint PDF is a density that needs to be integrated to obtain the probability, where the integration over a region A is necessary to find the probability that XY is in set A.', 'In the case of a rectangle region A, the double integral simplifies to the integral of the integral, making it similar to doing two integrals, and the interpretation of probability is proportional to area in the uniform distribution case.', 'The concept of integrating over a blob is described as a complicated multivariable calculus problem, which is not relevant for the course, and it is emphasized that probability is not interesting in this context.']}, {'end': 504.214, 'start': 347.528, 'title': 'Joint and marginal distributions', 'summary': 'Discusses joint and marginal distributions, emphasizing the process of obtaining marginal pdf of x and y through integration and ensuring that the double integral of a joint pdf equals 1.', 'duration': 156.686, 'highlights': ['The process of obtaining the marginal PDF of X and Y involves integrating out the other variable, resulting in a function that depends only on the marginalized variable. By integrating out the Y, the marginal PDF of X is obtained, and vice versa. This process simplifies the joint distribution into individual marginal distributions that depend only on the specified variable.', 'The necessity for the double integral of a joint PDF to equal 1, signifying the validity of the marginal PDF. The double integral of a joint PDF must equal 1, ensuring that the marginal PDF is a valid probability density function. This requirement is essential for the interpretation and application of the marginal distributions.', 'Emphasizing the need for the integral of the marginal PDF to equal 1, indicating the validity of the marginal distribution as a probability density function. The integral of the marginal PDF should equal 1, representing the total probability for the specified variable. This condition ensures that the marginal distribution is a proper probability density function, enabling meaningful probabilistic analysis.']}], 'duration': 294.101, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ210113.jpg', 'highlights': ['The double integral of a joint PDF must equal 1, ensuring the validity of the marginal PDF.', 'The process of obtaining the marginal PDF involves integrating out the other variable, resulting in a function that depends only on the marginalized variable.', 'The interpretation of probability in the uniform distribution case emphasizes that probability is proportional to area.']}, {'end': 1015.091, 'segs': [{'end': 586.068, 'src': 'embed', 'start': 532.169, 'weight': 0, 'content': [{'end': 539.353, 'text': "Sometimes we'd put a subscript of Y given X just for emphasis, and sometimes we may leave out the subscript just because it's clear from the context.", 'start': 532.169, 'duration': 7.184}, {'end': 549.798, 'text': 'Conditional PDF, think of it as the PDF where we get to pretend that we know what x is.', 'start': 540.953, 'duration': 8.845}, {'end': 551.939, 'text': 'We get to observe what x is okay?', 'start': 549.818, 'duration': 2.121}, {'end': 558.122, 'text': 'Given that information that we now know the value of x, what is the appropriate PDF for y??', 'start': 552.239, 'duration': 5.883}, {'end': 571.34, 'text': 'Well, we could think of that as being the joint, The joint density divided by the marginal density of X.', 'start': 559.282, 'duration': 12.058}, {'end': 580.203, 'text': 'That what I just wrote down just looks like the definition of conditional probability, right?', 'start': 572.878, 'duration': 7.325}, {'end': 582.325, 'text': 'The probability of this given.', 'start': 580.444, 'duration': 1.881}, {'end': 586.068, 'text': 'this is the probability of this and this, divided by the probability of this thing.', 'start': 582.325, 'duration': 3.743}], 'summary': 'Conditional pdf is a way to determine the appropriate pdf for y given the value of x, using joint and marginal densities.', 'duration': 53.899, 'max_score': 532.169, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ532169.jpg'}, {'end': 782.59, 'src': 'embed', 'start': 724.83, 'weight': 2, 'content': [{'end': 733.254, 'text': 'So usually the best way to think of it is independent means that the joint PDF is the product of the marginal PDFs.', 'start': 724.83, 'duration': 8.424}, {'end': 737.957, 'text': 'And that has to hold for all X and Y.', 'start': 735.456, 'duration': 2.501}, {'end': 747.426, 'text': "It's not too hard to show that that's equivalent to having the CDFs factor.", 'start': 742.985, 'duration': 4.441}, {'end': 754.228, 'text': "Cuz basically, if the CDFs factor, you could take the derivative, this derivative thing, and you'll get this.", 'start': 747.486, 'duration': 6.742}, {'end': 757.589, 'text': 'You could take this thing and integrate and go back there.', 'start': 754.648, 'duration': 2.941}, {'end': 758.949, 'text': "So it's basically equivalent.", 'start': 757.609, 'duration': 1.34}, {'end': 760.71, 'text': 'Intuitively, it should be equivalent.', 'start': 759.009, 'duration': 1.701}, {'end': 766.191, 'text': "All right, so let's come back to this uniform example.", 'start': 762.21, 'duration': 3.981}, {'end': 770.6, 'text': "because I wanted to write what's the conditional.", 'start': 768.438, 'duration': 2.162}, {'end': 774.143, 'text': "We wrote down the joint PDF last time, I'll remind you.", 'start': 771.461, 'duration': 2.682}, {'end': 782.59, 'text': 'That is, we have the distribution that was uniform on a circle, or inside the disc.', 'start': 774.824, 'duration': 7.766}], 'summary': 'Independence means joint pdf is product of marginal pdfs; equivalent to cdfs factoring.', 'duration': 57.76, 'max_score': 724.83, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ724830.jpg'}, {'end': 985.575, 'src': 'embed', 'start': 942.045, 'weight': 4, 'content': [{'end': 944.266, 'text': 'We have to be very, very careful with limits of integration.', 'start': 942.045, 'duration': 2.221}, {'end': 949.427, 'text': "You're not actually ever gonna have to do any difficult integral in this course,", 'start': 944.826, 'duration': 4.601}, {'end': 952.288, 'text': 'but sometimes you have to think carefully about the limits of integration.', 'start': 949.427, 'duration': 2.861}, {'end': 960.389, 'text': 'Okay, so this is just saying these are the bounds on y, for which I should have 1 over pi rather than 0 here.', 'start': 953.086, 'duration': 7.303}, {'end': 964.31, 'text': "okay?. So if you get the limits of integration wrong, then it's just completely wrong.", 'start': 960.389, 'duration': 3.921}, {'end': 968.992, 'text': 'All right, this is a very easy integral, just integral of a constant.', 'start': 964.33, 'duration': 4.662}, {'end': 972.613, 'text': "It's just the constant times the length of the interval.", 'start': 969.792, 'duration': 2.821}, {'end': 985.575, 'text': "So that's just 2 over pi square root of 1 minus x squared, and that's valid for minus 1 less than or equal to x less than or equal to 1.", 'start': 973.554, 'duration': 12.021}], 'summary': 'Careful limits of integration crucial in easy integrals. valid for -1 <= x <= 1.', 'duration': 43.53, 'max_score': 942.045, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ942045.jpg'}], 'start': 505.635, 'title': 'Conditional distributions and probabilities', 'summary': "Covers conditional probability density functions, bayes' rule, marginal density functions, and integration limits, emphasizing the importance of understanding conditional processes and precision in determining appropriate pdfs and integration bounds.", 'chapters': [{'end': 558.122, 'start': 505.635, 'title': 'Conditional distribution and pdf', 'summary': 'Discusses the concept of conditional probability density function (pdf) and its application in determining the appropriate pdf for a given value of x, emphasizing the need to understand and remember the analogous conditioning process.', 'duration': 52.487, 'highlights': ['The chapter explains the concept of conditional PDF, which involves determining the appropriate PDF for y given the known value of x, making it a crucial tool for conditional distribution analysis.', 'The speaker emphasizes the need to understand and remember the analogous conditioning process, highlighting its relevance in comprehending the conditional PDF concept and its practical applications.']}, {'end': 886.11, 'start': 559.282, 'title': "Conditional probability and bayes' rule", 'summary': "Discusses the concept of conditional probability, bayes' rule, and independence in the context of joint and marginal densities, emphasizing the analogy to discrete cases and the uniform distribution example.", 'duration': 326.828, 'highlights': ["The joint density divided by the marginal density of X looks like the definition of conditional probability, analogous to Bayes' rule. The joint density divided by the marginal density of X resembles the definition of conditional probability, similar to Bayes' rule, providing an analogy between continuous and discrete cases.", "The conditional PDF of X given Y is obtained by swapping the x and the y in something that looks like Bayes' rule, with the proof involving the use of Bayes' rule and taking a limit. The conditional PDF of X given Y involves a process resembling Bayes' rule, with the proof relying on the use of Bayes' rule and limit operations.", 'Independence in the continuous case is explained as the joint PDF being the product of the marginal PDFs for all X and Y, with an emphasis on the equivalence to the factorization of CDFs. Independence in the continuous case is described as the joint PDF being the product of the marginal PDFs for all X and Y, highlighting the equivalence to the factorization of CDFs.', 'The example of a uniform distribution on a circle is used to illustrate the concept of conditional and marginal densities, emphasizing the proportional relationship of probability to area. The example of a uniform distribution on a circle is employed to illustrate the concept of conditional and marginal densities, emphasizing the proportional relationship between probability and area.']}, {'end': 1015.091, 'start': 887.191, 'title': 'Marginal density function and integration limits', 'summary': 'Discusses finding the marginal density function and integrating it, emphasizing the importance of being careful with the limits of integration. it also highlights the simplicity of the integrals and the need for precision in determining the bounds, with an example of a straightforward integral result.', 'duration': 127.9, 'highlights': ['The process of finding the marginal density function and integrating it is crucial, and the chapter emphasizes the importance of being careful with the limits of integration.', 'The simplicity of the integrals in this context is highlighted, as it involves integrating a constant, resulting in 2 over pi square root of 1 minus x squared, valid for -1 <= x <= 1.', 'The need for precision in determining the bounds of integration is stressed, as making mistakes in this area can lead to completely incorrect results.', 'The example of a straightforward integral result serves as a reminder that while the integrals in the course may not be difficult, precision in determining the bounds of integration is essential.']}], 'duration': 509.456, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ505635.jpg', 'highlights': ['The chapter explains the concept of conditional PDF, crucial for conditional distribution analysis.', "The joint density divided by the marginal density of X resembles the definition of conditional probability, similar to Bayes' rule.", 'Independence in the continuous case is described as the joint PDF being the product of the marginal PDFs for all X and Y.', 'The example of a uniform distribution on a circle is employed to illustrate the concept of conditional and marginal densities.', 'The process of finding the marginal density function and integrating it is crucial, emphasizing the importance of being careful with the limits of integration.', 'The simplicity of the integrals in this context is highlighted, involving integrating a constant, resulting in 2 over pi square root of 1 minus x squared, valid for -1 <= x <= 1.', 'The need for precision in determining the bounds of integration is stressed, as making mistakes in this area can lead to completely incorrect results.']}, {'end': 1539.595, 'segs': [{'end': 1050.259, 'src': 'embed', 'start': 1015.931, 'weight': 1, 'content': [{'end': 1017.713, 'text': 'So okay, so that does integrate to one.', 'start': 1015.931, 'duration': 1.782}, {'end': 1022.476, 'text': "So that's the marginal PDF.", 'start': 1020.295, 'duration': 2.181}, {'end': 1024.838, 'text': 'Notice that this does not look like a uniform.', 'start': 1022.517, 'duration': 2.321}, {'end': 1030.323, 'text': "So it's certainly false to say that it's uniform between minus one and one.", 'start': 1025.118, 'duration': 5.205}, {'end': 1036.433, 'text': 'The point x, y is uniform, but the marginals are not uniform, right?', 'start': 1031.29, 'duration': 5.143}, {'end': 1040.535, 'text': 'And in fact you can see that this is largest when x is 0, which kind of makes sense.', 'start': 1036.573, 'duration': 3.962}, {'end': 1043.156, 'text': 'Cuz, if you imagine a random point here,', 'start': 1040.595, 'duration': 2.561}, {'end': 1050.259, 'text': "then kind of near the center seems like there's more space for stuff to happen and it seems a little less likely to be further out, okay?", 'start': 1043.156, 'duration': 7.103}], 'summary': 'The marginal pdf does not look uniform, with the largest value occurring when x is 0.', 'duration': 34.328, 'max_score': 1015.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1015931.jpg'}, {'end': 1108.582, 'src': 'embed', 'start': 1073.783, 'weight': 0, 'content': [{'end': 1079.625, 'text': "Okay, so let's do the PDF of Y given X.", 'start': 1073.783, 'duration': 5.842}, {'end': 1087.368, 'text': "So that's just gonna be, That's just gonna be the joint PDF divided by the marginal PDF of x.", 'start': 1079.625, 'duration': 7.743}, {'end': 1097.697, 'text': "So it's just gonna be 1 over pi divided by 2 over pi square root 1 minus x squared.", 'start': 1087.368, 'duration': 10.329}, {'end': 1104.242, 'text': 'I just took the joint PDF divided by the marginal PDF, and we have to be careful where is this non-zero.', 'start': 1097.837, 'duration': 6.405}, {'end': 1108.582, 'text': "I'm thinking of y as fixed right now.", 'start': 1106.941, 'duration': 1.641}], 'summary': 'Joint pdf of y given x is 1 over 2pi square root 1 minus x squared.', 'duration': 34.799, 'max_score': 1073.783, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1073783.jpg'}, {'end': 1295.497, 'src': 'embed', 'start': 1255.669, 'weight': 3, 'content': [{'end': 1256.55, 'text': "It's just notation.", 'start': 1255.669, 'duration': 0.881}, {'end': 1264.514, 'text': "Okay, so that says it's conditionally uniform over some interval.", 'start': 1258.251, 'duration': 6.263}, {'end': 1270.658, 'text': "Notice that that's the appropriate interval, cuz as soon as you specify what x is, we know y has to be between here and here.", 'start': 1264.815, 'duration': 5.843}, {'end': 1271.979, 'text': "This says it's uniform.", 'start': 1270.678, 'duration': 1.301}, {'end': 1279.012, 'text': 'So similarly, you could do f given y.', 'start': 1274.13, 'duration': 4.882}, {'end': 1281.352, 'text': "And you can see that they're not independent.", 'start': 1279.012, 'duration': 2.34}, {'end': 1295.497, 'text': 'because well, one way to see it is f does not equal the product of the marginal PDFs in general here, right?', 'start': 1281.352, 'duration': 14.145}], 'summary': 'Notation indicates conditional uniformity over an interval, showing non-independence with f not equaling the product of marginal pdfs.', 'duration': 39.828, 'max_score': 1255.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1255669.jpg'}, {'end': 1352.471, 'src': 'embed', 'start': 1317.598, 'weight': 4, 'content': [{'end': 1329.221, 'text': 'That is, learning X gives us information, okay? All right, so those are these basic concepts, joint, conditional, marginal.', 'start': 1317.598, 'duration': 11.623}, {'end': 1337.415, 'text': "I wanted to mention one more thing that's analogous to- the one-dimensional case, and that's what I call the 2D LOTUS.", 'start': 1329.521, 'duration': 7.894}, {'end': 1342, 'text': "And it's completely analogous.", 'start': 1340.738, 'duration': 1.262}, {'end': 1350.629, 'text': 'So we wanna do LOTUS where we have a function of more than one variable.', 'start': 1346.825, 'duration': 3.804}, {'end': 1352.471, 'text': "So let's let x, y.", 'start': 1351.309, 'duration': 1.162}], 'summary': 'Learning about joint, conditional, and marginal concepts. introducing 2d lotus.', 'duration': 34.873, 'max_score': 1317.598, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1317598.jpg'}, {'end': 1475.152, 'src': 'heatmap', 'start': 1442.196, 'weight': 0.781, 'content': [{'end': 1447.278, 'text': 'Completely analogous.', 'start': 1442.196, 'duration': 5.082}, {'end': 1450.079, 'text': "But let's do a couple examples.", 'start': 1447.438, 'duration': 2.641}, {'end': 1452.12, 'text': 'how is this fact useful?', 'start': 1450.079, 'duration': 2.041}, {'end': 1461.665, 'text': "So here's an important fact that I already needed this fact once and we didn't prove it yet.", 'start': 1453.48, 'duration': 8.185}, {'end': 1470.29, 'text': 'Which was, we were talking about the fact that the MGF of a sum of independent random variables is the product of the MGFs.', 'start': 1461.965, 'duration': 8.325}, {'end': 1475.152, 'text': 'And at some point we need to say E of something times something is E of something, E of the other thing.', 'start': 1470.63, 'duration': 4.522}], 'summary': 'Mgf of sum of independent random variables is product of mgfs, useful for calculations.', 'duration': 32.956, 'max_score': 1442.196, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1442196.jpg'}], 'start': 1015.931, 'title': 'Conditional pdf and joint probability analysis', 'summary': 'Delves into conditional pdf calculation, marginal pdf analysis, conditional density, and joint probability, emphasizing non-uniformity, conditional uniform distribution, and the theorem of expected value. it also explores the largest point in the distribution and the relationship between independence and correlation, providing insights into 2d lotus and the conditional pdf of y given x.', 'chapters': [{'end': 1143.415, 'start': 1015.931, 'title': 'Conditional pdf calculation and marginal pdf analysis', 'summary': 'Explains the calculation of conditional pdf and analyzes the marginal pdf, highlighting the non-uniformity and the largest point in the distribution, with a focus on the conditional pdf of y given x.', 'duration': 127.484, 'highlights': ['The conditional PDF is calculated by dividing the joint PDF by the marginal PDF of x, resulting in 1 over pi divided by 2 over pi square root 1 minus x squared, with constraints on the possible values of y for each x.', 'The analysis reveals the non-uniformity of the marginal PDF, particularly highlighted by the largest point when x is 0, indicating a higher likelihood near the center.', "The chapter emphasizes that the point x, y is uniform, but the marginals are not uniform, providing insights into the distribution's behavior and the likelihood of occurrences."]}, {'end': 1539.595, 'start': 1144.355, 'title': 'Conditional density and joint probability', 'summary': 'Explains the concept of conditional density and joint probability, emphasizing the conditional uniform distribution and the relationship between independence and correlation. it also introduces the concept of 2d lotus for functions of two variables and proves the theorem that the expected value of the product of independent random variables is the product of their expected values.', 'duration': 395.24, 'highlights': ['The chapter explains the concept of conditional density and joint probability, emphasizing the conditional uniform distribution and the relationship between independence and correlation. The chapter delves into the concept of conditional density and joint probability, particularly highlighting the conditional uniform distribution of y given x as uniform between minus root 1 minus x squared and root 1 minus x squared. It also discusses the relationship between independence and correlation, demonstrating that the conditional distribution of Y given X is not the same as the unconditional distribution of Y.', 'Introduces the concept of 2D LOTUS for functions of two variables and proves the theorem that the expected value of the product of independent random variables is the product of their expected values. The chapter introduces the concept of 2D LOTUS for functions of two variables, demonstrating how to calculate the expected value of a function of x and y directly in terms of the joint PDF. Additionally, it proves the theorem that if X and Y are independent, then the expected value of their product is the product of their expected values, emphasizing the relationship between independence and correlation.']}], 'duration': 523.664, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1015931.jpg', 'highlights': ['The conditional PDF is calculated by dividing the joint PDF by the marginal PDF of x, resulting in 1 over pi divided by 2 over pi square root 1 minus x squared, with constraints on the possible values of y for each x.', 'The analysis reveals the non-uniformity of the marginal PDF, particularly highlighted by the largest point when x is 0, indicating a higher likelihood near the center.', "The chapter emphasizes that the point x, y is uniform, but the marginals are not uniform, providing insights into the distribution's behavior and the likelihood of occurrences.", 'The chapter explains the concept of conditional density and joint probability, emphasizing the conditional uniform distribution and the relationship between independence and correlation.', 'Introduces the concept of 2D LOTUS for functions of two variables and proves the theorem that the expected value of the product of independent random variables is the product of their expected values.']}, {'end': 2382.857, 'segs': [{'end': 1626.924, 'src': 'embed', 'start': 1569.506, 'weight': 0, 'content': [{'end': 1574.391, 'text': "2D Lotus just says, well, that's just a function of xy, I'm just gonna use Lotus and it's gonna be easy.", 'start': 1569.506, 'duration': 4.885}, {'end': 1577.834, 'text': 'So e of xy equals.', 'start': 1575.412, 'duration': 2.422}, {'end': 1580.625, 'text': 'How do we do this??', 'start': 1579.884, 'duration': 0.741}, {'end': 1593.1, 'text': 'Well, just write down double integral minus infinity to infinity, minus infinity to infinity x y times the joint PDF.', 'start': 1580.725, 'duration': 12.375}, {'end': 1599.188, 'text': "But since we assume that they're independent, the joint PDF is just the product of the marginal PDFs.", 'start': 1593.801, 'duration': 5.387}, {'end': 1605.271, 'text': 'So independence means the joint PDF just factors like that.', 'start': 1601.249, 'duration': 4.022}, {'end': 1610.975, 'text': "And that's what makes this actually easy to deal with.", 'start': 1606.652, 'duration': 4.323}, {'end': 1619.44, 'text': 'Because this function has just separated out, this is a function of x, function of y, function of x, function of y, very nice.', 'start': 1611.775, 'duration': 7.665}, {'end': 1624.582, 'text': 'So now, what do we actually do?', 'start': 1621.861, 'duration': 2.721}, {'end': 1626.924, 'text': 'Well, what this says to do is take this.', 'start': 1624.622, 'duration': 2.302}], 'summary': 'Using independence, the joint pdf factors into the product of marginal pdfs, making the integration easier.', 'duration': 57.418, 'max_score': 1569.506, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1569506.jpg'}, {'end': 1768.486, 'src': 'embed', 'start': 1705.525, 'weight': 2, 'content': [{'end': 1712.068, 'text': "What's left, integral of y times the PDF of y, that's just e of y.", 'start': 1705.525, 'duration': 6.543}, {'end': 1719.131, 'text': "So that's immediately just e of x, e of y.", 'start': 1712.068, 'duration': 7.063}, {'end': 1725.794, 'text': "So basically this amounts to, that's e of x times e of y.", 'start': 1719.131, 'duration': 6.663}, {'end': 1731.096, 'text': "All this amounts to doing is just taking out things that you're treating as constant and then it factors.", 'start': 1725.794, 'duration': 5.302}, {'end': 1737.953, 'text': "Okay, so that's a useful fact, and it would be a nightmare to try to prove this without having LOTUS available.", 'start': 1732.369, 'duration': 5.584}, {'end': 1740.435, 'text': 'But with LOTUS, then we could do that pretty quickly.', 'start': 1737.993, 'duration': 2.442}, {'end': 1753.164, 'text': "All right, there's another problem I like to do with the 2D LOTUS, and that's expected distance between two points.", 'start': 1743.617, 'duration': 9.547}, {'end': 1764.443, 'text': "So let's start with the uniform case, okay? I talk about this on the strategic practice too that you can look at later.", 'start': 1756.018, 'duration': 8.425}, {'end': 1768.486, 'text': 'But I think this is a useful point for everyone to see this now.', 'start': 1764.483, 'duration': 4.003}], 'summary': 'Using lotus can simplify calculations and make them quicker.', 'duration': 62.961, 'max_score': 1705.525, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1705525.jpg'}, {'end': 1930.908, 'src': 'embed', 'start': 1907.921, 'weight': 4, 'content': [{'end': 1918.044, 'text': "Now if you think of the symmetry of the problem, this problem is completely symmetrical because they're iid and this is a symmetrical function.", 'start': 1907.921, 'duration': 10.123}, {'end': 1920.185, 'text': 'I could have changed this to y- x.', 'start': 1918.064, 'duration': 2.121}, {'end': 1922.666, 'text': "So really there's no point in doing two double integrals.", 'start': 1920.185, 'duration': 2.481}, {'end': 1925.006, 'text': "Let's just do one double integral and double it.", 'start': 1922.746, 'duration': 2.26}, {'end': 1930.908, 'text': "Then we have to do two integrals instead of four, so that's much nicer.", 'start': 1926.207, 'duration': 4.701}], 'summary': "The problem's symmetry allows simplifying calculations by using one double integral instead of two, reducing the workload by half.", 'duration': 22.987, 'max_score': 1907.921, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1907921.jpg'}, {'end': 2009.757, 'src': 'embed', 'start': 1982.074, 'weight': 5, 'content': [{'end': 1984.635, 'text': 'so these outer limits have to just be numbers.', 'start': 1982.074, 'duration': 2.561}, {'end': 1989.953, 'text': 'As you move inward, the limits can start depending on other variables.', 'start': 1985.77, 'duration': 4.183}, {'end': 1993.755, 'text': 'So these inner limits can depend on y.', 'start': 1990.233, 'duration': 3.522}, {'end': 1995.496, 'text': 'in fact they have to depend on y.', 'start': 1993.755, 'duration': 1.741}, {'end': 2005.343, 'text': "okay?. It would not work to go 0 to 1 here, because we know x has to be between 0 and 1, but we also know that we're only integrating over x greater than y.", 'start': 1995.496, 'duration': 9.847}, {'end': 2009.757, 'text': 'So x has to be greater than y, so we go from y to 1.', 'start': 2005.343, 'duration': 4.414}], 'summary': 'Outer limits are numbers, inner limits depend on y, integrating over x greater than y.', 'duration': 27.683, 'max_score': 1982.074, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1982074.jpg'}, {'end': 2108.264, 'src': 'heatmap', 'start': 2064.007, 'weight': 7, 'content': [{'end': 2065.728, 'text': 'And if you simplify that, you get 1 third.', 'start': 2064.007, 'duration': 1.721}, {'end': 2072.351, 'text': 'So the average distance between two uniforms is 1 third.', 'start': 2069.891, 'duration': 2.46}, {'end': 2078.801, 'text': "Let's draw a little picture to see whether that makes intuitive sense to us.", 'start': 2074.516, 'duration': 4.285}, {'end': 2088.469, 'text': "So we have this interval 0 to 1, okay? And we're picking two uniformly random points in this interval.", 'start': 2079.601, 'duration': 8.868}, {'end': 2095.116, 'text': "Let's say there and there, completely random.", 'start': 2088.929, 'duration': 6.187}, {'end': 2103.78, 'text': "But notice that the distance between them is one-third because That's one-third, two-thirds, the distance is one-third.", 'start': 2095.416, 'duration': 8.364}, {'end': 2106.122, 'text': 'That sort of looks like your stereotypical.', 'start': 2103.94, 'duration': 2.182}, {'end': 2108.264, 'text': 'if you had to guess something, what would it look like?', 'start': 2106.122, 'duration': 2.142}], 'summary': 'The average distance between two uniformly random points in an interval of 0 to 1 is 1 third.', 'duration': 31.109, 'max_score': 2064.007, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2064007.jpg'}, {'end': 2151.953, 'src': 'embed', 'start': 2118.593, 'weight': 6, 'content': [{'end': 2123.998, 'text': "For me, at least, that actually makes the result one-third easy to remember, even though that's not a proof, obviously.", 'start': 2118.593, 'duration': 5.405}, {'end': 2128.062, 'text': 'But that actually does suggest another way to look at this problem.', 'start': 2124.539, 'duration': 3.523}, {'end': 2137.126, 'text': "Which is, I'm picking these two random points, and there's gonna be a point on the left and a point on the right.", 'start': 2130.683, 'duration': 6.443}, {'end': 2141.388, 'text': 'So that suggests reinterpreting this in terms of the max and the min.', 'start': 2137.426, 'duration': 3.962}, {'end': 2151.953, 'text': "So another way to look at this would be to let, let's say, M equals maximum of x, y.", 'start': 2141.828, 'duration': 10.125}], 'summary': 'Picking two random points suggests reinterpreting in terms of the max and min.', 'duration': 33.36, 'max_score': 2118.593, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2118593.jpg'}, {'end': 2391.26, 'src': 'embed', 'start': 2360.463, 'weight': 8, 'content': [{'end': 2361.783, 'text': 'So you could go in either direction.', 'start': 2360.463, 'duration': 1.32}, {'end': 2363.724, 'text': 'So I was just doing it.', 'start': 2363.023, 'duration': 0.701}, {'end': 2367.545, 'text': "I actually don't like doing double integrals, and I'm not gonna do a lot of double integrals.", 'start': 2363.824, 'duration': 3.721}, {'end': 2368.505, 'text': "You won't have to do many.", 'start': 2367.585, 'duration': 0.92}, {'end': 2371.466, 'text': 'I felt I should do one for practice with the 2D Lotus.', 'start': 2368.805, 'duration': 2.661}, {'end': 2377.235, 'text': 'Okay, but in general, I would rather think more in this way.', 'start': 2373.433, 'duration': 3.802}, {'end': 2382.857, 'text': 'Use linearity, use the CDFs, things like that, and not do a lot of integrals.', 'start': 2377.795, 'duration': 5.062}, {'end': 2386.518, 'text': 'Okay, so those are continuous examples.', 'start': 2384.298, 'duration': 2.22}, {'end': 2391.26, 'text': 'I want to do one discrete example for the rest of today.', 'start': 2386.579, 'duration': 4.681}], 'summary': 'Prefer using linearity and cdfs over double integrals. only do one for practice with 2d lotus. will focus on discrete examples.', 'duration': 30.797, 'max_score': 2360.463, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2360463.jpg'}], 'start': 1539.635, 'title': 'Expected values in probability', 'summary': 'Covers the use of 2d lotus in the continuous case, simplifying the calculation of the expected value of the product of two random variables. it also discusses finding the expected distance between two uniform iid points and the expected values of the maximum and minimum of the two points, which are 2 thirds and 1 third respectively.', 'chapters': [{'end': 1737.953, 'start': 1539.635, 'title': 'Continuous case in 2d lotus', 'summary': 'Explains the use of 2d lotus in the continuous case to simplify the calculation of the expected value of the product of two random variables, which reduces to the product of their individual expected values.', 'duration': 198.318, 'highlights': ['The 2D LOTUS simplifies the calculation of the expected value of the product of two random variables, reducing it to the product of their individual expected values, thus saving a lot of effort.', 'Independence of the random variables allows the joint PDF to factorize, making the calculation easier by separating the function into individual components.', 'The process involves treating certain variables as constants, pulling constants out of integrals, and ultimately simplifies to the product of the individual expected values, demonstrating the usefulness of LOTUS in such calculations.']}, {'end': 2063.967, 'start': 1737.993, 'title': 'Expected distance between two points', 'summary': 'Discusses the method of finding the expected distance between two uniform iid points using double integrals and the application of lotus, resulting in a symmetrical function and a simplified approach to solving the problem.', 'duration': 325.974, 'highlights': ['The application of LOTUS is used to find the expected distance between two uniform IID points, where the double integral of X minus Y is calculated, resulting in a symmetrical function and a simplified approach to solving the problem. Application of LOTUS, calculation of double integral of X minus Y, symmetrical function, simplified approach', 'The method involves splitting the integral into pieces based on the relationship between x and y, resulting in two integrals instead of four, thus simplifying the process. Splitting the integral, reducing integrals from four to two', 'The process involves careful consideration of the limits of integration, with the outer limits referring to y from 0 to 1, and the inner limits depending on y and ranging from y to 1. Consideration of limits of integration, outer limits referring to y (0 to 1), inner limits depending on y (y to 1)']}, {'end': 2382.857, 'start': 2064.007, 'title': 'Expected max and min', 'summary': 'Discusses the average distance between two uniformly random points in an interval 0 to 1, and through calculations, it is shown that the expected values of the maximum and minimum of the two points are 2 thirds and 1 third respectively, with further insights on different problem-solving approaches and techniques.', 'duration': 318.85, 'highlights': ['The expected value of the maximum of the two random points is calculated to be 2 thirds, and the expected value of the minimum is 1 third, providing a deeper understanding of the problem. The chapter delves into the calculation of the expected values of the maximum and minimum of the two random points, yielding 2 thirds and 1 third respectively, which contributes to a comprehensive grasp of the problem.', 'The average distance between two uniformly random points in the interval 0 to 1 is shown to be 1 third, offering an intuitive insight into the distribution of the points. Through a visual representation and calculation, it is demonstrated that the average distance between two uniformly random points in the interval 0 to 1 is 1 third, providing an intuitive understanding of the distribution of the points.', 'Different problem-solving approaches and techniques are discussed, including the use of linearity and CDFs instead of extensive double integrals, presenting alternative methods for tackling similar problems. The chapter explores various problem-solving approaches, such as the utilization of linearity and CDFs over extensive double integrals, offering alternative methods for addressing similar problems.']}], 'duration': 843.222, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ1539635.jpg', 'highlights': ['The 2D LOTUS simplifies the calculation of the expected value of the product of two random variables, reducing it to the product of their individual expected values, thus saving a lot of effort.', 'Independence of the random variables allows the joint PDF to factorize, making the calculation easier by separating the function into individual components.', 'The process involves treating certain variables as constants, pulling constants out of integrals, and ultimately simplifies to the product of the individual expected values, demonstrating the usefulness of LOTUS in such calculations.', 'The application of LOTUS is used to find the expected distance between two uniform IID points, where the double integral of X minus Y is calculated, resulting in a symmetrical function and a simplified approach to solving the problem.', 'The method involves splitting the integral into pieces based on the relationship between x and y, resulting in two integrals instead of four, thus simplifying the process.', 'The process involves careful consideration of the limits of integration, with the outer limits referring to y from 0 to 1, and the inner limits depending on y and ranging from y to 1.', 'The expected value of the maximum of the two random points is calculated to be 2 thirds, and the expected value of the minimum is 1 third, providing a deeper understanding of the problem.', 'The average distance between two uniformly random points in the interval 0 to 1 is shown to be 1 third, offering an intuitive insight into the distribution of the points.', 'Different problem-solving approaches and techniques are discussed, including the use of linearity and CDFs instead of extensive double integrals, presenting alternative methods for tackling similar problems.']}, {'end': 3008.58, 'segs': [{'end': 2470.773, 'src': 'embed', 'start': 2442.828, 'weight': 2, 'content': [{'end': 2451.151, 'text': "The twist to this problem is that the number of eggs is random, right? Because chicken doesn't always lay exactly the same number of eggs.", 'start': 2442.828, 'duration': 8.323}, {'end': 2453.652, 'text': "So let's assume that it's Poisson lambda.", 'start': 2451.471, 'duration': 2.181}, {'end': 2455.193, 'text': "That's the number of eggs.", 'start': 2454.432, 'duration': 0.761}, {'end': 2461.928, 'text': 'Okay, now each one either hatches or fails to hatch.', 'start': 2457.586, 'duration': 4.342}, {'end': 2470.773, 'text': 'So each hatches with probability p.', 'start': 2463.509, 'duration': 7.264}], 'summary': 'Problem involves random number of eggs with poisson lambda distribution and hatching probability p.', 'duration': 27.945, 'max_score': 2442.828, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2442828.jpg'}, {'end': 2625.501, 'src': 'embed', 'start': 2590.463, 'weight': 0, 'content': [{'end': 2593.924, 'text': "That is, if you have a lot of eggs that hatch, then there's not so many left that don't hatch.", 'start': 2590.463, 'duration': 3.461}, {'end': 2599.087, 'text': "But we haven't yet proven whether they're independent or not, because this is equal n.", 'start': 2594.585, 'duration': 4.502}, {'end': 2601.648, 'text': "All right, so now let's find the joint PMF.", 'start': 2599.087, 'duration': 2.561}, {'end': 2610.632, 'text': "So just by definition, the joint PMF is the probability that x equals something, let's say i y equals j.", 'start': 2602.888, 'duration': 7.744}, {'end': 2615.614, 'text': "Could use little x and little y, but I'm using i and j just to remind us that they're integers.", 'start': 2610.632, 'duration': 4.982}, {'end': 2625.501, 'text': 'Now, to do that, Somehow we have to bring in this Poisson thing.', 'start': 2617.774, 'duration': 7.727}], 'summary': 'The transcript discusses the joint pmf and poisson distribution.', 'duration': 35.038, 'max_score': 2590.463, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2590463.jpg'}, {'end': 2665.966, 'src': 'embed', 'start': 2636.889, 'weight': 1, 'content': [{'end': 2639.131, 'text': 'What do you condition on? What we wish that we knew.', 'start': 2636.889, 'duration': 2.242}, {'end': 2642.033, 'text': 'I wish I knew the number of eggs.', 'start': 2640.392, 'duration': 1.641}, {'end': 2644.515, 'text': "then it's an easy binomial problem, right?", 'start': 2642.033, 'duration': 2.482}, {'end': 2645.956, 'text': 'Conditional on the number of eggs.', 'start': 2644.575, 'duration': 1.381}, {'end': 2647.397, 'text': 'just a binomial, okay?', 'start': 2645.956, 'duration': 1.441}, {'end': 2648.778, 'text': "So we're gonna condition on n.", 'start': 2647.778, 'duration': 1}, {'end': 2651.26, 'text': 'The law of total probability says.', 'start': 2649.699, 'duration': 1.561}, {'end': 2658.203, 'text': 'we can just write this as the sum of x equals i, probably x equals i, y equals j.', 'start': 2651.26, 'duration': 6.943}, {'end': 2665.966, 'text': 'given n equals n times the probability that n equals n, summed over all n from 0 to infinity.', 'start': 2658.203, 'duration': 7.763}], 'summary': 'Conditioning on number of eggs for binomial problem.', 'duration': 29.077, 'max_score': 2636.889, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2636889.jpg'}, {'end': 2739.923, 'src': 'embed', 'start': 2705.298, 'weight': 3, 'content': [{'end': 2709.059, 'text': "That is, if I said what's the probability, this is just kind of some scratch work.", 'start': 2705.298, 'duration': 3.761}, {'end': 2722.938, 'text': "What's the probability that x equals 3, y equals 5, given n equals 10, what's that? 0, because there's ten eggs, three hatched, five didn't hatch.", 'start': 2709.399, 'duration': 13.539}, {'end': 2724.86, 'text': 'Someone stole the other two eggs.', 'start': 2723.479, 'duration': 1.381}, {'end': 2726.441, 'text': "I mean, it doesn't make any sense.", 'start': 2724.94, 'duration': 1.501}, {'end': 2729.082, 'text': "So it's impossible, 0.", 'start': 2726.801, 'duration': 2.281}, {'end': 2739.923, 'text': "What's the probability that x equals 3, y equals 5, given n equals 2?", 'start': 2729.082, 'duration': 10.841}], 'summary': "Probability of x=3, y=5 given n=10 is 0, as 3 hatched and 5 didn't hatch out of 10 eggs. probability of x=3, y=5 given n=2 is undefined.", 'duration': 34.625, 'max_score': 2705.298, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2705298.jpg'}, {'end': 2899.454, 'src': 'embed', 'start': 2864.143, 'weight': 4, 'content': [{'end': 2876.67, 'text': "So that's i plus j, thanks, that's i plus j choose i, right, from the binomial, times p to the i times, because we're assuming binomial np.", 'start': 2864.143, 'duration': 12.527}, {'end': 2880.072, 'text': 'So p to the i and q, as usual, q is 1 minus p.', 'start': 2876.69, 'duration': 3.382}, {'end': 2893.48, 'text': 'So i successes, j failures, q to the j, and then times the Poisson PMF, e to the minus lambda, lambda to the i plus j over i plus j factorial.', 'start': 2880.072, 'duration': 13.408}, {'end': 2899.454, 'text': "Let's just simplify this quickly, i plus j factorials cancel.", 'start': 2895.632, 'duration': 3.822}], 'summary': 'The transcript discusses the binomial and poisson distribution in the context of i successes and j failures.', 'duration': 35.311, 'max_score': 2864.143, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2864143.jpg'}, {'end': 2979.166, 'src': 'heatmap', 'start': 2947.571, 'weight': 1, 'content': [{'end': 2954.812, 'text': 'So actually it factored, so actually that shows that they are independent.', 'start': 2947.571, 'duration': 7.241}, {'end': 2959.934, 'text': 'So that says that x and y are independent.', 'start': 2956.033, 'duration': 3.901}, {'end': 2968.076, 'text': 'And x is also Poisson, x is Poisson lambda p, and y is Poisson lambda q.', 'start': 2961.454, 'duration': 6.622}, {'end': 2972.22, 'text': 'Which sounds like impossible at first.', 'start': 2969.738, 'duration': 2.482}, {'end': 2973.441, 'text': 'how could they be independent?', 'start': 2972.22, 'duration': 1.221}, {'end': 2979.166, 'text': "And if your intuition was that they are not independent, you shouldn't feel bad about that,", 'start': 2974.182, 'duration': 4.984}], 'summary': 'X and y are independent, both poisson with respective lambdas and p, q.', 'duration': 31.595, 'max_score': 2947.571, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2947571.jpg'}, {'end': 3008.58, 'src': 'embed', 'start': 2989.294, 'weight': 5, 'content': [{'end': 2992.977, 'text': "It happens to be true for the Poisson, we just proved that they're independent.", 'start': 2989.294, 'duration': 3.683}, {'end': 2996.841, 'text': "That is, you think, well, you have more eggs that hatch, there's less that didn't hatch.", 'start': 2993.198, 'duration': 3.643}, {'end': 3002.71, 'text': 'But the number of eggs is random, and that randomness for the Poisson exactly makes them independent.', 'start': 2997.341, 'duration': 5.369}, {'end': 3004.954, 'text': "So that's just an example of a joint PMF.", 'start': 3002.99, 'duration': 1.964}, {'end': 3006.076, 'text': "It's also a nice story.", 'start': 3005.074, 'duration': 1.002}, {'end': 3008.58, 'text': 'And have a good weekend.', 'start': 3006.777, 'duration': 1.803}], 'summary': 'Poisson distribution proves independence of events, illustrating a joint pmf.', 'duration': 19.286, 'max_score': 2989.294, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2989294.jpg'}], 'start': 2384.298, 'title': 'Probability and poisson distribution', 'summary': 'Covers the probability of hatching eggs given the total number of eggs, utilizing binomial and poisson pmf, and showcases the independence between two poisson distributions. it also addresses the chicken and egg problem involving a random number of eggs and the probability of hatching, aiming to find the joint pmf of x and y and determine their independence, with a strategy to condition on the number of eggs and utilize the law of total probability.', 'chapters': [{'end': 2704.778, 'start': 2384.298, 'title': 'Chicken and egg problem', 'summary': 'Covers the chicken and egg problem involving a random number of eggs and the probability of hatching, aiming to find the joint pmf of x and y and determine their independence, with a strategy to condition on the number of eggs and utilize the law of total probability.', 'duration': 320.48, 'highlights': ['The chapter aims to find the joint PMF of X and Y and determine their independence. This is the main focus of the problem and the ultimate goal of the discussion.', 'The strategy involves conditioning on the number of eggs and utilizing the law of total probability. This highlights the approach taken to solve the problem, providing a clear strategy for the audience to follow.', 'The number of eggs is random, following a Poisson distribution with parameter lambda. This provides important information about the nature of the problem and sets the foundation for the subsequent analysis.']}, {'end': 3008.58, 'start': 2705.298, 'title': 'Probability and poisson distribution', 'summary': 'Discusses the probability of hatching eggs given the total number of eggs, evaluating the probability using binomial and poisson pmf, and showcases the special property of independence between two poisson distributions.', 'duration': 303.282, 'highlights': ["The probability of x equals 3, y equals 5, given n equals 10 is 0, as there are 10 eggs, 3 hatched, 5 didn't hatch, and someone stole the other 2 eggs, illustrating the impossibility.", 'The chapter evaluates the joint probability by using the binomial PMF for x and Poisson PMF for n, simplifying the calculation to lambda p to the i over i factorial and lambda q to the j over j factorial.', 'It is revealed that x and y are independent, showcasing a special property of the Poisson distribution, where the randomness of the number of eggs exactly makes them independent.']}], 'duration': 624.282, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J70dP_AECzQ/pics/J70dP_AECzQ2384298.jpg', 'highlights': ['The chapter aims to find the joint PMF of X and Y and determine their independence. This is the main focus of the problem and the ultimate goal of the discussion.', 'The strategy involves conditioning on the number of eggs and utilizing the law of total probability. This highlights the approach taken to solve the problem, providing a clear strategy for the audience to follow.', 'The number of eggs is random, following a Poisson distribution with parameter lambda. This provides important information about the nature of the problem and sets the foundation for the subsequent analysis.', "The probability of x equals 3, y equals 5, given n equals 10 is 0, as there are 10 eggs, 3 hatched, 5 didn't hatch, and someone stole the other 2 eggs, illustrating the impossibility.", 'The chapter evaluates the joint probability by using the binomial PMF for x and Poisson PMF for n, simplifying the calculation to lambda p to the i over i factorial and lambda q to the j over j factorial.', 'It is revealed that x and y are independent, showcasing a special property of the Poisson distribution, where the randomness of the number of eggs exactly makes them independent.']}], 'highlights': ['Understanding joint CDF and PDF is crucial for working with multiple random variables, as it extends to as many variables as needed and can be discrete, continuous, or mixtures of both.', 'The chapter emphasizes the importance of understanding joint, conditional, and marginal distributions and how these concepts relate.', 'The big theme revolves around joint, conditional, and marginal distributions, and it is essential for everyone to grasp how these concepts interrelate.', 'The process of obtaining the joint PDF from the joint CDF involves taking two partial derivatives and is analogous to the one-dimensional case, where the derivative of the CDF gives the PDF.', 'The double integral of a joint PDF must equal 1, ensuring the validity of the marginal PDF.', 'The process of obtaining the marginal PDF involves integrating out the other variable, resulting in a function that depends only on the marginalized variable.', 'The interpretation of probability in the uniform distribution case emphasizes that probability is proportional to area.', 'The chapter explains the concept of conditional PDF, crucial for conditional distribution analysis.', "The joint density divided by the marginal density of X resembles the definition of conditional probability, similar to Bayes' rule.", 'Independence in the continuous case is described as the joint PDF being the product of the marginal PDFs for all X and Y.', 'The example of a uniform distribution on a circle is employed to illustrate the concept of conditional and marginal densities.', 'The process of finding the marginal density function and integrating it is crucial, emphasizing the importance of being careful with the limits of integration.', 'The simplicity of the integrals in this context is highlighted, involving integrating a constant, resulting in 2 over pi square root of 1 minus x squared, valid for -1 <= x <= 1.', 'The need for precision in determining the bounds of integration is stressed, as making mistakes in this area can lead to completely incorrect results.', 'The conditional PDF is calculated by dividing the joint PDF by the marginal PDF of x, resulting in 1 over pi divided by 2 over pi square root 1 minus x squared, with constraints on the possible values of y for each x.', 'The analysis reveals the non-uniformity of the marginal PDF, particularly highlighted by the largest point when x is 0, indicating a higher likelihood near the center.', "The chapter emphasizes that the point x, y is uniform, but the marginals are not uniform, providing insights into the distribution's behavior and the likelihood of occurrences.", 'The chapter explains the concept of conditional density and joint probability, emphasizing the conditional uniform distribution and the relationship between independence and correlation.', 'Introduces the concept of 2D LOTUS for functions of two variables and proves the theorem that the expected value of the product of independent random variables is the product of their expected values.', 'The 2D LOTUS simplifies the calculation of the expected value of the product of two random variables, reducing it to the product of their individual expected values, thus saving a lot of effort.', 'Independence of the random variables allows the joint PDF to factorize, making the calculation easier by separating the function into individual components.', 'The process involves treating certain variables as constants, pulling constants out of integrals, and ultimately simplifies to the product of the individual expected values, demonstrating the usefulness of LOTUS in such calculations.', 'The application of LOTUS is used to find the expected distance between two uniform IID points, where the double integral of X minus Y is calculated, resulting in a symmetrical function and a simplified approach to solving the problem.', 'The method involves splitting the integral into pieces based on the relationship between x and y, resulting in two integrals instead of four, thus simplifying the process.', 'The process involves careful consideration of the limits of integration, with the outer limits referring to y from 0 to 1, and the inner limits depending on y and ranging from y to 1.', 'The expected value of the maximum of the two random points is calculated to be 2 thirds, and the expected value of the minimum is 1 third, providing a deeper understanding of the problem.', 'The average distance between two uniformly random points in the interval 0 to 1 is shown to be 1 third, offering an intuitive insight into the distribution of the points.', 'Different problem-solving approaches and techniques are discussed, including the use of linearity and CDFs instead of extensive double integrals, presenting alternative methods for tackling similar problems.', 'The chapter aims to find the joint PMF of X and Y and determine their independence. This is the main focus of the problem and the ultimate goal of the discussion.', 'The strategy involves conditioning on the number of eggs and utilizing the law of total probability. This highlights the approach taken to solve the problem, providing a clear strategy for the audience to follow.', 'The number of eggs is random, following a Poisson distribution with parameter lambda. This provides important information about the nature of the problem and sets the foundation for the subsequent analysis.', "The probability of x equals 3, y equals 5, given n equals 10 is 0, as there are 10 eggs, 3 hatched, 5 didn't hatch, and someone stole the other 2 eggs, illustrating the impossibility.", 'The chapter evaluates the joint probability by using the binomial PMF for x and Poisson PMF for n, simplifying the calculation to lambda p to the i over i factorial and lambda q to the j over j factorial.', 'It is revealed that x and y are independent, showcasing a special property of the Poisson distribution, where the randomness of the number of eggs exactly makes them independent.']}