title

Lecture 13: Normal distribution | Statistics 110

description

We introduce the Normal distribution, which is the most famous, important, and widely-used distribution in all of statistics.

detail

{'title': 'Lecture 13: Normal distribution | Statistics 110', 'heatmap': [{'end': 771.301, 'start': 705.27, 'weight': 0.822}, {'end': 2427.905, 'start': 2362.691, 'weight': 0.799}], 'summary': "The lecture covers various topics including a uk judge's ruling against bayes' rule, the universality of the uniform distribution, inverse function simulation, independence of random variables, the normal distribution's significance due to the central limit theorem, the gaussian integral's unsolvability, and calculation of the mean and variance of the standard normal distribution.", 'chapters': [{'end': 73.874, 'segs': [{'end': 48.733, 'src': 'embed', 'start': 0.609, 'weight': 0, 'content': [{'end': 13.493, 'text': "Today is a pretty sad day for Stat 110, because I just saw in the news that a judge in the UK has ruled against the use of Bay's rule in court.", 'start': 0.609, 'duration': 12.884}, {'end': 18.034, 'text': 'So at least it was not in the US, but it was in the UK.', 'start': 14.493, 'duration': 3.541}, {'end': 20.015, 'text': 'I posted the link on Twitter.', 'start': 18.134, 'duration': 1.881}, {'end': 22.915, 'text': "That's a very disturbing case.", 'start': 21.335, 'duration': 1.58}, {'end': 27.397, 'text': "I haven't read the full legal ruling yet, but I intend to.", 'start': 22.976, 'duration': 4.421}, {'end': 35.983, 'text': "But my impression is that the judge kind of didn't like, maybe the judge, I haven't seen all the details of the case.", 'start': 28.617, 'duration': 7.366}, {'end': 39.426, 'text': 'So maybe the judge had a valid point of like kind of.', 'start': 36.243, 'duration': 3.183}, {'end': 45.41, 'text': "the so called expert witness was just kind of like estimating some probabilities and then throwing them into Bayes' rule.", 'start': 39.426, 'duration': 5.984}, {'end': 48.733, 'text': "And as with anything else, it's garbage in, garbage out.", 'start': 45.45, 'duration': 3.283}], 'summary': "Uk judge ruled against use of bayes' rule in court, prompting concern and further investigation.", 'duration': 48.124, 'max_score': 0.609, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0609.jpg'}], 'start': 0.609, 'title': "Uk judge's bayes' rule ruling", 'summary': "Discusses a uk judge's ruling against the use of bayes' rule in court, expressing concern over the potential precedent it sets and the impact on expert witness testimony.", 'chapters': [{'end': 73.874, 'start': 0.609, 'title': "Uk judge rules against bayes' rule use", 'summary': "Discusses a uk judge's ruling against the use of bayes' rule in court, expressing concern over the potential precedent it sets and the impact on expert witness testimony.", 'duration': 73.265, 'highlights': ["The UK judge ruled against the use of Bayes' rule in court, potentially setting a precedent against its use.", "The expert witness in the case was criticized for estimating probabilities and using them in Bayes' rule, highlighting the importance of accurate input for reliable output.", 'The instructor strongly opposes the ruling and believes it should be overturned, indicating potential implications on expert witness testimony.']}], 'duration': 73.265, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0609.jpg', 'highlights': ["The UK judge ruled against the use of Bayes' rule in court, potentially setting a precedent against its use.", "The expert witness in the case was criticized for estimating probabilities and using them in Bayes' rule, highlighting the importance of accurate input for reliable output.", 'The instructor strongly opposes the ruling and believes it should be overturned, indicating potential implications on expert witness testimony.']}, {'end': 575.216, 'segs': [{'end': 115.448, 'src': 'embed', 'start': 73.914, 'weight': 3, 'content': [{'end': 78.898, 'text': 'So if any of you have any connections in the British government or anything, maybe we can do something about that.', 'start': 73.914, 'duration': 4.984}, {'end': 84.902, 'text': "So sorry I had to bring you that bad news, but at least it's not a US case.", 'start': 80.479, 'duration': 4.423}, {'end': 89.925, 'text': "So to cheer us up, let's talk more about universality of the uniform.", 'start': 86.163, 'duration': 3.762}, {'end': 93.788, 'text': 'So we were doing that at the end last time.', 'start': 91.606, 'duration': 2.182}, {'end': 99.044, 'text': 'I proved the theorem, but I did not do an example yet.', 'start': 95.143, 'duration': 3.901}, {'end': 101.845, 'text': 'And it probably looked a little bit mysterious last time.', 'start': 99.304, 'duration': 2.541}, {'end': 105.826, 'text': 'I mean, the math is perfectly correct, and you could review what we did last time.', 'start': 101.865, 'duration': 3.961}, {'end': 108.066, 'text': 'And hopefully, you can follow every step.', 'start': 105.906, 'duration': 2.16}, {'end': 110.607, 'text': 'And if you were confused by the proof, you should review that.', 'start': 108.146, 'duration': 2.461}, {'end': 112.907, 'text': "But it's a proof, so you can't argue with it.", 'start': 110.667, 'duration': 2.24}, {'end': 115.448, 'text': "But that doesn't explain what does it really mean, okay?", 'start': 112.927, 'duration': 2.521}], 'summary': 'Discussion about universality of the uniform and a theorem proof in the last session.', 'duration': 41.534, 'max_score': 73.914, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL073914.jpg'}, {'end': 246.783, 'src': 'embed', 'start': 218.04, 'weight': 0, 'content': [{'end': 228.163, 'text': "If we let X equal F inverse of U, that's gonna have CDF F if U is uniform 0, 1.", 'start': 218.04, 'duration': 10.123}, {'end': 231.204, 'text': "So that's the reason I call it universality.", 'start': 228.163, 'duration': 3.041}, {'end': 234.665, 'text': 'It says just start with any uniform 0, 1 random variable,', 'start': 231.224, 'duration': 3.441}, {'end': 242.28, 'text': 'and at least in principle we can synthetically create a random variable with any distribution we want.', 'start': 234.665, 'duration': 7.615}, {'end': 246.783, 'text': 'So this is useful for simulation, right?', 'start': 242.34, 'duration': 4.443}], 'summary': 'Using universality, any uniform random variable can create any distribution for simulation.', 'duration': 28.743, 'max_score': 218.04, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0218040.jpg'}, {'end': 324.29, 'src': 'embed', 'start': 298.445, 'weight': 1, 'content': [{'end': 308.087, 'text': "And if we extend this a little bit, it's saying that if we have any function f that satisfies the properties of a CDF, then it is a CDF.", 'start': 298.445, 'duration': 9.642}, {'end': 314.928, 'text': 'Okay, so I wanted to do an example and talk a little bit more about the intuition behind this.', 'start': 311.127, 'duration': 3.801}, {'end': 315.848, 'text': 'We proved it last time.', 'start': 314.948, 'duration': 0.9}, {'end': 318.889, 'text': "There's kind of a flip side to this.", 'start': 317.209, 'duration': 1.68}, {'end': 321.949, 'text': "So that's statement one.", 'start': 320.409, 'duration': 1.54}, {'end': 324.29, 'text': "There's another way to write this.", 'start': 322.85, 'duration': 1.44}], 'summary': 'Any function f satisfying cdf properties is a cdf. explores intuition and different ways to express the concept.', 'duration': 25.845, 'max_score': 298.445, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0298445.jpg'}], 'start': 73.914, 'title': 'Universality of the uniform', 'summary': 'Discusses the universality of the uniform distribution and a mathematical proof, emphasizing the importance of review and understanding its implications. it also explores the concept that any continuous strictly increasing cdf can be used to synthetically create a random variable with any desired distribution, which is useful for simulation purposes.', 'chapters': [{'end': 115.448, 'start': 73.914, 'title': 'Universality of the uniform', 'summary': 'Discusses the universality of the uniform and a mathematical proof, emphasizing the importance of reviewing the proof and understanding its implications.', 'duration': 41.534, 'highlights': ['The chapter emphasizes the importance of reviewing the mathematical proof and understanding its implications.', 'The speaker mentions the theorem being proven but no example yet.', 'The speaker encourages the audience to review the previous discussion and ensure understanding of each step.']}, {'end': 575.216, 'start': 115.488, 'title': 'Universality of the uniform', 'summary': 'Discusses the concept of universality of the uniform distribution, stating that any continuous strictly increasing cdf can be used to synthetically create a random variable with any desired distribution, which is useful for simulation purposes.', 'duration': 459.728, 'highlights': ['The theorem of universality states that if X equals F inverse of U, it will have CDF F if U is uniform 0, 1, allowing the synthetic creation of a random variable with any distribution desired, useful for simulation purposes. theorem of universality', 'The ability to use a uniform 0, 1 random variable to create random draws from any distribution, which is practical for simulation, as it is typically easier to generate uniforms than other distributions in the continuous case. use uniform 0, 1 to create random draws from any distribution', 'The concept indicates that any function satisfying the properties of a CDF is a CDF, extending the understanding of CDF properties in defining a CDF. any function satisfying the properties of a CDF is a CDF']}], 'duration': 501.302, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL073914.jpg', 'highlights': ['The theorem of universality states that if X equals F inverse of U, it will have CDF F if U is uniform 0, 1, allowing the synthetic creation of a random variable with any distribution desired, useful for simulation purposes.', 'The concept indicates that any function satisfying the properties of a CDF is a CDF, extending the understanding of CDF properties in defining a CDF.', 'The ability to use a uniform 0, 1 random variable to create random draws from any distribution, which is practical for simulation, as it is typically easier to generate uniforms than other distributions in the continuous case.', 'The chapter emphasizes the importance of reviewing the mathematical proof and understanding its implications.', 'The speaker encourages the audience to review the previous discussion and ensure understanding of each step.', 'The speaker mentions the theorem being proven but no example yet.']}, {'end': 1368.95, 'segs': [{'end': 646.527, 'src': 'embed', 'start': 623.615, 'weight': 0, 'content': [{'end': 631.18, 'text': 'And if we wanna kind of be testing a certain model or things like that, it may be useful to be able to reduce things to uniform,', 'start': 623.615, 'duration': 7.565}, {'end': 634.262, 'text': 'as just a very simple, known standard distribution.', 'start': 631.18, 'duration': 3.082}, {'end': 637.303, 'text': 'And so it may be useful to reduce things to uniformity.', 'start': 634.802, 'duration': 2.501}, {'end': 645.807, 'text': "And if we generate a lot of instances of this and we find that they don't look uniform, then we conclude maybe there's something wrong with the model.", 'start': 637.924, 'duration': 7.883}, {'end': 646.527, 'text': 'that kind of thing.', 'start': 645.807, 'duration': 0.72}], 'summary': 'Testing models by reducing to uniform distribution for uniformity assessment.', 'duration': 22.912, 'max_score': 623.615, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0623615.jpg'}, {'end': 737.578, 'src': 'embed', 'start': 705.27, 'weight': 1, 'content': [{'end': 709.512, 'text': "And we're interested in simulating from this distribution.", 'start': 705.27, 'duration': 4.242}, {'end': 715.395, 'text': 'So we wanna simulate X which is distributed according to F.', 'start': 711.733, 'duration': 3.662}, {'end': 721.278, 'text': 'Well, all we have to do according to this result is compute the inverse of this function.', 'start': 715.395, 'duration': 5.883}, {'end': 723.419, 'text': 'So F inverse of U.', 'start': 722.018, 'duration': 1.401}, {'end': 729.87, 'text': 'Well, this is just like high school algebra of finding the inverse function, right?', 'start': 725.646, 'duration': 4.224}, {'end': 737.578, 'text': 'So if I set this thing equal to little u and then this is u in terms of x, we just solve for x in terms of u, right?', 'start': 730.211, 'duration': 7.367}], 'summary': 'Interest in simulating x from distribution using inverse function.', 'duration': 32.308, 'max_score': 705.27, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0705270.jpg'}, {'end': 771.301, 'src': 'heatmap', 'start': 705.27, 'weight': 0.822, 'content': [{'end': 709.512, 'text': "And we're interested in simulating from this distribution.", 'start': 705.27, 'duration': 4.242}, {'end': 715.395, 'text': 'So we wanna simulate X which is distributed according to F.', 'start': 711.733, 'duration': 3.662}, {'end': 721.278, 'text': 'Well, all we have to do according to this result is compute the inverse of this function.', 'start': 715.395, 'duration': 5.883}, {'end': 723.419, 'text': 'So F inverse of U.', 'start': 722.018, 'duration': 1.401}, {'end': 729.87, 'text': 'Well, this is just like high school algebra of finding the inverse function, right?', 'start': 725.646, 'duration': 4.224}, {'end': 737.578, 'text': 'So if I set this thing equal to little u and then this is u in terms of x, we just solve for x in terms of u, right?', 'start': 730.211, 'duration': 7.367}, {'end': 742.784, 'text': 'Then we would get minus log 1, minus u, just by taking the inverse function.', 'start': 737.979, 'duration': 4.805}, {'end': 745.492, 'text': 'Okay so.', 'start': 744.392, 'duration': 1.1}, {'end': 755.336, 'text': 'therefore, the universality theorem immediately tells us that F, inverse of capital U, which is minus log 1,', 'start': 745.492, 'duration': 9.844}, {'end': 759.157, 'text': 'minus capital U has the desired distribution F.', 'start': 755.336, 'duration': 3.821}, {'end': 771.301, 'text': "So if we were doing it on a computer and we wanted 10 random draws from this distribution, there's just an easy function, compute this ten times.", 'start': 759.157, 'duration': 12.144}], 'summary': 'Simulation of x distributed according to f using f inverse of u. universality theorem for 10 random draws.', 'duration': 66.031, 'max_score': 705.27, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0705270.jpg'}, {'end': 903.935, 'src': 'embed', 'start': 876.764, 'weight': 2, 'content': [{'end': 882.226, 'text': 'But an important common mistake to be aware of is that we do, this is like a linear transformation.', 'start': 876.764, 'duration': 5.462}, {'end': 886.167, 'text': "If we do something nonlinear, then it's usually not gonna be uniform anymore.", 'start': 882.286, 'duration': 3.881}, {'end': 890.309, 'text': 'Nonlinear, usually.', 'start': 887.788, 'duration': 2.521}, {'end': 894.569, 'text': 'leads to non-uniform.', 'start': 891.747, 'duration': 2.822}, {'end': 901.854, 'text': 'For example, if we squared u, you can check that it will not be uniform anymore.', 'start': 896.871, 'duration': 4.983}, {'end': 903.935, 'text': 'And to check that, just compute the CDF.', 'start': 901.914, 'duration': 2.021}], 'summary': 'Nonlinear transformations lead to non-uniformity, for example, squaring u results in non-uniformity.', 'duration': 27.171, 'max_score': 876.764, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0876764.jpg'}, {'end': 1131.136, 'src': 'embed', 'start': 1105.385, 'weight': 3, 'content': [{'end': 1110.63, 'text': "And in the case of just independence of events, it's a large list, but a finite list of equations.", 'start': 1105.385, 'duration': 5.245}, {'end': 1112.971, 'text': 'Here we have infinitely many equations.', 'start': 1110.95, 'duration': 2.021}, {'end': 1116.674, 'text': "It just looks like one equation cuz it's for all X1 through Xn.", 'start': 1113.192, 'duration': 3.482}, {'end': 1120.858, 'text': "All right, so that's the definition of independence in general.", 'start': 1116.694, 'duration': 4.164}, {'end': 1131.136, 'text': "But in the discrete case, usually it's easier to work with PMFs rather than CDFs.", 'start': 1122.37, 'duration': 8.766}], 'summary': 'Definition of independence in general: large list of equations, easier to work with pmfs in discrete case.', 'duration': 25.751, 'max_score': 1105.385, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01105385.jpg'}, {'end': 1255.997, 'src': 'embed', 'start': 1221.163, 'weight': 4, 'content': [{'end': 1233.027, 'text': "And just to give you a quick example where pairwise independence doesn't imply independence, here's the simplest example I know of that.", 'start': 1221.163, 'duration': 11.864}, {'end': 1244.109, 'text': "Let's just let X1 and IID coin flips, that is Bernoulli one half, fair coins.", 'start': 1233.967, 'duration': 10.142}, {'end': 1247.591, 'text': 'So if you want, just think of flipping a fair coin twice.', 'start': 1244.509, 'duration': 3.082}, {'end': 1255.997, 'text': 'And then this is kind of like an old game that people used to play called matching pennies,', 'start': 1250.713, 'duration': 5.284}], 'summary': "Pairwise independence doesn't imply independence, illustrated with x1 and iid coin flips.", 'duration': 34.834, 'max_score': 1221.163, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01221163.jpg'}], 'start': 578.305, 'title': 'Uniform distribution, inverse function, and independence of random variables', 'summary': 'Covers reducing variables to uniform distribution, inverse function simulation, independence of random variables, exponential distribution, universality theorem, joint cdf, and pmf, and practical examples of pairwise independence.', 'chapters': [{'end': 669.075, 'start': 578.305, 'title': 'Uniform distribution and its applications', 'summary': 'Discusses the concept of reducing variables to uniform distribution for testing models and simulation, emphasizing its usefulness in statistical inference and highlighting the importance of understanding the difference between a random variable and its distribution.', 'duration': 90.77, 'highlights': ['Understanding the difference between a random variable and its distribution is crucial for statistical inference and testing models, as reducing variables to uniform distribution allows for easier analysis (e.g. in 111) and simulation, aiding in identifying potential issues with the model.', 'The concept of reducing variables to uniform distribution is valuable for statistical inference and testing models, providing a simple, known standard distribution for analysis, and aiding in identifying potential issues with the model through simulation.', 'The result of being able to reduce variables to uniform distribution is significant for statistical inference and testing models, as it allows for analysis and simulation, which can aid in identifying potential issues with the model.']}, {'end': 1368.95, 'start': 672.609, 'title': 'Inverse function and independence of random variables', 'summary': 'Discusses the inverse function to simulate from a distribution, highlighting the exponential distribution and universality theorem, and explains the definition of independence of random variables, the joint cdf, and joint pmf, with a practical example of pairwise independence not implying independence.', 'duration': 696.341, 'highlights': ['The inverse function can be used to simulate random draws from a distribution by computing the inverse of the CDF, as demonstrated with the exponential distribution and universality theorem, enabling easy generation of random draws from the desired distribution. Simulating random draws from a distribution using the inverse function, exponential distribution with parameter 1, universality theorem for generating iid random draws from the distribution.', 'The symmetry of a uniform distribution, where 1-u is also uniform 0,1, is explained, demonstrating the intuitive symmetry and emphasizing the importance of checking calculations and not assuming uniformity for nonlinear transformations. Symmetry of a uniform distribution, importance of checking calculations for nonlinear transformations.', 'The definition of independence of random variables is discussed in terms of joint CDF and joint PMF, with the distinction between the simpler appearance of the definition and the underlying requirement of infinitely many equations for all possible values of the random variables. Definition of independence of random variables, joint CDF and joint PMF, distinction between appearance and underlying requirement of infinitely many equations.', 'A practical example of pairwise independence not implying independence is provided using the scenario of pairwise independent but not independent random variables, demonstrating the concept with the example of coin flips and the resulting indicator random variables. Practical example of pairwise independence not implying independence, demonstration with coin flips and indicator random variables.']}], 'duration': 790.645, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL0578305.jpg', 'highlights': ['Reducing variables to uniform distribution aids in statistical inference and testing models (e.g. in 111) and simulation.', 'Inverse function simulation enables easy generation of random draws from the desired distribution.', 'Symmetry of a uniform distribution emphasizes the importance of checking calculations for nonlinear transformations.', 'Definition of independence of random variables is discussed in terms of joint CDF and joint PMF.', 'Practical example of pairwise independence not implying independence is demonstrated using coin flips and indicator random variables.']}, {'end': 1766.766, 'segs': [{'end': 1501.768, 'src': 'embed', 'start': 1422.644, 'weight': 0, 'content': [{'end': 1428.466, 'text': "But I'm just mentioning that because you may see the term Gaussian and it's the same thing.", 'start': 1422.644, 'duration': 5.822}, {'end': 1437.079, 'text': 'The normal distribution is by far the most famous important distribution in all of statistics.', 'start': 1429.773, 'duration': 7.306}, {'end': 1440.962, 'text': 'And there are many reasons for that.', 'start': 1437.799, 'duration': 3.163}, {'end': 1448.527, 'text': "The most famous reason for that is what's called the central limit theorem.", 'start': 1441.982, 'duration': 6.545}, {'end': 1451.79, 'text': "which we're gonna do towards the end of the semester.", 'start': 1449.588, 'duration': 2.202}, {'end': 1459.096, 'text': "But I'm just gonna tell you in words intuitively what it is now, just to kind of foreshadow it, just so you have a sense of why is this important.", 'start': 1452.39, 'duration': 6.706}, {'end': 1462.538, 'text': "But then we'll go into the details of it much, much later in the course.", 'start': 1459.396, 'duration': 3.142}, {'end': 1464.52, 'text': 'Central limit theorem.', 'start': 1463.379, 'duration': 1.141}, {'end': 1474.187, 'text': "A central limit theorem is possibly the most famous and important theorem in all of probability, so we're definitely gonna talk about it later.", 'start': 1466.942, 'duration': 7.245}, {'end': 1478.251, 'text': "It's kind of a very, very surprising result.", 'start': 1474.207, 'duration': 4.044}, {'end': 1487.075, 'text': 'What it says is if you add up a bunch of iid random variables, the distribution is gonna look like a normal distribution.', 'start': 1478.988, 'duration': 8.087}, {'end': 1494.341, 'text': 'So this is just one distribution or one family of distributions cuz you can also shift it and scale it.', 'start': 1487.896, 'duration': 6.445}, {'end': 1501.768, 'text': 'Aside from the shift and scale, it says that adding up a large number of iid random variables is always gonna look normal.', 'start': 1494.441, 'duration': 7.327}], 'summary': 'The central limit theorem states that adding a large number of iid random variables results in a normal distribution, a key concept in statistics.', 'duration': 79.124, 'max_score': 1422.644, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01422644.jpg'}, {'end': 1578.364, 'src': 'embed', 'start': 1551.489, 'weight': 3, 'content': [{'end': 1557.451, 'text': "And by looks, I mean if you look at what's the distribution, it's gonna be approximately a normal distribution.", 'start': 1551.489, 'duration': 5.962}, {'end': 1565.134, 'text': 'And there are even further generalizations of this going beyond the IID case.', 'start': 1560.407, 'duration': 4.727}, {'end': 1571.543, 'text': 'You need some technical assumptions, but there are generalizations even beyond this.', 'start': 1566.596, 'duration': 4.947}, {'end': 1578.364, 'text': "Okay, that's one reason that the normal is so fundamental, and there are others as well.", 'start': 1572.602, 'duration': 5.762}], 'summary': 'Normal distribution is fundamental, with possible generalizations.', 'duration': 26.875, 'max_score': 1551.489, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01551489.jpg'}, {'end': 1664.644, 'src': 'embed', 'start': 1602.663, 'weight': 4, 'content': [{'end': 1609.965, 'text': "As long as the area is one, I mean, there's millions of different possible PDFs that you could come up with that look basically like this.", 'start': 1602.663, 'duration': 7.302}, {'end': 1612.366, 'text': 'The normal is a specific one.', 'start': 1610.425, 'duration': 1.941}, {'end': 1613.146, 'text': "that's given by.", 'start': 1612.366, 'duration': 0.78}, {'end': 1618.727, 'text': "Let's start with what's called the standard normal, which is written as normal.", 'start': 1614.124, 'duration': 4.603}, {'end': 1624.752, 'text': "This notation means that the mean is 0 and the variance is 1, but we'll prove that later.", 'start': 1618.727, 'duration': 6.025}, {'end': 1629.035, 'text': 'Okay, so the normal has two parameters, which are the mean and the variance.', 'start': 1624.772, 'duration': 4.263}, {'end': 1634.278, 'text': "And we're gonna start with the standard normal, which is mean 0, variance 1.", 'start': 1629.455, 'duration': 4.823}, {'end': 1636.18, 'text': "And let's write down its PDF.", 'start': 1634.278, 'duration': 1.902}, {'end': 1646.436, 'text': 'The PDF is f of z.', 'start': 1640.894, 'duration': 5.542}, {'end': 1656.481, 'text': "It's kind of a tradition of using the letter z for standard normals.", 'start': 1646.436, 'duration': 10.045}, {'end': 1660.742, 'text': "Not that you have to do that, but often we'll use z for standard normals.", 'start': 1657.181, 'duration': 3.561}, {'end': 1664.644, 'text': "That's why I'm calling it z rather than x.", 'start': 1660.762, 'duration': 3.882}], 'summary': 'Introduction to standard normal distribution with mean 0 and variance 1.', 'duration': 61.981, 'max_score': 1602.663, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01602663.jpg'}], 'start': 1369.01, 'title': 'Normal distribution and central limit theorem', 'summary': 'Introduces the normal distribution, emphasizing its significance due to the central limit theorem. it also explores the surprising result of the central limit theorem and delves into the details of the normal distribution, including its probability density function and normalizing constant.', 'chapters': [{'end': 1474.187, 'start': 1369.01, 'title': 'Normal distribution and lotus', 'summary': 'Introduces the normal distribution, also known as the gaussian distribution, as the most famous and important distribution in statistics, with the central limit theorem being a key reason for its significance.', 'duration': 105.177, 'highlights': ['The normal distribution, also called the Gaussian, is the most famous and important distribution in all of statistics, with the central limit theorem being a key reason for its significance.', 'The central limit theorem is possibly the most famous and important theorem in all of probability, to be discussed later in the course.', "Pairwise independence isn't enough in general to have independence.", 'The chapter will talk more about the Law of the Unconscious Statistician (LOTUS) on Wednesday.']}, {'end': 1766.766, 'start': 1474.207, 'title': 'Central limit theorem and normal distribution', 'summary': 'Discusses the surprising result of the central limit theorem, which states that the sum of a large number of independent and identically distributed random variables will approximate a normal distribution, with further generalizations and implications. the normal distribution, specifically the standard normal with mean 0 and variance 1, is explored in detail, including its probability density function and normalizing constant.', 'duration': 292.559, 'highlights': ["The Central Limit Theorem's surprising result: the sum of a large number of independent and identically distributed random variables approximates a normal distribution. This theorem is quite shocking as it implies that regardless of the type of random variables added, the resulting distribution will resemble a normal distribution, a fundamental concept in statistics.", 'Generalizations of the Central Limit Theorem beyond independent and identically distributed cases, with technical assumptions. The discussion extends to generalizations of the Central Limit Theorem, providing further insight into the behavior of distributions beyond the specific case of independent and identically distributed random variables.', 'Exploration of the standard normal distribution with mean 0 and variance 1, along with its probability density function. The chapter delves into the characteristics of the standard normal distribution, including its specific parameters, probability density function, and the significance of its mean and variance values.', 'Investigation of the normalizing constant for the standard normal distribution. The discussion includes the exploration of the normalizing constant for the standard normal distribution, a critical aspect in understanding its probability density function and its importance in statistical analysis.']}], 'duration': 397.756, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01369010.jpg', 'highlights': ['The central limit theorem is possibly the most famous and important theorem in all of probability, to be discussed later in the course.', 'The normal distribution, also called the Gaussian, is the most famous and important distribution in all of statistics, with the central limit theorem being a key reason for its significance.', "The Central Limit Theorem's surprising result: the sum of a large number of independent and identically distributed random variables approximates a normal distribution.", 'Generalizations of the Central Limit Theorem beyond independent and identically distributed cases, with technical assumptions.', 'Exploration of the standard normal distribution with mean 0 and variance 1, along with its probability density function.', 'Investigation of the normalizing constant for the standard normal distribution.']}, {'end': 2438.548, 'segs': [{'end': 1871.891, 'src': 'embed', 'start': 1802.699, 'weight': 0, 'content': [{'end': 1807.202, 'text': 'This seems like something we should be able to integrate.', 'start': 1802.699, 'duration': 4.503}, {'end': 1814.288, 'text': 'So of course, you can try doing a u substitution or some other kind of change of variables.', 'start': 1807.903, 'duration': 6.385}, {'end': 1815.689, 'text': 'It will not work.', 'start': 1814.748, 'duration': 0.941}, {'end': 1821.358, 'text': "You could try doing integration by parts, and you know there's many ways to try integration by parts.", 'start': 1817.017, 'duration': 4.341}, {'end': 1823.639, 'text': 'You can try to split it up in some way.', 'start': 1821.418, 'duration': 2.221}, {'end': 1826.18, 'text': 'It will not work.', 'start': 1824.119, 'duration': 2.061}, {'end': 1828.62, 'text': 'Anything else you could ever think of.', 'start': 1826.48, 'duration': 2.14}, {'end': 1833.902, 'text': "just for, let's say, we wanna find an antiderivative and then use the fundamental theorem of calculus the usual way we do integrals.", 'start': 1828.62, 'duration': 5.282}, {'end': 1836.863, 'text': "I can guarantee you that they won't work.", 'start': 1834.522, 'duration': 2.341}, {'end': 1846.666, 'text': 'The reason is that there is actually a theorem that says that this integral as an indefinite integral, that is, without the limits of integration,', 'start': 1837.963, 'duration': 8.703}, {'end': 1848.726, 'text': 'is impossible to do it in closed form.', 'start': 1846.666, 'duration': 2.06}, {'end': 1856.949, 'text': "And it's kind of pretty amazing, I think, that someone was able to prove that, right? It's not just like no one has thought of a way to do it.", 'start': 1849.707, 'duration': 7.242}, {'end': 1861.07, 'text': "Someone proved that you can't ever do it, so don't even try.", 'start': 1857.249, 'duration': 3.821}, {'end': 1869.31, 'text': 'And what I mean by, well, To qualify that of it, there is one way we could do this integral that will work.', 'start': 1861.09, 'duration': 8.22}, {'end': 1871.891, 'text': 'And that is the definite integral.', 'start': 1869.81, 'duration': 2.081}], 'summary': 'It is impossible to find a closed form indefinite integral for the given function, except through definite integral.', 'duration': 69.192, 'max_score': 1802.699, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01802699.jpg'}, {'end': 1940.776, 'src': 'embed', 'start': 1915.039, 'weight': 3, 'content': [{'end': 1920.181, 'text': "okay?. So when I say this integral is impossible, it means that it's impossible to do it in closed form.", 'start': 1915.039, 'duration': 5.142}, {'end': 1923.962, 'text': 'that is just as a finite sum in terms of what are called elementary functions.', 'start': 1920.181, 'duration': 3.781}, {'end': 1931.604, 'text': 'By elementary, I mean just the familiar functions, sine and cosine and exponential and log and polynomials and stuff.', 'start': 1924.042, 'duration': 7.562}, {'end': 1934.965, 'text': "And anything you can write down in terms of standard functions, you can't do.", 'start': 1931.624, 'duration': 3.341}, {'end': 1940.776, 'text': 'Okay, so this is impossible as a definite integral, as an indefinite integral.', 'start': 1936.934, 'duration': 3.842}], 'summary': 'The integral is impossible to solve in closed form using elementary functions.', 'duration': 25.737, 'max_score': 1915.039, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01915039.jpg'}, {'end': 2039.693, 'src': 'embed', 'start': 2011.053, 'weight': 4, 'content': [{'end': 2017.148, 'text': "And that method is, we have this problem that we can't do, so we write it down a second time.", 'start': 2011.053, 'duration': 6.095}, {'end': 2032.208, 'text': "And that solution may just look like kind of like banging your head against the wall that you can't do the integral,", 'start': 2027.264, 'duration': 4.944}, {'end': 2034.269, 'text': 'so you keep writing it down over and over again.', 'start': 2032.208, 'duration': 2.061}, {'end': 2037.352, 'text': "That doesn't seem like it would help the situation.", 'start': 2034.289, 'duration': 3.063}, {'end': 2039.693, 'text': 'Actually, this solves the whole problem.', 'start': 2038.012, 'duration': 1.681}], 'summary': 'Repeating problem-solving method helps solve the whole problem.', 'duration': 28.64, 'max_score': 2011.053, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02011053.jpg'}, {'end': 2427.905, 'src': 'heatmap', 'start': 2362.691, 'weight': 0.799, 'content': [{'end': 2368.061, 'text': 'So this integral is just 1.', 'start': 2362.691, 'duration': 5.37}, {'end': 2372.677, 'text': "So what are we doing? We're integrating 1 from 0 to 2 pi, that's 2 pi.", 'start': 2368.061, 'duration': 4.616}, {'end': 2382.596, 'text': "And lastly, we just noticed, well, that's not the integral we started with.", 'start': 2378.832, 'duration': 3.764}, {'end': 2384.978, 'text': "That's the square of the integral we started with.", 'start': 2382.616, 'duration': 2.362}, {'end': 2392.525, 'text': 'So therefore, the integral from minus infinity to infinity of e to the minus z squared over 2 dz.', 'start': 2385.499, 'duration': 7.026}, {'end': 2395.188, 'text': 'Since we wrote it down twice, we got 2 pi.', 'start': 2393.166, 'duration': 2.022}, {'end': 2398.131, 'text': "If we had written it down once, we'd get the square root of 2 pi.", 'start': 2395.528, 'duration': 2.603}, {'end': 2402.155, 'text': "So that's what we need for the normalizing constant.", 'start': 2399.552, 'duration': 2.603}, {'end': 2406.074, 'text': 'All right, so now we know what this C is.', 'start': 2403.692, 'duration': 2.382}, {'end': 2408.776, 'text': 'C equals 1 over square root of 2 pi.', 'start': 2406.775, 'duration': 2.001}, {'end': 2415.614, 'text': 'Kind of amazing, first of all, that this trick worked.', 'start': 2412.991, 'duration': 2.623}, {'end': 2423.341, 'text': "And secondly, we're integrating an exponential, and suddenly we get square root of pi.", 'start': 2416.234, 'duration': 7.107}, {'end': 2427.905, 'text': 'Where did the pi come from? Where did the circle come? Pi makes you think of circles.', 'start': 2423.521, 'duration': 4.384}], 'summary': 'Integrating an exponential, the constant c equals 1/sqrt(2pi), resulting from a notable trick.', 'duration': 65.214, 'max_score': 2362.691, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02362691.jpg'}, {'end': 2415.614, 'src': 'embed', 'start': 2385.499, 'weight': 5, 'content': [{'end': 2392.525, 'text': 'So therefore, the integral from minus infinity to infinity of e to the minus z squared over 2 dz.', 'start': 2385.499, 'duration': 7.026}, {'end': 2395.188, 'text': 'Since we wrote it down twice, we got 2 pi.', 'start': 2393.166, 'duration': 2.022}, {'end': 2398.131, 'text': "If we had written it down once, we'd get the square root of 2 pi.", 'start': 2395.528, 'duration': 2.603}, {'end': 2402.155, 'text': "So that's what we need for the normalizing constant.", 'start': 2399.552, 'duration': 2.603}, {'end': 2406.074, 'text': 'All right, so now we know what this C is.', 'start': 2403.692, 'duration': 2.382}, {'end': 2408.776, 'text': 'C equals 1 over square root of 2 pi.', 'start': 2406.775, 'duration': 2.001}, {'end': 2415.614, 'text': 'Kind of amazing, first of all, that this trick worked.', 'start': 2412.991, 'duration': 2.623}], 'summary': 'Integral of e^(-z^2/2) from -∞ to ∞ yields 2π, aiding normalization constant calculation.', 'duration': 30.115, 'max_score': 2385.499, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02385499.jpg'}], 'start': 1767.787, 'title': 'Gaussian integral and double integrals', 'summary': 'Discusses the impossibility of solving the gaussian integral in closed form, introduces the method of writing down the integral twice, and demonstrates the use of double integrals and polar coordinates, leading to the discovery of the normalizing constant as 1 over square root of 2 pi.', 'chapters': [{'end': 1871.891, 'start': 1767.787, 'title': 'The gaussian integral challenge', 'summary': 'Explains the impossibility of solving the famous gaussian integral in closed form, despite various attempts using different integration methods, as it was proven to be unsolvable except for the definite integral.', 'duration': 104.104, 'highlights': ['The theorem that says that the indefinite integral of the Gaussian function is impossible to do in closed form was proven by someone, making it an unsolvable problem. The indefinite integral of the Gaussian function, e to the power of negative z squared over 2, from minus infinity to infinity, is proven to be impossible to solve in closed form, as a theorem has been established to support this claim.', 'Attempts to solve the integral using various integration methods such as u substitution, change of variables, and integration by parts were unsuccessful. Efforts to solve the integral using u substitution, change of variables, and integration by parts all failed, indicating the difficulty and complexity of the Gaussian integral problem.', 'The only way to solve the Gaussian integral is through the definite integral method. The only successful approach to solving the Gaussian integral is through the definite integral method, as opposed to attempting to find an antiderivative and using the fundamental theorem of calculus.']}, {'end': 2438.548, 'start': 1872.571, 'title': 'Area under curve and double integrals', 'summary': 'Discusses the impossibility of solving a specific integral in closed form, introduces the method of writing down the integral twice, and demonstrates the use of double integrals and polar coordinates to solve the problem, leading to the discovery of the normalizing constant as 1 over square root of 2 pi.', 'duration': 565.977, 'highlights': ['The chapter discusses the impossibility of solving a specific integral in closed form The speaker explains that the integral cannot be solved in closed form, ruling out the use of elementary functions like sine, cosine, exponential, log, and polynomials.', "Introduces the method of writing down the integral twice The method of writing down the same integral twice is introduced as an 'incredibly stupid and incredibly brilliant way' to solve the impossible integral, leading to the discovery of the normalizing constant.", 'Demonstrates the use of double integrals and polar coordinates to solve the problem The use of double integrals and polar coordinates is demonstrated as a method to convert the original integral into a form that allows for an easier solution, leading to the discovery of the normalizing constant as 1 over square root of 2 pi.', 'Discovery of the normalizing constant as 1 over square root of 2 pi The discovery of the normalizing constant as 1 over square root of 2 pi is highlighted as a significant outcome of the problem-solving process using double integrals and polar coordinates.']}], 'duration': 670.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL01767787.jpg', 'highlights': ['The indefinite integral of the Gaussian function is impossible to solve in closed form, as proven by a theorem.', 'Efforts to solve the integral using u substitution, change of variables, and integration by parts all failed.', 'The only successful approach to solving the Gaussian integral is through the definite integral method.', 'The integral cannot be solved in closed form, ruling out the use of elementary functions like sine, cosine, exponential, log, and polynomials.', "The method of writing down the same integral twice is introduced as an 'incredibly stupid and incredibly brilliant way' to solve the impossible integral.", 'The use of double integrals and polar coordinates is demonstrated as a method to convert the original integral into a form that allows for an easier solution, leading to the discovery of the normalizing constant as 1 over square root of 2 pi.']}, {'end': 3069.013, 'segs': [{'end': 2501.252, 'src': 'embed', 'start': 2438.848, 'weight': 0, 'content': [{'end': 2441.25, 'text': "So that's the standard normal distribution.", 'start': 2438.848, 'duration': 2.402}, {'end': 2450.155, 'text': "Let's compute its mean and variance, and then we can talk a little bit about the general normal as opposed to the standard normal.", 'start': 2442.33, 'duration': 7.825}, {'end': 2454.438, 'text': "Okay, so that's the standard normal.", 'start': 2450.175, 'duration': 4.263}, {'end': 2457.74, 'text': "Let's compute its mean and variance.", 'start': 2455.759, 'duration': 1.981}, {'end': 2462.805, 'text': "I've already claimed that the mean is 0.", 'start': 2457.98, 'duration': 4.825}, {'end': 2465.586, 'text': "And the variance is 1, but we haven't checked that yet.", 'start': 2462.805, 'duration': 2.781}, {'end': 2474.327, 'text': "So let's verify that, okay? So first of all, let's get the mean.", 'start': 2469.826, 'duration': 4.501}, {'end': 2477.768, 'text': 'The mean is easy.', 'start': 2474.347, 'duration': 3.421}, {'end': 2480.749, 'text': "So we're gonna let Z be standard normal.", 'start': 2478.688, 'duration': 2.061}, {'end': 2486.55, 'text': "And, sorry for the pun, we're gonna compute EZ, which I said is easy.", 'start': 2482.309, 'duration': 4.241}, {'end': 2491.211, 'text': "Why is it easy? It's easy because of symmetry.", 'start': 2486.73, 'duration': 4.481}, {'end': 2501.252, 'text': 'By definition, the mean is the expected value of z times.', 'start': 2496.488, 'duration': 4.764}], 'summary': 'Compute mean and variance of standard normal distribution, mean=0, variance=1.', 'duration': 62.404, 'max_score': 2438.848, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02438848.jpg'}, {'end': 2651.916, 'src': 'embed', 'start': 2622.07, 'weight': 1, 'content': [{'end': 2624.632, 'text': 'Variance is gonna take a little more work.', 'start': 2622.07, 'duration': 2.562}, {'end': 2636.547, 'text': 'So the variance of z is e of z squared minus e of z squared the other way.', 'start': 2629.062, 'duration': 7.485}, {'end': 2641.209, 'text': "But that second part we just showed is 0, so that's just e of z squared.", 'start': 2637.367, 'duration': 3.842}, {'end': 2645.732, 'text': "So now is where we're starting to need LOTUS again.", 'start': 2642.13, 'duration': 3.602}, {'end': 2650.315, 'text': "So I'll remind you, we haven't proven LOTUS yet, but we'll talk about that on Wednesday.", 'start': 2646.112, 'duration': 4.203}, {'end': 2651.916, 'text': 'Here I just wanna show how to use it.', 'start': 2650.395, 'duration': 1.521}], 'summary': 'Variance of z is e of z squared, and lotus will be discussed later.', 'duration': 29.846, 'max_score': 2622.07, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02622070.jpg'}, {'end': 2993.422, 'src': 'embed', 'start': 2956.047, 'weight': 2, 'content': [{'end': 2960.931, 'text': "All right, so a couple more very quick things about the normal, and then we'll continue next time.", 'start': 2956.047, 'duration': 4.884}, {'end': 2967.917, 'text': 'Just for an important piece of notation, this is standard notation in statistics.', 'start': 2961.832, 'duration': 6.085}, {'end': 2980.383, 'text': 'The notation is for the CDF, capital Phi is the standard normal CDF.', 'start': 2972.011, 'duration': 8.372}, {'end': 2993.422, 'text': "Because this distribution is so important and yet so hard to deal with in the sense that it's a lot of work to do that integral.", 'start': 2982.927, 'duration': 10.495}], 'summary': 'The standard normal cdf, capital phi, is an important notation in statistics.', 'duration': 37.375, 'max_score': 2956.047, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02956047.jpg'}], 'start': 2438.848, 'title': 'Standard normal distribution and variance calculation', 'summary': 'Covers the mean and variance of the standard normal distribution, emphasizing mean as 0, variance as 1, and introduces variance calculation using lotus and integration by parts, resulting in a variance of 1 and standard normal cdf notation, phi(z).', 'chapters': [{'end': 2594.587, 'start': 2438.848, 'title': 'Standard normal distribution', 'summary': 'Discusses the mean and variance of the standard normal distribution, highlighting the mean as 0 and the variance as 1, using symmetry and defining odd functions, with emphasis on the integral of odd functions resulting in 0.', 'duration': 155.739, 'highlights': ['The mean of the standard normal distribution is 0, computed using the expected value of z times the PDF, resulting in an easy integral that evaluates to 0 due to symmetry. Mean of the standard normal distribution is 0, computed using the expected value and symmetry, with the integral result of 0.', 'The variance of the standard normal distribution is 1, verified through computation, although not explicitly shown in the provided transcript. Variance of the standard normal distribution is 1, not explicitly shown in the transcript.', 'Explanation of odd functions and their symmetry property, where g of -x equals -g of x, defining the concept of odd functions in relation to symmetry. Detailed explanation of odd functions and their symmetry property.']}, {'end': 3069.013, 'start': 2596.894, 'title': 'Variance calculation and standard normal notation', 'summary': 'Discusses the calculation of the variance using lotus and integration by parts, resulting in a variance of 1, and introduces the standard normal cdf notation, phi(z).', 'duration': 472.119, 'highlights': ['Calculation of Variance using LOTUS and Integration by Parts The variance of z is calculated as e(z^2) - e(z)^2, resulting in a variance of 1 after integration by parts, showcasing the application of LOTUS and symmetry.', 'Introduction of Standard Normal CDF Notation The standard normal CDF notation, phi(z), is introduced as 1/sqrt(2*pi) * integral from -infinity to z of e^(-t^2/2)dt, providing a method for easy calculation and usage of the standard normal distribution.']}], 'duration': 630.165, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/72QjzHnYvL0/pics/72QjzHnYvL02438848.jpg', 'highlights': ['The variance of the standard normal distribution is 1, verified through computation, although not explicitly shown in the provided transcript.', 'Calculation of Variance using LOTUS and Integration by Parts The variance of z is calculated as e(z^2) - e(z)^2, resulting in a variance of 1 after integration by parts, showcasing the application of LOTUS and symmetry.', 'Introduction of Standard Normal CDF Notation The standard normal CDF notation, phi(z), is introduced as 1/sqrt(2*pi) * integral from -infinity to z of e^(-t^2/2)dt, providing a method for easy calculation and usage of the standard normal distribution.', 'The mean of the standard normal distribution is 0, computed using the expected value of z times the PDF, resulting in an easy integral that evaluates to 0 due to symmetry.']}], 'highlights': ["The UK judge ruled against the use of Bayes' rule in court, potentially setting a precedent against its use.", "The expert witness in the case was criticized for estimating probabilities and using them in Bayes' rule, highlighting the importance of accurate input for reliable output.", 'The instructor strongly opposes the ruling and believes it should be overturned, indicating potential implications on expert witness testimony.', 'The theorem of universality states that if X equals F inverse of U, it will have CDF F if U is uniform 0, 1, allowing the synthetic creation of a random variable with any distribution desired, useful for simulation purposes.', 'The concept indicates that any function satisfying the properties of a CDF is a CDF, extending the understanding of CDF properties in defining a CDF.', 'The ability to use a uniform 0, 1 random variable to create random draws from any distribution, which is practical for simulation, as it is typically easier to generate uniforms than other distributions in the continuous case.', 'Reducing variables to uniform distribution aids in statistical inference and testing models (e.g. in 111) and simulation.', 'Inverse function simulation enables easy generation of random draws from the desired distribution.', 'The central limit theorem is possibly the most famous and important theorem in all of probability, to be discussed later in the course.', 'The normal distribution, also called the Gaussian, is the most famous and important distribution in all of statistics, with the central limit theorem being a key reason for its significance.', 'The indefinite integral of the Gaussian function is impossible to solve in closed form, as proven by a theorem.', 'The variance of the standard normal distribution is 1, verified through computation, although not explicitly shown in the provided transcript.', 'Introduction of Standard Normal CDF Notation The standard normal CDF notation, phi(z), is introduced as 1/sqrt(2*pi) * integral from -infinity to z of e^(-t^2/2)dt, providing a method for easy calculation and usage of the standard normal distribution.', 'The mean of the standard normal distribution is 0, computed using the expected value of z times the PDF, resulting in an easy integral that evaluates to 0 due to symmetry.']}