title

Lecture 16: Exponential Distribution | Statistics 110

description

We introduce the Exponential distribution, which is characterized by the memoryless property.
Note: This lecture video is shorter than the other Stat 110 lecture
videos, since the first part of class that day was devoted to giving
back and discussing the midterm exam.

detail

{'title': 'Lecture 16: Exponential Distribution | Statistics 110', 'heatmap': [{'end': 1091.477, 'start': 1052.502, 'weight': 0.898}], 'summary': 'The lecture covers exponential distribution basics, including rate parameter, pdf, cdf, and special properties, relevant to the course and homework assignment, while also exploring standardizing exponential distribution, calculating mean and variance, and explaining the memoryless property with specific examples and reasoning provided.', 'chapters': [{'end': 241.152, 'segs': [{'end': 69.788, 'src': 'embed', 'start': 25.504, 'weight': 0, 'content': [{'end': 29.406, 'text': 'So the next most important one is called the exponential distribution.', 'start': 25.504, 'duration': 3.902}, {'end': 34.888, 'text': "And that's one of the biggest topics for the next homework, and it's one of the most important distributions in general.", 'start': 30.006, 'duration': 4.882}, {'end': 37.63, 'text': "Now, we've already seen the exponential distribution.", 'start': 35.269, 'duration': 2.361}, {'end': 41.323, 'text': 'A couple times.', 'start': 40.583, 'duration': 0.74}, {'end': 48.526, 'text': "it's just like as part of like on the previous homework, with waiting for the book to be released,", 'start': 41.323, 'duration': 7.203}, {'end': 50.947, 'text': 'and I used it to illustrate universality of the uniform.', 'start': 48.526, 'duration': 2.421}, {'end': 53.268, 'text': "But we haven't formally introduced it yet.", 'start': 51.247, 'duration': 2.021}, {'end': 59.403, 'text': 'So this will be our formal introduction of exponential distribution.', 'start': 55.681, 'duration': 3.722}, {'end': 63.405, 'text': 'So stat 110, this is the exponential distribution.', 'start': 60.563, 'duration': 2.842}, {'end': 67.347, 'text': 'Exponential distribution, this is stat 110.', 'start': 63.805, 'duration': 3.542}, {'end': 69.788, 'text': "Now you're friends with the exponential distribution.", 'start': 67.347, 'duration': 2.441}], 'summary': 'Exponential distribution is a key topic in stat 110, emphasized for the next homework and as one of the most important distributions.', 'duration': 44.284, 'max_score': 25.504, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns25504.jpg'}, {'end': 167.266, 'src': 'embed', 'start': 135.848, 'weight': 2, 'content': [{'end': 137.609, 'text': "So it's always positive.", 'start': 135.848, 'duration': 1.761}, {'end': 141.351, 'text': "It's a continuous positive random variable.", 'start': 138.93, 'duration': 2.421}, {'end': 142.732, 'text': "That's the PDF.", 'start': 141.831, 'duration': 0.901}, {'end': 150.462, 'text': "You can see why it's called the exponential distribution cuz just an exponential function with a couple constant stuff stuck in there,", 'start': 144.52, 'duration': 5.942}, {'end': 151.522, 'text': 'exponential decay.', 'start': 150.462, 'duration': 1.06}, {'end': 163.325, 'text': 'This is a valid PDF because if we integrate lambda e to the minus, lambda x, dx from 0 to infinity,', 'start': 151.542, 'duration': 11.783}, {'end': 167.266, 'text': "that's a very easy integral and we get one just integrating exponential.", 'start': 163.325, 'duration': 3.941}], 'summary': 'Exponential distribution is a continuous positive random variable with a valid pdf, integrating to one.', 'duration': 31.418, 'max_score': 135.848, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns135848.jpg'}], 'start': 0.089, 'title': 'Exponential distribution basics', 'summary': 'Introduces the exponential distribution, its rate parameter, pdf, cdf, and special properties, illustrating its relevance to the course and the next homework assignment.', 'chapters': [{'end': 69.788, 'start': 0.089, 'title': 'Exponential distribution introduction', 'summary': 'Introduces the exponential distribution, one of the most important continuous distributions, and its significance for the next homework assignment, illustrating its relevance to the course.', 'duration': 69.699, 'highlights': ['Exponential distribution is one of the most important distributions in general, and a significant topic for the next homework.', 'The chapter provides a formal introduction to the exponential distribution, emphasizing its importance for the course.', 'The exponential distribution is illustrated as part of previous homework and used for demonstrating the universality of the uniform distribution.']}, {'end': 241.152, 'start': 69.928, 'title': 'Exponential distribution basics', 'summary': "Explains the exponential distribution's rate parameter, pdf, cdf, and special properties, such as its valid pdf, cdf, mean, and variance.", 'duration': 171.224, 'highlights': ['The exponential distribution has a PDF given by lambda e to the minus lambda x for x positive and 0 otherwise, representing a continuous positive random variable.', 'The CDF of the exponential distribution is given by 1 minus e to the minus lambda x for x greater than 0, and it is a valid CDF as it is an increasing function and approaches 1 as x goes to infinity.']}], 'duration': 241.063, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns89.jpg', 'highlights': ['Exponential distribution is one of the most important distributions in general, and a significant topic for the next homework.', 'The chapter provides a formal introduction to the exponential distribution, emphasizing its importance for the course.', 'The exponential distribution has a PDF given by lambda e to the minus lambda x for x positive and 0 otherwise, representing a continuous positive random variable.', 'The CDF of the exponential distribution is given by 1 minus e to the minus lambda x for x greater than 0, and it is a valid CDF as it is an increasing function and approaches 1 as x goes to infinity.', 'The exponential distribution is illustrated as part of previous homework and used for demonstrating the universality of the uniform distribution.']}, {'end': 610.067, 'segs': [{'end': 300.536, 'src': 'embed', 'start': 267.781, 'weight': 0, 'content': [{'end': 270.782, 'text': 'In this case, the normal has two parameters, mu and sigma squared.', 'start': 267.781, 'duration': 3.001}, {'end': 274.103, 'text': "Here there's only one parameter lambda.", 'start': 271.482, 'duration': 2.621}, {'end': 294.313, 'text': "okay?. And we're gonna show right now that if we let y equal lambda times x, then we're gonna show that y is exponential of 1..", 'start': 274.103, 'duration': 20.21}, {'end': 300.536, 'text': "So that's kind of analogous to standardization of the standard normal, in that we're multiplying.", 'start': 294.313, 'duration': 6.223}], 'summary': 'Comparison between exponential distribution and standard normal distribution parameters.', 'duration': 32.755, 'max_score': 267.781, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns267781.jpg'}, {'end': 426.318, 'src': 'embed', 'start': 350.12, 'weight': 1, 'content': [{'end': 354.443, 'text': "But that's just the same as the CDF of x evaluated at y over lambda.", 'start': 350.12, 'duration': 4.323}, {'end': 364.768, 'text': "So if we take this CDF and plug in y over lambda, then we just get 1- e to the minus lambda, right? X, well, that's y there.", 'start': 354.463, 'duration': 10.305}, {'end': 370.59, 'text': 'an exponential of 1, right?', 'start': 368.55, 'duration': 2.04}, {'end': 375.151, 'text': 'I just plugged in y over lambda here and the lambdas cancel, okay?', 'start': 371.351, 'duration': 3.8}, {'end': 381.013, 'text': "So it's just that one short equation proves that this statement is true.", 'start': 375.472, 'duration': 5.541}, {'end': 391.815, 'text': "Okay, so let's first concentrate on the case where it has parameter 1 and find the mean and variance, and then we can get the general case okay?", 'start': 381.033, 'duration': 10.782}, {'end': 395.096, 'text': "So let's do that case now.", 'start': 393.616, 'duration': 1.48}, {'end': 405.923, 'text': "So we're letting y be exponential of 1.", 'start': 397.597, 'duration': 8.326}, {'end': 407.604, 'text': 'And we wanna find the mean invariance.', 'start': 405.923, 'duration': 1.681}, {'end': 416.371, 'text': "Okay? So let's get the mean first.", 'start': 412.628, 'duration': 3.743}, {'end': 426.318, 'text': 'Just by definition, expected value y is just the integral of y times the PDF.', 'start': 417.031, 'duration': 9.287}], 'summary': 'Analyzing exponential distribution with parameter 1 to find mean and variance.', 'duration': 76.198, 'max_score': 350.12, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns350120.jpg'}, {'end': 546.971, 'src': 'embed', 'start': 513.78, 'weight': 3, 'content': [{'end': 516.022, 'text': "It has to be 1, otherwise that wasn't a valid PDF.", 'start': 513.78, 'duration': 2.242}, {'end': 523.986, 'text': "So in this course, there are actually a lot of integrals we can do, not using calculus, but by recognizing, that's a PDF, or that kind of thing.", 'start': 516.662, 'duration': 7.324}, {'end': 525.527, 'text': "Of course, that's an easy integral anyway.", 'start': 524.147, 'duration': 1.38}, {'end': 529.37, 'text': 'Okay, therefore, the mean is 1.', 'start': 526.168, 'duration': 3.202}, {'end': 546.971, 'text': 'Now, for the variance, For the variance, we can just do e of y squared minus e of y squared the other way, as usual.', 'start': 529.37, 'duration': 17.601}], 'summary': 'Course covers integrals, mean is 1, variance calculation explained.', 'duration': 33.191, 'max_score': 513.78, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns513780.jpg'}, {'end': 630.922, 'src': 'embed', 'start': 595.361, 'weight': 4, 'content': [{'end': 599.123, 'text': "basically it's gonna reduce it back down to an integral that looks like this right?", 'start': 595.361, 'duration': 3.762}, {'end': 601.304, 'text': "It's gonna lower the power from two to one.", 'start': 599.163, 'duration': 2.141}, {'end': 603.965, 'text': 'So this is like one where you do integration by parts twice.', 'start': 601.684, 'duration': 2.281}, {'end': 608.426, 'text': "You don't actually have to do it twice cuz you do it once and then you reduce it to the one we just did.", 'start': 604.525, 'duration': 3.901}, {'end': 610.067, 'text': 'So you actually do integration by parts once.', 'start': 608.466, 'duration': 1.601}, {'end': 614.789, 'text': "And if you do that, you also get one, and that's pretty easy to remember.", 'start': 611.368, 'duration': 3.421}, {'end': 620.074, 'text': 'Okay, so now coming back to this general exponential.', 'start': 616.532, 'duration': 3.542}, {'end': 630.922, 'text': 'So just continuing the notation here, y equals lambda x, so x equals y over lambda.', 'start': 623.076, 'duration': 7.846}], 'summary': 'The process reduces the integral from power two to one, simplifying integration by parts to once, yielding y = lambda x and x = y / lambda.', 'duration': 35.561, 'max_score': 595.361, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns595361.jpg'}], 'start': 241.192, 'title': 'Calculating distributions and moments', 'summary': 'Explores standardizing exponential distribution and calculating mean and variance, as well as using integration by parts for moment calculation, with specific examples and reasoning provided.', 'chapters': [{'end': 426.318, 'start': 241.192, 'title': 'Calculating cdf and standardizing exponential distribution', 'summary': 'Discusses standardizing the exponential distribution by showing the relationship between a normal distribution and an exponential distribution, with a focus on calculating the mean and variance of the exponential distribution.', 'duration': 185.126, 'highlights': ['Showing the relationship between a normal distribution and an exponential distribution by standardizing the exponential distribution. The chapter explains the relationship between a normal distribution and an exponential distribution by demonstrating the process of standardizing the exponential distribution, akin to standardizing a normal distribution, to make the calculations easier.', 'Calculation of CDF of y, demonstrating y less than or equal to y/lambda, leading to 1- e to the minus lambda, which represents an exponential of 1. The chapter provides a detailed calculation of the CDF of y, illustrating the process of deriving 1- e to the minus lambda, which signifies an exponential of 1, simplifying the calculations and relationships within the distribution.', "Explanation of finding the mean and variance of the exponential distribution. The chapter delves into the process of finding the mean and variance of the exponential distribution, beginning with the computation of the mean through the integral of y times the PDF, leading to a comprehensive understanding of the distribution's parameters."]}, {'end': 610.067, 'start': 427.341, 'title': 'Integration by parts and moment calculation', 'summary': 'Discusses integration by parts to calculate the mean and variance of a random variable, with specific examples and reasoning behind the calculations.', 'duration': 182.726, 'highlights': ['The mean of the random variable is calculated as 1, based on the evaluation of the integral and recognizing it as a probability density function (PDF).', 'The process of calculating the variance involves the use of integration by parts and recognizing the second moment of the random variable, providing insights into alternative methods for obtaining moments.', 'The application of integration by parts is demonstrated in calculating the variance, showcasing the reduction of the integral through power reduction and its relationship to moments of random variables.']}], 'duration': 368.875, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns241192.jpg', 'highlights': ['The chapter explains the relationship between a normal distribution and an exponential distribution by demonstrating the process of standardizing the exponential distribution, akin to standardizing a normal distribution, to make the calculations easier.', 'The chapter provides a detailed calculation of the CDF of y, illustrating the process of deriving 1- e to the minus lambda, which signifies an exponential of 1, simplifying the calculations and relationships within the distribution.', "The chapter delves into the process of finding the mean and variance of the exponential distribution, beginning with the computation of the mean through the integral of y times the PDF, leading to a comprehensive understanding of the distribution's parameters.", 'The process of calculating the variance involves the use of integration by parts and recognizing the second moment of the random variable, providing insights into alternative methods for obtaining moments.', 'The application of integration by parts is demonstrated in calculating the variance, showcasing the reduction of the integral through power reduction and its relationship to moments of random variables.', 'The mean of the random variable is calculated as 1, based on the evaluation of the integral and recognizing it as a probability density function (PDF).']}, {'end': 1098.756, 'segs': [{'end': 646.393, 'src': 'embed', 'start': 611.368, 'weight': 0, 'content': [{'end': 614.789, 'text': "And if you do that, you also get one, and that's pretty easy to remember.", 'start': 611.368, 'duration': 3.421}, {'end': 620.074, 'text': 'Okay, so now coming back to this general exponential.', 'start': 616.532, 'duration': 3.542}, {'end': 630.922, 'text': 'So just continuing the notation here, y equals lambda x, so x equals y over lambda.', 'start': 623.076, 'duration': 7.846}, {'end': 636.947, 'text': "And that's gonna have the mean.", 'start': 632.363, 'duration': 4.584}, {'end': 642.471, 'text': 'Just take the expected value of this, lambda is a constant, that comes out.', 'start': 638.888, 'duration': 3.583}, {'end': 646.393, 'text': 'So it has mean 1 over lambda, and it has variance.', 'start': 643.251, 'duration': 3.142}], 'summary': 'Exponential function has mean 1/lambda and variance.', 'duration': 35.025, 'max_score': 611.368, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns611368.jpg'}, {'end': 854.441, 'src': 'embed', 'start': 825.082, 'weight': 1, 'content': [{'end': 829.703, 'text': "Because you've started over with a fresh exponential distribution with the same parameter.", 'start': 825.082, 'duration': 4.621}, {'end': 832.123, 'text': "That's the definition of the memoryless property.", 'start': 830.243, 'duration': 1.88}, {'end': 838.305, 'text': 'but I hope that this equation kind of intuitively makes sense to capture that intuition of memorylessness.', 'start': 832.123, 'duration': 6.182}, {'end': 843.846, 'text': "Okay, so now let's prove that in fact that this equation is satisfied by the exponential.", 'start': 839.485, 'duration': 4.361}, {'end': 849.476, 'text': 'So for the exponential lambda, this is the general definition.', 'start': 844.952, 'duration': 4.524}, {'end': 854.441, 'text': "But now we're assuming X is exponential lambda, and let's check that that equation holds.", 'start': 849.697, 'duration': 4.744}], 'summary': 'Proving that the exponential distribution satisfies the memoryless property.', 'duration': 29.359, 'max_score': 825.082, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns825082.jpg'}, {'end': 1091.477, 'src': 'heatmap', 'start': 1052.502, 'weight': 0.898, 'content': [{'end': 1058.167, 'text': "that you've waited at least A minutes for what's the expected value of X given that information?", 'start': 1052.502, 'duration': 5.665}, {'end': 1063.839, 'text': 'Well, we could think of that as A plus E, of X minus A.', 'start': 1058.857, 'duration': 4.982}, {'end': 1067.541, 'text': 'just linearity, pull out an A.', 'start': 1063.839, 'duration': 3.702}, {'end': 1075.884, 'text': 'The cool thing about the memoryless property is, given that X is greater than A, this thing X minus A just becomes a fresh exponential right?', 'start': 1067.541, 'duration': 8.343}, {'end': 1078.805, 'text': 'Because this is the additional waiting time.', 'start': 1075.904, 'duration': 2.901}, {'end': 1087.769, 'text': "You waited A, but it started over again, right? So it's immediate that this is just A plus 1 over lambda by the memoryless property.", 'start': 1078.825, 'duration': 8.944}, {'end': 1091.477, 'text': "Otherwise you'd have to do some calculus, do integration and stuff,", 'start': 1088.789, 'duration': 2.688}], 'summary': 'Expected value of x given a minutes wait: a + 1/lambda by memoryless property.', 'duration': 38.975, 'max_score': 1052.502, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns1052502.jpg'}], 'start': 611.368, 'title': 'Exponential distribution', 'summary': 'Covers the properties of exponential distribution, including mean and variance, with mean being 1 over lambda and variance being 1 over lambda squared. it also explains the memoryless property in exponential distribution, demonstrating its satisfaction by the distribution and its unique characterization, along with its usefulness in calculating conditional expectation.', 'chapters': [{'end': 673.629, 'start': 611.368, 'title': 'Exponential distribution properties', 'summary': 'Discusses the properties of exponential distribution, including its mean and variance, where mean is 1 over lambda and variance is 1 over lambda squared.', 'duration': 62.261, 'highlights': ["The exponential distribution's mean is 1 over lambda, and its variance is 1 over lambda squared.", 'The chapter explains the importance of the exponential distribution and its relevance in different contexts.']}, {'end': 1098.756, 'start': 674.33, 'title': 'Memoryless property in exponential distribution', 'summary': 'Explains the memoryless property in exponential distribution, demonstrating how it is satisfied by the exponential distribution and how it is the only distribution that completely characterizes the memoryless property, along with a corollary that showcases the usefulness of the memoryless property in calculating conditional expectation.', 'duration': 424.426, 'highlights': ['The chapter explains the memoryless property in exponential distribution and demonstrates how it is satisfied by the exponential distribution. Demonstrates the property through equations and calculations.', 'It is revealed that the exponential distribution is the only distribution that completely characterizes the memoryless property. States that the exponential distribution is the only distribution that satisfies the memoryless property.', 'A corollary is presented that showcases the usefulness of the memoryless property in calculating conditional expectation. Demonstrates how the memoryless property simplifies the calculation of conditional expectation without the need for integration.']}], 'duration': 487.388, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/bM6nFDjvEns/pics/bM6nFDjvEns611368.jpg', 'highlights': ["The exponential distribution's mean is 1 over lambda, and its variance is 1 over lambda squared.", 'The chapter explains the memoryless property in exponential distribution and demonstrates how it is satisfied by the exponential distribution.', 'A corollary is presented that showcases the usefulness of the memoryless property in calculating conditional expectation.']}], 'highlights': ['The exponential distribution has a PDF given by lambda e to the minus lambda x for x positive and 0 otherwise, representing a continuous positive random variable.', 'The CDF of the exponential distribution is given by 1 minus e to the minus lambda x for x greater than 0, and it is a valid CDF as it is an increasing function and approaches 1 as x goes to infinity.', 'The chapter explains the relationship between a normal distribution and an exponential distribution by demonstrating the process of standardizing the exponential distribution, akin to standardizing a normal distribution, to make the calculations easier.', "The chapter delves into the process of finding the mean and variance of the exponential distribution, beginning with the computation of the mean through the integral of y times the PDF, leading to a comprehensive understanding of the distribution's parameters.", "The exponential distribution's mean is 1 over lambda, and its variance is 1 over lambda squared.", 'The chapter explains the memoryless property in exponential distribution and demonstrates how it is satisfied by the exponential distribution.']}