title
Convolutions | Why X+Y in probability is a beautiful mess

description
Adding random variables, with connections to the central limit theorem. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share the videos. 0:00 - Intro quiz 2:24 - Discrete case, diagonal slices 6:49 - Discrete case, flip-and-slide 8:41 - The discrete formula 10:58 - Continuous case, flip-and-slide 15:53 - Example with uniform distributions 18:42 - Central limit theorem 20:50 - Continuous case, diagonal slices 25:26 - Returning to the intro quiz ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown

detail
{'title': 'Convolutions | Why X+Y in probability is a beautiful mess', 'heatmap': [{'end': 677.728, 'start': 661.232, 'weight': 0.899}], 'summary': 'Delves into summing and convolution of random variables, explaining the interpretation of normal distribution curve, discussing convolution in probability with examples, and exploring symbolic representation of pairwise products, continuous distributions, integral, and uniform distributions, emphasizing the central limit theorem and visualization techniques for rolling dice computation.', 'chapters': [{'end': 282.643, 'segs': [{'end': 53.405, 'src': 'embed', 'start': 18.51, 'weight': 0, 'content': [{'end': 25.294, 'text': 'what the function actually means is that if you want the probability that your sample falls within a given range of values,', 'start': 18.51, 'duration': 6.784}, {'end': 32.756, 'text': 'say the probability that it ends up between negative one and two, well, that would equal the area under this curve in that range of values.', 'start': 25.294, 'duration': 7.462}, {'end': 34.618, 'text': "That's what the curve actually means.", 'start': 32.957, 'duration': 1.661}, {'end': 41.4, 'text': "I'll also pull up a second random variable, also following a normal distribution, but maybe this time a little more spread out,", 'start': 35.298, 'duration': 6.102}, {'end': 42.981, 'text': 'a slightly bigger standard deviation.', 'start': 41.4, 'duration': 1.581}, {'end': 44.302, 'text': "And here's the quiz for you.", 'start': 43.381, 'duration': 0.921}, {'end': 49.804, 'text': 'If you repeatedly sample both of these variables and at each iteration you add up the two results.', 'start': 44.902, 'duration': 4.902}, {'end': 53.405, 'text': 'Well then, that sum behaves like its own random variable.', 'start': 50.444, 'duration': 2.961}], 'summary': 'The probability of a sample falling within a certain range can be calculated as the area under the curve. when adding up results from two random variables with normal distributions, the sum behaves like its own random variable.', 'duration': 34.895, 'max_score': 18.51, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M18510.jpg'}, {'end': 139.6, 'src': 'embed', 'start': 89.023, 'weight': 3, 'content': [{'end': 96.027, 'text': 'this is actually meant to be a much more general lesson about how you add two different random variables regardless of their distribution,', 'start': 89.023, 'duration': 7.004}, {'end': 98.249, 'text': 'not necessarily just the normally distributed ones.', 'start': 96.027, 'duration': 2.222}, {'end': 104.406, 'text': 'This amounts to a special operation that you apply to the distributions underlying those variables.', 'start': 99.164, 'duration': 5.242}, {'end': 107.568, 'text': "The operation has a special name, it's called a convolution.", 'start': 104.727, 'duration': 2.841}, {'end': 122.095, 'text': 'And the primary thing you and I will do today is motivate and build up two distinct ways to visualize what a convolution looks like for continuous functions and then to talk about how these two different visualizations can each be helpful in different ways,', 'start': 107.968, 'duration': 14.127}, {'end': 123.996, 'text': 'with a special focus on the central limit theorem.', 'start': 122.095, 'duration': 1.901}, {'end': 131.277, 'text': 'After we do the general lesson, I want to return to the opening quiz and offer an unusually satisfying way to answer it.', 'start': 124.975, 'duration': 6.302}, {'end': 138.02, 'text': "As a quick side note, regular viewers among you might know there's already a video about convolutions on this channel,", 'start': 132.218, 'duration': 5.802}, {'end': 139.6, 'text': 'but that one had a pretty different focus.', 'start': 138.02, 'duration': 1.58}], 'summary': 'Lesson on convolutions for random variables, with visualizations and focus on central limit theorem.', 'duration': 50.577, 'max_score': 89.023, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M89023.jpg'}, {'end': 296.254, 'src': 'embed', 'start': 265.065, 'weight': 6, 'content': [{'end': 269.289, 'text': "of course, the die rolls should be independent from each other, but it's a point worth emphasizing,", 'start': 265.065, 'duration': 4.224}, {'end': 274.835, 'text': "because everything that we're going to do from here, moving forward from this simple example, all the way up to the central limit,", 'start': 269.289, 'duration': 5.546}, {'end': 278.138, 'text': 'theorem assumes that the random variables are independent.', 'start': 274.835, 'duration': 3.303}, {'end': 282.643, 'text': 'In the real world you want to keep a sharp eye out for if this assumption actually holds.', 'start': 278.799, 'duration': 3.844}, {'end': 288.768, 'text': "Now what I'm going to do is take this grid of all possible outcomes, but start filling it in with some numbers.", 'start': 283.584, 'duration': 5.184}, {'end': 296.254, 'text': "Maybe we'll put the numbers for all the probabilities of the blue die down on the bottom, all the probabilities for the red die over here on the left,", 'start': 289.169, 'duration': 7.085}], 'summary': 'Emphasizing the independence of die rolls for future statistical calculations.', 'duration': 31.189, 'max_score': 265.065, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M265065.jpg'}], 'start': 0.089, 'title': 'Summing and convolution of random variables', 'summary': 'Explores summing two normal distributions, emphasizing interpretation of normal distribution curve, and discusses convolution for adding random variables, focusing on central limit theorem, real-world applications, and independence of variables.', 'chapters': [{'end': 89.023, 'start': 0.089, 'title': 'Summing random variables distribution', 'summary': 'Explores the behavior of summing two random variables drawn from normal distributions, emphasizing the interpretation of the normal distribution curve and the resulting distribution of the sum.', 'duration': 88.934, 'highlights': ['The sum of two random variables drawn from normal distributions also follows a normal distribution, showcasing the significance of understanding the behavior and interpretation of normal distributions in probability.', "The interpretation of the normal distribution curve lies in the area under the curve representing the probability of the sample falling within a given range of values, providing a fundamental understanding of the distribution's significance in probability.", 'Understanding the distribution that describes the sum of two random variables drawn from normal distributions is crucial, emphasizing the need to comprehend the underlying reasons for the resulting distribution.', 'The behavior of summing two random variables drawn from normal distributions serves as a pivotal concept in understanding the special function of normal distributions in probability, highlighting the importance of a profound comprehension of the topic.']}, {'end': 282.643, 'start': 89.023, 'title': 'Understanding convolution and visualizing distributions', 'summary': 'Discusses the concept of convolution for adding different random variables, using two distinct visualizations and focusing on the central limit theorem, with a special emphasis on the independence of random variables and their applications in real-world scenarios.', 'duration': 193.62, 'highlights': ['The chapter explains the concept of convolution for adding different random variables and emphasizes its application beyond normally distributed ones. ', 'It describes two distinct ways to visualize convolutions for continuous functions and highlights their respective usefulness, particularly in the context of the central limit theorem. ', "The video offers a satisfying way to answer the opening quiz and mentions the previous video's different focus on the discrete case of convolutions. ", 'It discusses the importance of independence of random variables in various contexts, including the central limit theorem, and advises vigilance in ensuring this assumption holds in real-world scenarios. ']}], 'duration': 282.554, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M89.jpg', 'highlights': ['The sum of two random variables drawn from normal distributions also follows a normal distribution, showcasing the significance of understanding the behavior and interpretation of normal distributions in probability.', "The interpretation of the normal distribution curve lies in the area under the curve representing the probability of the sample falling within a given range of values, providing a fundamental understanding of the distribution's significance in probability.", 'The behavior of summing two random variables drawn from normal distributions serves as a pivotal concept in understanding the special function of normal distributions in probability, highlighting the importance of a profound comprehension of the topic.', 'Understanding the distribution that describes the sum of two random variables drawn from normal distributions is crucial, emphasizing the need to comprehend the underlying reasons for the resulting distribution.', 'The chapter explains the concept of convolution for adding different random variables and emphasizes its application beyond normally distributed ones.', 'It describes two distinct ways to visualize convolutions for continuous functions and highlights their respective usefulness, particularly in the context of the central limit theorem.', 'It discusses the importance of independence of random variables in various contexts, including the central limit theorem, and advises vigilance in ensuring this assumption holds in real-world scenarios.', "The video offers a satisfying way to answer the opening quiz and mentions the previous video's different focus on the discrete case of convolutions."]}, {'end': 565.975, 'segs': [{'end': 359.925, 'src': 'embed', 'start': 296.254, 'weight': 0, 'content': [{'end': 306.162, 'text': 'and then we will fill in the grid where the probability for every outcome inside the grid looks like some product between one number from the blue distribution and one number from the red distribution.', 'start': 296.254, 'duration': 9.908}, {'end': 310.106, 'text': "Another way to think about it is we're basically constructing a multiplication table.", 'start': 306.683, 'duration': 3.423}, {'end': 312.903, 'text': 'To be a little more visual about all of this,', 'start': 310.782, 'duration': 2.121}, {'end': 319.646, 'text': 'we could plot each one of these probabilities as the height of a bar above the square in this sort of three-dimensional plot.', 'start': 312.903, 'duration': 6.743}, {'end': 325.528, 'text': 'In some sense, this three-dimensional plot carries all the data that we would need to know about rolling a pair of dice.', 'start': 320.206, 'duration': 5.322}, {'end': 330.03, 'text': 'And so the question is how do we extract the thing that we want to know?', 'start': 326.388, 'duration': 3.642}, {'end': 332.191, 'text': 'the probabilities for various different sums?', 'start': 330.03, 'duration': 2.161}, {'end': 341.299, 'text': 'Well, if you highlight all of the outcomes with a certain sum, say a sum of six, Notice how all of those end up on a certain diagonal.', 'start': 333.611, 'duration': 7.688}, {'end': 345.042, 'text': 'Same deal if I highlight all the pairs where the sum is 7.', 'start': 341.7, 'duration': 3.342}, {'end': 346.784, 'text': 'They sit along a different diagonal.', 'start': 345.042, 'duration': 1.742}, {'end': 354.651, 'text': 'So to compute the probability of each possible sum, what you do is you add together all of the entries that sit on one of these diagonals.', 'start': 347.284, 'duration': 7.367}, {'end': 359.925, 'text': 'Pulling up the 3D plot.', 'start': 358.444, 'duration': 1.481}], 'summary': 'Constructing a multiplication table to compute probabilities for dice outcomes.', 'duration': 63.671, 'max_score': 296.254, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M296254.jpg'}, {'end': 525.223, 'src': 'embed', 'start': 483.22, 'weight': 4, 'content': [{'end': 483.72, 'text': 'In general.', 'start': 483.22, 'duration': 0.5}, {'end': 484.701, 'text': 'from this point of view,', 'start': 483.72, 'duration': 0.981}, {'end': 493.167, 'text': 'computing the full distribution for the sum looks like sliding that bottom distribution into various different positions and computing this dot product along the way.', 'start': 484.701, 'duration': 8.466}, {'end': 499.751, 'text': 'It is precisely the same operation as the diagonal slices we were looking at earlier.', 'start': 495.108, 'duration': 4.643}, {'end': 503.694, 'text': "They're just two different ways to visualize the same underlying operation.", 'start': 500.272, 'duration': 3.422}, {'end': 509.799, 'text': 'And however you choose to visualize it,', 'start': 507.478, 'duration': 2.321}, {'end': 518.861, 'text': 'this operation that takes in two different distributions and spits out a new one describing the sum of the relevant random variables is called a convolution,', 'start': 509.799, 'duration': 9.062}, {'end': 520.782, 'text': 'and we often denote it with this asterisk.', 'start': 518.861, 'duration': 1.921}, {'end': 525.223, 'text': 'Really, the way you want to think about it, especially as we set up for the continuous case,', 'start': 521.381, 'duration': 3.842}], 'summary': 'Computing the full distribution for the sum involves convolution, denoted by asterisk.', 'duration': 42.003, 'max_score': 483.22, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M483220.jpg'}], 'start': 283.584, 'title': 'Probability and convolution', 'summary': 'Explains calculating probabilities for dice sums using a multiplication table and visual representation, and discusses convolution in probability with examples and visualization techniques for rolling dice computation.', 'chapters': [{'end': 396.281, 'start': 283.584, 'title': 'Probability of dice sums', 'summary': 'Explains how to calculate the probabilities for different sums of rolling a pair of dice by constructing a multiplication table and visually representing the probabilities as a three-dimensional plot, where the probabilities for each sum can be computed by adding up the entries along specific diagonals.', 'duration': 112.697, 'highlights': ['The probabilities for each outcome inside the grid are calculated as a product between one number from the blue distribution and one number from the red distribution. The method of calculating probabilities for each outcome inside the grid is discussed, emphasizing the product of probabilities from the blue and red distributions.', 'The visualization of probabilities as a three-dimensional plot represents all the data needed to understand the rolling of a pair of dice. The three-dimensional plot is described as a comprehensive representation of the probabilities for different outcomes when rolling a pair of dice.', 'The computation of the probability of each possible sum involves adding together all the entries that sit on specific diagonals within the grid. The process of computing the probability of each possible sum is explained, focusing on the addition of entries along specific diagonals within the grid.']}, {'end': 565.975, 'start': 396.981, 'title': 'Understanding convolution in probability', 'summary': 'Discusses the concept of convolution in probability, highlighting how it can be visualized through diagonal slices and the dot product, with examples demonstrating the computation process for rolling dice.', 'duration': 168.994, 'highlights': ['The concept of convolution in probability is visualized through diagonal slices and the dot product, demonstrating the computation process for rolling dice. It explains how to visualize the convolution operation through diagonal slices and the dot product, providing examples of computing the probabilities for rolling dice.', 'Sliding the bottom distribution into various positions and computing the dot product along the way is precisely the same operation as the diagonal slices, offering a different visualization for the underlying operation of convolution. It highlights the equivalence of sliding the bottom distribution and computing the dot product to the diagonal slices, presenting it as an alternative visualization for the convolution operation.', 'The convolution operation in probability takes in two different distributions and generates a new one describing the sum of the relevant random variables, denoted with an asterisk and illustrated as combining two different functions to produce a new function. It introduces the concept of the convolution operation in probability, depicting it as a process that combines two different distributions to produce a new one, denoted with an asterisk and visualized as combining functions.']}], 'duration': 282.391, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M283584.jpg', 'highlights': ['The visualization of probabilities as a three-dimensional plot represents all the data needed to understand the rolling of a pair of dice.', 'The probabilities for each outcome inside the grid are calculated as a product between one number from the blue distribution and one number from the red distribution.', 'The method of calculating probabilities for each outcome inside the grid is discussed, emphasizing the product of probabilities from the blue and red distributions.', 'The computation of the probability of each possible sum involves adding together all the entries that sit on specific diagonals within the grid.', 'The concept of convolution in probability is visualized through diagonal slices and the dot product, demonstrating the computation process for rolling dice.', 'The convolution operation in probability takes in two different distributions and generates a new one describing the sum of the relevant random variables, denoted with an asterisk and illustrated as combining two different functions to produce a new function.', 'Sliding the bottom distribution into various positions and computing the dot product along the way is precisely the same operation as the diagonal slices, offering a different visualization for the underlying operation of convolution.']}, {'end': 798.242, 'segs': [{'end': 603.357, 'src': 'embed', 'start': 566.455, 'weight': 0, 'content': [{'end': 570.457, 'text': "You've seen two different ways to visualize it, but how do we actually write it down in symbols?", 'start': 566.455, 'duration': 4.002}, {'end': 576.121, 'text': "To get your bearings, maybe it's helpful to write down a specific example, like the case of plugging in a 4,", 'start': 570.998, 'duration': 5.123}, {'end': 582.404, 'text': 'where you add up over all the different pairwise products corresponding to pairs of inputs that add up to a 4..', 'start': 576.121, 'duration': 6.283}, {'end': 584.385, 'text': "And more generally, here's how it might look.", 'start': 582.404, 'duration': 1.981}, {'end': 591.95, 'text': "This new function takes as an input a possible sum for your random variables, which I'll call s, and what it outputs.", 'start': 585.046, 'duration': 6.904}, {'end': 595.992, 'text': 'looks like a sum over a bunch of pairs of values for x and y.', 'start': 591.95, 'duration': 4.042}, {'end': 603.357, 'text': "Except the usual way it's written is not to write with x and y, but instead we just focus on one of those variables, in this case x,", 'start': 595.992, 'duration': 7.365}], 'summary': 'Exploring symbol representation and example of plugging in a 4 for pairwise products.', 'duration': 36.902, 'max_score': 566.455, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M566455.jpg'}, {'end': 677.728, 'src': 'heatmap', 'start': 622.168, 'weight': 1, 'content': [{'end': 631.374, 'text': 'For example, if you plug in a given value like s equals 4, and you unpack this sum, letting x range over all the possible values going from 1 up to 6,', 'start': 622.168, 'duration': 9.206}, {'end': 636.997, 'text': "then sometimes that corresponding y value drops below the domain of what we've explicitly defined.", 'start': 631.374, 'duration': 5.623}, {'end': 641.42, 'text': 'For example, you plug in 0 and negative 1 and negative 2.', 'start': 637.378, 'duration': 4.042}, {'end': 642.861, 'text': "It's not actually that big a deal.", 'start': 641.42, 'duration': 1.441}, {'end': 645.664, 'text': 'Essentially, you would just say all of these values are zero.', 'start': 643.002, 'duration': 2.662}, {'end': 648.106, 'text': "So all these later terms don't get counted.", 'start': 645.924, 'duration': 2.182}, {'end': 649.707, 'text': 'And that should kind of make sense.', 'start': 648.706, 'duration': 1.001}, {'end': 654.832, 'text': "What is the probability that the red die rolls to become a negative one? Well, it's zero.", 'start': 649.927, 'duration': 4.905}, {'end': 656.333, 'text': 'That is an impossible outcome.', 'start': 654.972, 'duration': 1.361}, {'end': 665.116, 'text': "As a next step, let's turn our attention towards continuous distributions,", 'start': 661.232, 'duration': 3.884}, {'end': 671.002, 'text': 'where your random variable can take on values anywhere in an infinite continuum, like all possible real numbers.', 'start': 665.116, 'duration': 5.886}, {'end': 677.728, 'text': "Maybe you're doing weather modeling and trying to predict the temperature tomorrow at noon, or you're doing some financial projections,", 'start': 671.502, 'duration': 6.226}], 'summary': 'Explains handling of values in a given scenario, including zero probabilities and continuous distributions.', 'duration': 42.948, 'max_score': 622.168, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M622168.jpg'}, {'end': 739.454, 'src': 'embed', 'start': 703.45, 'weight': 3, 'content': [{'end': 711.036, 'text': 'Essentially, the probability that a sample of your variable falls within a given range looks like the area under the curve in that range.', 'start': 703.45, 'duration': 7.586}, {'end': 715.919, 'text': 'The function describing this curve is commonly called a probability density function,', 'start': 711.636, 'duration': 4.283}, {'end': 719.781, 'text': "a common enough phrase that it's frequently just given the abbreviation pdf.", 'start': 715.919, 'duration': 3.862}, {'end': 729.207, 'text': 'And so the proper way to write all of this down would be to say that the probability that your sample falls within a given range looks like the integral of your pdf.', 'start': 720.282, 'duration': 8.925}, {'end': 731.949, 'text': 'the probability density function in that range.', 'start': 729.207, 'duration': 2.742}, {'end': 739.454, 'text': 'As a general rule of thumb, anytime that you see a sum in the discrete case, you would use an integral in the continuous case.', 'start': 732.89, 'duration': 6.564}], 'summary': 'Probability of sample falling within range is integral of pdf curve.', 'duration': 36.004, 'max_score': 703.45, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M703450.jpg'}], 'start': 566.455, 'title': 'Symbolic representation of pairwise products and continuous distributions', 'summary': 'Discusses symbolic representation of pairwise products with a specific example of plugging in a 4 and the range of x from 1 to 6, and continuous distributions and probability density functions, emphasizing the use of integrals in the continuous case and the analogy between the discrete and continuous cases.', 'chapters': [{'end': 642.861, 'start': 566.455, 'title': 'Symbolic representation of pairwise products', 'summary': 'Discusses the symbolic representation of pairwise products in a new function, highlighting the manipulation of variables and domain considerations, with a specific example of plugging in a 4 and the range of x from 1 to 6.', 'duration': 76.406, 'highlights': ['The new function represents pairwise products as a sum over pairs of values for x and y, with the focus on manipulating variables to ensure the sum equals a specific value, such as s.', 'The example of plugging in s equals 4 demonstrates the manipulation of variables x and y, where the range of x from 1 to 6 is considered, despite some corresponding y values falling outside the explicitly defined domain.', 'The chapter explains the manipulation of variables to represent the sum over pairs of values, emphasizing the symbolic representation of pairwise products and the considerations for ensuring the sum equals a specific value.']}, {'end': 798.242, 'start': 643.002, 'title': 'Continuous distributions and probability density functions', 'summary': 'Discusses continuous distributions and probability density functions, emphasizing that in continuous distributions, the probability of specific outcomes is zero, and the probability density function is used to describe the likelihood of a variable falling within a range. it also highlights the analogy between the discrete and continuous cases and the use of integrals in the continuous case.', 'duration': 155.24, 'highlights': ['In continuous distributions, the probability of specific outcomes, such as the red die rolling to become a negative one, is zero, emphasizing the impossibility of certain outcomes.', 'The chapter explains that in continuous distributions, the probability density function is used to describe the likelihood of a variable falling within a given range, where the area under the curve represents the probability.', 'Analogous to the discrete case, the continuous case utilizes integrals instead of sums, where the integral of the probability density function describes the probability of a sample falling within a given range.']}], 'duration': 231.787, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M566455.jpg', 'highlights': ['The new function represents pairwise products as a sum over pairs of values for x and y, with the focus on manipulating variables to ensure the sum equals a specific value, such as s.', 'In continuous distributions, the probability of specific outcomes, such as the red die rolling to become a negative one, is zero, emphasizing the impossibility of certain outcomes.', 'The example of plugging in s equals 4 demonstrates the manipulation of variables x and y, where the range of x from 1 to 6 is considered, despite some corresponding y values falling outside the explicitly defined domain.', 'The chapter explains that in continuous distributions, the probability density function is used to describe the likelihood of a variable falling within a given range, where the area under the curve represents the probability.', 'The chapter explains the manipulation of variables to represent the sum over pairs of values, emphasizing the symbolic representation of pairwise products and the considerations for ensuring the sum equals a specific value.', 'Analogous to the discrete case, the continuous case utilizes integrals instead of sums, where the integral of the probability density function describes the probability of a sample falling within a given range.']}, {'end': 1077.969, 'segs': [{'end': 852.285, 'src': 'embed', 'start': 832.375, 'weight': 3, 'content': [{'end': 843.42, 'text': 'the way you want to think about this integral is that what it wants to do is iterate over all possible pairs of values x and y that are constrained to a given sum s.', 'start': 832.375, 'duration': 11.045}, {'end': 847.202, 'text': "We don't really have great notation for doing that symmetrically, so instead,", 'start': 843.42, 'duration': 3.782}, {'end': 852.285, 'text': 'the way we commonly write it down gives this artificial emphasis to one of the variables, in this case x.', 'start': 847.202, 'duration': 5.083}], 'summary': 'Integral iterates over pairs of x and y to a given sum s.', 'duration': 19.91, 'max_score': 832.375, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M832375.jpg'}, {'end': 942.816, 'src': 'embed', 'start': 898.696, 'weight': 0, 'content': [{'end': 904.198, 'text': 'The real fun comes from graphing the entire contents of the integral, the product between these two graphs.', 'start': 898.696, 'duration': 5.502}, {'end': 912.141, 'text': 'This is analogous to the list of pairwise products that we saw earlier, but in this case, instead of adding up all of those pairwise products,', 'start': 904.898, 'duration': 7.243}, {'end': 917.463, 'text': 'we want to integrate them together, which you would interpret as the area underneath this product graph.', 'start': 912.141, 'duration': 5.322}, {'end': 924.166, 'text': 'As I shift around this value of s, the shape of that product graph changes, and so does the corresponding area.', 'start': 918.184, 'duration': 5.982}, {'end': 933.329, 'text': 'Keep in mind, for all three graphs on the left, the input is x, and the number s is just a parameter.', 'start': 927.284, 'duration': 6.045}, {'end': 935.51, 'text': 'But for the final graph on the right.', 'start': 933.829, 'duration': 1.681}, {'end': 942.816, 'text': 'for the resulting convolution itself, this number s is the input to that function and the corresponding output is,', 'start': 935.51, 'duration': 7.306}], 'summary': "Graphing the integral's product changes with s, affecting the area underneath.", 'duration': 44.12, 'max_score': 898.696, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M898696.jpg'}, {'end': 1077.969, 'src': 'embed', 'start': 1041.91, 'weight': 1, 'content': [{'end': 1049.635, 'text': 'There, if you add up two different uniformly distributed variables, then the distribution for the sum has a certain wedge shape.', 'start': 1041.91, 'duration': 7.725}, {'end': 1054.237, 'text': 'Probabilities increase until they max out at a 7, and then they decrease back down again.', 'start': 1049.935, 'duration': 4.302}, {'end': 1062.462, 'text': 'Where this gets a lot more fun is if, instead of asking for a sum of two uniformly distributed variables,', 'start': 1056.36, 'duration': 6.102}, {'end': 1066.743, 'text': 'I ask you what it looks like if we add up three different uniformly distributed variables.', 'start': 1062.462, 'duration': 4.281}, {'end': 1072.405, 'text': "At first you might say, I don't know, we'd need some new way to visualize combining three things instead of two.", 'start': 1067.323, 'duration': 5.082}, {'end': 1077.969, 'text': 'But really what you can do here is think about the sum of the first two as their own variable,', 'start': 1073.425, 'duration': 4.544}], 'summary': 'Adding three uniformly distributed variables creates a wedge-shaped distribution, with probabilities increasing until maxing out at 7 and then decreasing.', 'duration': 36.059, 'max_score': 1041.91, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M1041910.jpg'}], 'start': 798.783, 'title': 'Integral, convolution, and uniform distributions', 'summary': 'Delves into the concepts of integral and convolution, demonstrating the process of iterating over pairs of values and graphing the product, as well as exploring the combination and visualization of uniform distributions for the sum of random variables, including the extension to adding three uniformly distributed variables.', 'chapters': [{'end': 942.816, 'start': 798.783, 'title': 'Understanding integral and convolution', 'summary': 'Explores the concept of integral and convolution by iterating over all possible pairs of values x and y constrained to a given sum, graphing the product between two graphs and interpreting it as the area underneath the product graph.', 'duration': 144.033, 'highlights': ['The integral iterates over all possible pairs of values x and y constrained to a given sum s. The integral iterates over all possible pairs of values x and y constrained to a given sum s.', 'Graphing the product between two graphs and interpreting it as the area underneath the product graph. Graphing the entire contents of the integral, which is the product between the graphs, and interpreting it as the area underneath the product graph.', 'Shifting the value of s changes the shape of the product graph and the corresponding area. Shifting the value of s changes the shape of the product graph and the corresponding area.']}, {'end': 1077.969, 'start': 942.816, 'title': 'Combining uniform distributions', 'summary': 'Explores the combination of uniform distributions and visualizes the distribution for the sum of two random variables, showcasing a wedge-shaped pattern as they overlap, and extends the concept to adding up three uniformly distributed variables.', 'duration': 135.153, 'highlights': ['The distribution for the sum of two uniformly distributed variables forms a wedge shape as the graphs overlap, increasing linearly until they reach a maximum and then decreasing linearly again.', 'When adding up three different uniformly distributed variables, the concept of combining the first two as their own variable is introduced for visualization.']}], 'duration': 279.186, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M798783.jpg', 'highlights': ['Delves into the concepts of integral and convolution, demonstrating the process of iterating over pairs of values and graphing the product.', 'Exploring the combination and visualization of uniform distributions for the sum of random variables, including the extension to adding three uniformly distributed variables.', 'Graphing the product between two graphs and interpreting it as the area underneath the product graph.', 'The integral iterates over all possible pairs of values x and y constrained to a given sum s.', 'Shifting the value of s changes the shape of the product graph and the corresponding area.', 'The distribution for the sum of two uniformly distributed variables forms a wedge shape as the graphs overlap, increasing linearly until they reach a maximum and then decreasing linearly again.', 'When adding up three different uniformly distributed variables, the concept of combining the first two as their own variable is introduced for visualization.']}, {'end': 1627.879, 'segs': [{'end': 1240.781, 'src': 'embed', 'start': 1198.56, 'weight': 0, 'content': [{'end': 1205.584, 'text': 'That, as you take repeated convolutions like this, representing bigger and bigger sums of a given random variable,', 'start': 1198.56, 'duration': 7.024}, {'end': 1212.748, 'text': 'then the distribution describing that sum, which might start off looking very different from a normal distribution over time,', 'start': 1205.584, 'duration': 7.164}, {'end': 1217.371, 'text': 'smooths out more and more until it gets arbitrarily close to a normal distribution.', 'start': 1212.748, 'duration': 4.623}, {'end': 1223.783, 'text': "It's as if a bell curve is, in some loose manner of speaking, the smoothest possible distribution,", 'start': 1218.197, 'duration': 5.586}, {'end': 1230.83, 'text': 'an attractive fixed point in the space of all possible functions, as we apply this process of repeated smoothing through the convolution.', 'start': 1223.783, 'duration': 7.047}, {'end': 1238.578, 'text': 'Naturally, you might wonder why normal distributions??', 'start': 1235.635, 'duration': 2.943}, {'end': 1240.781, 'text': 'Why this function and not some other one?', 'start': 1238.879, 'duration': 1.902}], 'summary': 'Repeated convolutions of random variables approach normal distribution over time.', 'duration': 42.221, 'max_score': 1198.56, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M1198560.jpg'}, {'end': 1353.554, 'src': 'embed', 'start': 1327.537, 'weight': 2, 'content': [{'end': 1333.781, 'text': 'But if we look at it from another angle, emphasizing the change in the y direction, it takes on the shape of our second graph.', 'start': 1327.537, 'duration': 6.244}, {'end': 1337.743, 'text': 'This three-dimensional graph encodes all of the information we need.', 'start': 1334.301, 'duration': 3.442}, {'end': 1341.066, 'text': 'It shows all the probability densities for every possible outcome.', 'start': 1338.084, 'duration': 2.982}, {'end': 1348.291, 'text': 'And if you want to limit your view just to those outcomes where x plus y is constrained to be a given sum,', 'start': 1341.886, 'duration': 6.405}, {'end': 1353.554, 'text': 'what that looks like is limiting our view to a diagonal slice, specifically a slice over the line.', 'start': 1348.291, 'duration': 5.263}], 'summary': 'A three-dimensional graph encodes all probability densities for every possible outcome, with a diagonal slice representing specific outcomes.', 'duration': 26.017, 'max_score': 1327.537, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M1327537.jpg'}, {'end': 1425.562, 'src': 'embed', 'start': 1396.28, 'weight': 3, 'content': [{'end': 1400.684, 'text': 'the areas of these slices give us the values of the convolution.', 'start': 1396.28, 'duration': 4.404}, {'end': 1408.171, 'text': "In fact, all of these slices that we're looking at are precisely the same as the product graph that we were looking at earlier.", 'start': 1401.425, 'duration': 6.746}, {'end': 1417.517, 'text': "Here. to emphasize this point, let me pull up both visualizations side by side, and I'm going to slowly decrease the value of s,", 'start': 1409.392, 'duration': 8.125}, {'end': 1425.562, 'text': "which on the left means we're looking at different slices and on the right means we're shifting around the modified graph of g.", 'start': 1417.517, 'duration': 8.045}], 'summary': 'Slices represent convolution values, matching the product graph.', 'duration': 29.282, 'max_score': 1396.28, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M1396280.jpg'}, {'end': 1499.003, 'src': 'embed', 'start': 1471.345, 'weight': 4, 'content': [{'end': 1477.069, 'text': "The nice thing about the diagonal slice visualization is that it makes it much more clear that it's a symmetric operation.", 'start': 1471.345, 'duration': 5.724}, {'end': 1484.035, 'text': "It's much more obvious that F convolved with G is the same thing as G convolved with F.", 'start': 1477.71, 'duration': 6.325}, {'end': 1487.458, 'text': 'Technically, the diagonal slices are not exactly the same shape.', 'start': 1484.035, 'duration': 3.423}, {'end': 1491.86, 'text': "They've actually been stretched out by a factor of the square root of 2.", 'start': 1488.078, 'duration': 3.782}, {'end': 1499.003, 'text': 'The basic reason is that if you imagine taking some small step along one of these lines, where x plus y equals a constant,', 'start': 1491.86, 'duration': 7.143}], 'summary': 'Diagonal slice visualization shows symmetric operation, stretched by square root of 2.', 'duration': 27.658, 'max_score': 1471.345, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M1471345.jpg'}], 'start': 1077.969, 'title': 'Convolution and central limit theorem', 'summary': 'Discusses the central limit theorem and convolution, showing how repeated convolutions result in the distribution of sums of random variables approaching a normal distribution. it also covers generalizing convolution to the continuous case and its visualization as a 3d graph, along with its relation to probability densities and symmetric operations.', 'chapters': [{'end': 1240.781, 'start': 1077.969, 'title': 'Central limit theorem and convolution', 'summary': 'Discusses the central limit theorem and convolution, demonstrating how repeated convolutions result in the distribution of sums of random variables approaching a normal distribution, with the bell curve being the smoothest possible distribution.', 'duration': 162.812, 'highlights': ['The chapter discusses the central limit theorem and convolution. The chapter explores the central limit theorem and the process of convolution.', 'Repeated convolutions result in the distribution of sums of random variables approaching a normal distribution. The repeated convolution process leads to the distribution of sums of random variables smoothing out over time and approaching a normal distribution.', 'The bell curve is the smoothest possible distribution. The bell curve is described as the smoothest possible distribution and an attractive fixed point in the space of all possible functions.']}, {'end': 1627.879, 'start': 1241.684, 'title': 'Visualizing convolution in continuous case', 'summary': 'Discusses generalizing convolution to the continuous case, showcasing the visualization of convolution as a 3d graph and its relation to probability densities and symmetric operations.', 'duration': 386.195, 'highlights': ['The visualization of convolution as a 3D graph encodes all the probability densities for every possible outcome. The 3D graph represents all possible outcomes when sampling from both distributions and shows the probability densities for each outcome, based on the product of the two functions.', 'The areas of the diagonal slices of the 3D graph give the values of the convolution, up to a constant factor. The areas of the diagonal slices correspond to the values of the convolution, providing a method to interpret the combination of probability densities along these slices, with a subtle detail regarding a factor of the square root of 2.', 'The symmetrical nature of the convolution operation is made clearer through the diagonal slice visualization. The diagonal slice visualization highlights the symmetric nature of the convolution operation, emphasizing that F convolved with G is the same as G convolved with F, with a technical distinction involving stretching by a factor of the square root of 2.', 'The visualizations play a front and center role in a more fun method for proving the convolution of two normal distributions, particularly with diagonal slices. Visualizations, specifically the diagonal slices, are utilized for a more engaging proof of the convolution of two normal distributions, where the 3D graph and its properties are crucial for the proof, especially when both distributions have the same standard deviation.']}], 'duration': 549.91, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/IaSGqQa5O-M/pics/IaSGqQa5O-M1077969.jpg', 'highlights': ['Repeated convolutions result in the distribution of sums of random variables approaching a normal distribution.', 'The bell curve is the smoothest possible distribution.', 'The visualization of convolution as a 3D graph encodes all the probability densities for every possible outcome.', 'The areas of the diagonal slices of the 3D graph give the values of the convolution, up to a constant factor.', 'The symmetrical nature of the convolution operation is made clearer through the diagonal slice visualization.', 'The visualizations play a front and center role in a more fun method for proving the convolution of two normal distributions, particularly with diagonal slices.']}], 'highlights': ['The sum of two random variables drawn from normal distributions also follows a normal distribution, showcasing the significance of understanding the behavior and interpretation of normal distributions in probability.', "The interpretation of the normal distribution curve lies in the area under the curve representing the probability of the sample falling within a given range of values, providing a fundamental understanding of the distribution's significance in probability.", 'The behavior of summing two random variables drawn from normal distributions serves as a pivotal concept in understanding the special function of normal distributions in probability, highlighting the importance of a profound comprehension of the topic.', 'Understanding the distribution that describes the sum of two random variables drawn from normal distributions is crucial, emphasizing the need to comprehend the underlying reasons for the resulting distribution.', 'The chapter explains the concept of convolution for adding different random variables and emphasizes its application beyond normally distributed ones.', 'The visualization of probabilities as a three-dimensional plot represents all the data needed to understand the rolling of a pair of dice.', 'The new function represents pairwise products as a sum over pairs of values for x and y, with the focus on manipulating variables to ensure the sum equals a specific value, such as s.', 'Delves into the concepts of integral and convolution, demonstrating the process of iterating over pairs of values and graphing the product.', 'Repeated convolutions result in the distribution of sums of random variables approaching a normal distribution.', 'The bell curve is the smoothest possible distribution.']}