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The video discusses why pi appears in the formula for a normal distribution beyond integral tricks. The speaker goes beyond the classic proof that explains the presence of pi and connects it to the Central Limit Theorem. They explain how to find the volume underneath a bell surface and relate it to the two-dimensional graph of a bell curve. The speaker introduces John Hershel's theory of using the e^-x^2 function to describe a probability distribution in two-dimensional space. They simplify the problem of deriving the normal distribution of numbers using Herschel's two properties and a helper function that satisfies a property of addition to multiplication. The video also explores how the Herschel/Maxwell derivation converges with the Gaussian distribution's defining property of circular symmetry and offers insight into how a convolution theorem and Fourier transforms help explain the e^(-x^2) function in the Central Limit Theorem.
In this section, the speaker discusses the story about two classmates, where one tried to explain to the other the pi symbol used in population trends and the meaning of the symbols for actual population and average population. The speaker then focuses on explaining why pi is inside the formula for a normal distribution. They delve into the classic proof that explains the pi inside a normal distribution formula, wherein pi originates from the area underneath a curve that comes out to be the square root of pi. The speaker aims to go beyond the classic proof and find a connection between the proof that shows why pi shows up and the Central Limit Theorem.
In this section, the speaker explains how finding the area under a bell curve requires a new strategy, which involves bumping things up one dimension to find the volume underneath a bell surface. By respecting the symmetry of the volume, we can integrate together thin cylinders to get the volume of the surface that is equal to pi. By analyzing the volume in a second different way, by chopping it up into slices parallel to one of the axes, we can directly relate the three-dimensional graph to our two-dimensional graph by analyzing each slice as a bell curve.
In this section, the speaker explains how to find the area underneath the bell curve using integrals. They factor out the e^-x^2 function and show how it can be used to find the volume under the surface. The integral of the area of each slice is multiplied by a little thickness, dy, to give each slice a little bit of volume. The volume under the bell surface turns out to be a mystery constant squared, which was later found to be equal to pi. The speaker then introduces John Hershel's theory, which explains how to describe a probability distribution in two-dimensional space using the e^-x^2 function with the shape of a bell curve.
In this section, the speaker describes how the normal distribution of numbers can be derived using Herschel's two properties. A functional equation arises that the speaker refers to as one of the most challenging yet fascinating types of equations to solve. The speaker simplifies the problem by introducing a helper function that satisfies a property of addition to multiplication and simplifies the function's analysis to an exponential form. Ultimately, the speaker concludes that the helper function must be an exponential function for all real-number inputs.
In this section, the video delves into how the Herschel/Maxwell derivation led to the e^(-x^2) function in a multi-dimensional distribution and how it converged with the Gaussian distribution's defining property of circular symmetry. It also offers an explanation, based on circular symmetry, for why that particular function arises in the Central Limit Theorem and offers insight into how a convolution theorem and Fourier transforms help explain the e^(-x^2) function in the Central Limit Theorem. The video suggests that a combination of Convolutions, the Central Limit Theorem, and the Herschel/Maxwell characterization of normal distributions may help explain the e^(-x^2) function in a more satisfying way and invites the viewers for the next video in the series.
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