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Gaussian Integration by Parts – Ethan N. Epperly

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Gaussian random variables are wonderful, and there are lots of clever tricks for doing computations with them. One particularly nice tool is the Gaussian integration by parts formula, which I learned from my PhD advisor Joel Tropp. Here it is: Gaussian integration by parts. Let be a standard Gaussian random variable. Then . This formula makes many basic computations effortless. For instance, to compute the second moment of a standard Gaussian random variable , we apply the formula with to obtain The fourth moment is no harder to compute. Using , we compute Iterating this trick, we can compute all the even moments of a standard Gaussian random variable. Indeed, As a spicier application, let us now compute . To do so, we choose to be the sign function: As an application of the Gaussian integration by parts formula, we can analyze the famous power method for eigenvalue computations with a (Gaussian) random initialization. This discussion is adapted from the tutorial of Kireeva and Tro

Gaussian random variables are wonderful, and there are lots of clever tricks for doing computations with them. One particularly nice tool is the Gaussian integration by parts formula , which I learned from my PhD advisor Joel Tropp . Here it is: Gaussian integration by parts. Let be a standard Gaussian random variable. Then . This formula makes many basic computations effortless. For instance, to compute the second moment of a standard Gaussian random variable , we apply the formula with to obtain Therefore, the second moment is one. Since has mean zero, this also means that the variance of a

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