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Understanding Moments

gregorygundersen.com · 13,176 words · saved by 1 readers

While drafting another post, I realized that I didn’t fully understand the idea of a distribution’s moments. I could write down the equation for the kth moment of a random variable X with a density function f(x), μ k ​ =E[X k ]=∫ −∞ ∞ ​ x k f(x)dx, (1) and I understood that the first moment is a random variable’s mean, the second (central moment) is its variance, and so forth. Yet the concept felt slippery because I had too many unanswered questions. Why is it called a “moment”? Is it related to the concept of a moment in physics? Why do the first four moments—mean, variance, skewness, and kurtosis—have commonly used names but higher-order moments do not? Is there a probabilistic interpretation of the seventy-second moment? Why does a moment-generating function uniquely specify a probability distribution? The goal of this post is to explore these questions in detail. This post is long, and I spent many hours writing it. However, I have consistently found that the most time-

Understanding Moments --> Home Blog RSS Understanding Moments Why are a distribution's moments called "moments"? How does the equation for a moment capture the shape of a distribution? Why do we typically only study four moments? I explore these and other questions in detail. Published 11 April 2020 While drafting another post, I realized that I didn’t fully understand the idea of a distribution’s moments. I could write down the equation for the k k k th moment of a random variable X X X with a density function f ( x ) f(x) f ( x ) , μ k = E [ X k ] = ∫ − ∞ ∞ x k f ( x ) d x , (1) \mu_k = \mat

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