> For the complete documentation index, see [llms.txt](https://wiki.hanzheteng.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wiki.hanzheteng.com/math/probability/moment.md). # Moment ## Moment in Statistics * In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. * If the function is a probability distribution, then the first moment is the expected value, the second **central** moment is the variance, the third **standardized** moment is the skewness, and the fourth **standardized** moment is the kurtosis. * The mathematical concept is closely related to the concept of moment in physics. ### Definitions Suppose that we have a random variable X and the moments considered below have convergence (< +inf.), then * E\[X^k] is the k-th order moment to the origin, * E\[(X-E\[X])^k] is the k-th order central moment, * E\[(X-E\[X])^k]/sigma^k is the k-th order standardized moment, which is the k-th order central moment normalized by sigma^k. ### References *