# 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

* [https://en.wikipedia.org/wiki/Moment\_(mathematics)](https://en.wikipedia.org/wiki/Moment_\(mathematics\))
