Moment

Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$. The $k$-th moment of $X$ is $$m_k = E[X^k].$$ The $k$-th absolute moment is $E[|X|^k]$.

Remarks

By the law of the unconscious statistician, $m_k = \int x^k \, \mu_X(dx)$. So the moments depend only on the law $\mu_X$ of $X$. The first moment is the mean. The variance is $m_2 - m_1^2$. If $E[|X|^k] < \infty$, then $E[|X|^j] < \infty$ for each $j \le k$. The Cauchy distribution has no moments of order $k \ge 1$.

Explanations

L1 human

Moments are numbers that sum up the shape of a distribution. The first gives the center, the second with the first gives the spread, and the higher ones describe asymmetry and the weight of the tails.