Moment ($\mathbb{E}[X^n]$)

The $n$th raw moment of a random variable $X$ is $\mathbb{E}[X^n]$ when this expectation exists.

More generally, a moment is an integral weighted by a power of a coordinate against a measure or density; raw, central, and standardized moments are distinguished by the coordinate and normalization used.

The $r$th raw moment of a random variable $X$ is $\mathbb{E}[X^r]$, when it exists. A central moment instead uses $\mathbb{E}[(X-\mathbb{E}[X]{)}^r]$.

The fourth moment is $\mathbb{E}[X^4]$. Its finiteness is equivalent to square integrability of $X^2$. For centered variables, dividing it by the squared variance gives kurtosis.

A probability distribution is moment-determinate if its moments uniquely specify it among probability distributions. A moment-generating function finite in a neighbourhood of zero is a sufficient condition. The standard normal distribution is moment-determinate.

The second moment of a real random variable is $\mathbb{E}[X^2]$. Its variance is $\mathbb{E}[X^2]-(\mathbb{E}X{)}^2$ when these quantities are finite.

The absolute moment of order $p>0$ of a random variable $X$ is $\mathbb{E}[|X{|}^p]$ when this expectation is finite.

The $n$th central moment is $\mathbb{E}[(X-\mathbb{E}X{)}^n]$. The first central moment is zero and the second is the variance.

Skewness is the standardized third central moment of a probability distribution. A symmetric distribution with finite third moment has zero skewness. Cosmology also uses the reduced skewness $S_3=⟨{\delta }^3⟩/{\sigma }^4$, which differs from the standardized skewness by a factor of $\sigma$.