Moment of a distribution

The $k^\text{th}$ moment of a distribution $p(x)$ is written $\avg{x^k}$ and defined as

Intuitively, we can think of the $k^\text{th}$ moment as the average value of $x^k$. This interpretation works because the distribution must be normalized, so we don’t need to devide by the sum as we would normally to find the average.

Useful statistics about a distribution can be written in terms of its moments. For example, the first moment is the mean $\mu = \avg{x}$.

The variance can be written as $\Var(x) = \avg{x^2} - \avg{x}^2$. The standard deviation $\sigma = \sqrt{\Var}$.