Moments

Given a random variable $X$ , the $k$ th moment of $X$ is the value $E[X^k]$ , if the expectation exists.

Note that the expected value is the first moment of a random variable, and the variance is the second moment minus the first moment squared.

The $k$ th moment of $X$ is usually obtained by using the moment generating function.

Central moments

Given a random variable $X$ , the $k$ th central moment of $X$ is the value $E\big[(X-E[X])^k\big]$ , if the expectation exists. It is denoted by $\mu_k$ .

Note that the $\mu_1=0$ and $\mu_2=Var[X]=\sigma^2$ . The third central moment divided by the standard deviation cubed is called the skewness $\tau$ : $$\tau=\frac{\mu_3}{\sigma^3}$$ The skewness measures how ``symmetrical'', or rather, how ``skewed'', a distribution is with respect to its mode. A non-zero $\tau$ means there is some degree of skewness in the distribution. For example, $\tau>0$ means that the distribution has a longer positive tail.

The fourth central moment divided by the fourth power of the standard deviation is called the kurtosis $\kappa$ : $$\kappa=\frac{\mu_4}{\sigma^4}$$ The kurtosis measures how ``peaked'' a distribution is compared to the standard normal distribution. The standard normal distribution has $\kappa=3$ . $\kappa<3$ means that the distribution is ``flatter'' than then standard normal distribution, or platykurtic. On the other hand, a distribution with $\kappa>3$ can be characterized as being more ``peaked'' than $N(0,1)$ , or leptokurtic.