## Definition ==Moments== are expected values of powers of a random variable that characterize the shape of its distribution. The $k$-th raw moment is $E[X^k]$ and the $k$-th central moment is $E[(X - \mu)^k]$. The first raw moment is the mean, the second central moment is the variance, the third central moment relates to skewness, and the fourth to kurtosis. > [!example]- For a random variable $X$ with $E[X] = 3$, $E[X^2] = 13$, and $E[X^3] = 51$, what is the third central moment? > The third central moment is $E[(X - \mu)^3] = E[X^3] - 3\mu E[X^2] + 3\mu^2 E[X] - \mu^3$. > $ = 51 - 3(3)(13) + 3(9)(3) - 27 = 51 - 117 + 81 - 27 = -12 $ > A negative third central moment indicates the distribution is left-skewed.