Expectation, Variance, and Moments

Distributions are infinite-dimensional objects, but a handful of summary statistics — the moments — capture most of what we usually need.

The $k$-th raw moment is $E[X^k]$. The $k$-th central moment is $E[(X - \mu)^k]$. Four of them dominate everyday work:

• Mean (1st moment): $\mu = E[X]$ — where the distribution is centered.
• Variance (2nd central moment): $\sigma^2 = E[(X - \mu)^2]$ — how spread out it is.
• Skewness: $E[(X - \mu)^3] / \sigma^3$ — asymmetry. Positive skew means a right tail.
• Kurtosis: $E[(X - \mu)^4] / \sigma^4$ — tail heaviness. The Normal has kurtosis $3$; "excess kurtosis" subtracts $3$ for easier comparison.

In finance, asset return distributions are routinely summarized this way. Equity returns exhibit negative skew (bigger drawdowns than rallies of the same magnitude) and substantial excess kurtosis (fat tails). Risk models that ignore the higher moments will systematically underprice tail risk.