Variance

A variance of a random variable $X$ is the quantity

defined when $\mathbb E[X^2]<\infty$ (so in particular the expectation $\mathbb E[X]$ is finite). Equivalently,

Variance is the second centered moment of $X$. It is also the special case $\operatorname{Var}(X)=\operatorname{Cov}(X,X)$ of covariance , and it is used to normalize covariance into the correlation coefficient .

Examples:

• If $X$ is Bernoulli$(p)$ (so $\mathbb P(X=1)=p$, $\mathbb P(X=0)=1-p$), then $\operatorname{Var}(X)=p(1-p)$.
• If $X\sim N(\mu,\sigma^2)$, then $\operatorname{Var}(X)=\sigma^2$.