Variance - Imported Knowlpedia Content

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

\operatorname{Var}(X)=\mathbb E\big[(X-\mathbb E[X])^2\big],

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

Equivalent characterizations

Equivalently,

\operatorname{Var}(X)=\mathbb E[X^2]-\big(\mathbb E[X]\big)^2.

Remarks

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 $\operatorname{Var}(X)=\sigma^2$ .