We define the mean and variance of a continuous distribution in a similar way to those of a discrete distribution.

Definition

The mean of a continuous random variable is

\[ E[X] \;=\; \mu \]

and its variance is

\[ \Var(X) \;=\; \sigma^2 \;=\; E \left[(X - \mu)^2 \right] \]

Their interpretations are also similar.

The following result is often useful for evaluating a continuous distribution's variance.

Alternative formula for the variance

A continuous random variable's variance can be written as

\[ \Var (X) \;=\; E \left[(X - \mu)^2 \right] \;=\; E[X^2] - \left( E[X] \right)^2 \]