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