Definitions of mean and variance
The concepts of a continuous distribution's mean and variance are similar to those of discrete distributions.
Definition
The mean of a continuous random variable is
\[ E[X] \;=\; \mu \]and its variance is
\[ \Var(X) \;=\; \sigma^2 \;=\; E \left[(X - \mu)^2 \right] \]The mean can be interpreted as a 'typical value' from the distribution, and the square root of the variance (also called the distribution's standard deviation) is a 'typical distance from the mean'.
Finding variances
The following result is identical to that for discrete random variables, and is often useful for evaluating a continuous distribution's variance. Since its proof is almost identical to that for discrete variables, it is not repeated here.
Alternative formula for the variance
A continuous random variable's variance can be written as
\[ \Var (X) \;=\; E \left[(X - \mu)^2 \right] \;=\; E[X^2] - \left( E[X] \right)^2 \]