The mean and variance of a general normal distribution, can be found from those of the standard normal distribution.

Mean and variance of standard normal distribution

If \(Z \sim \NormalDistn(0,\; 1)\), its mean and variance are

\[ E[Z] \;=\; 0 \spaced{and} \Var(Z) \;=\; 1 \]

(Proved in full version)

A change of variable, \(z = \frac {x-\mu}{\sigma}\), can be used to find the mean and variance of a general normal distribution from this result.

Mean and variance of a general normal distribution

If \(X \sim \NormalDistn(\mu,\; \sigma^2)\), its mean and variance are

\[ E[X] \;=\; \mu \spaced{and} \Var(X) \;=\; \sigma^2 \]

(Proved in full version)

This explain why the symbols "\(\mu\)" and "\(\sigma^2\)" are used for the normal distribution's two parameters.