MeanAndVariance

If we have a continuous random variable $ X $ with a probability density function $ f(x) $, the mean and variance are given by:

$ \mu = E(X) = \int x f(x) dx $

(where the integral is over the range of x values)

The variance is given by

$ \sigma^2 = Var(X)^2 = Var(X) = \int (x - \mu)^2 f(x) dx $

This can be simplified:

$ Var(x) = \int (x^2 - 2 x \mu + \mu^2) f(x) dx $

$ = \int x^2 f(x) d - 2 \mu \int x f(x) dx + \mu^2 \int f(x) dx $

$ = \int x^2 f(x) d - 2 \mu^2 + \mu^2 $

$ = \int x^2 f(x) dx - \mu^2 $

which is equivalent to saying:

$ Var(X) = E(X^2) - E(X)^2 $