# Mixture Distributions

Finite:  A finite mixture distribution is described by the following cumulative distribution function:

$F(x) = \displaystyle \sum_{i=1}^n w_iF(x_i)$

Where $X$ is the mixture random variable, $X_i$ are the component random variables that make up the mixture, and $w_i$ is the weighting for each component.  The weights add to 1.

If $X$ is a mixture of 50% $X_1$ and 50% $X_2$, $F(x) = 0.5F(x_1) + 0.5F(x_2)$.  This is not the same as $X = 0.5X_1 +0.5X_2$.  The latter expression is a sum of random variables NOT a mixture!

Moments and Variance:

$\begin{array}{rl} E(X^t) &= \displaystyle \sum_{i=1}^n w_iE(X_i^t) \\ Var(X) &= E(X^2) - E(X)^2 \\ &= \displaystyle \sum_{i=1}^n w_iE(X^2) - \left(\sum_{i=1}^n w_iE(X)\right)^2 \end{array}$