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}


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