混合分布の期待値、分散

\begin{align*}
E[X]&= \sum_{i=1}^np_i\mu_i  \\

E[g(X)] &= \sum_{i=1}^np_iE[g(X_i)] \\

V[X] &= \sum_{i=1}^np_i\sigma_i^2 + \sum_{i=1}^{n-1} \sum_{j=i+1}^np_ip_j(\mu_i - \mu_j)^2 \\
&= \sum_{i=1}^np_i\sigma_i^2 + \frac{1}{2} \sum_{i=1}^n\sum_{j=1}^np_ip_j(\mu_i - \mu_j)^2
\end{align*}