Mathmatical expression approach
Probabilistic Graphical models
$$ F(x) = \sum_{k=1}^{K} w_kG_k(x),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \sum_{k=1}^{K}w_k = 1, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;G_1, \cdots,G_K \;\;are\;\;cdf $$
Mixture of normal distribution
$f(x) = \sum_{k=1}^{K}w_k\frac{1}{\sqrt{2\pi}\sigma}exp\left\{-\frac{1}{2\sigma^2}(x-\mu_k)^2 \right\}$ : Location mixture of normals
$f(x) = \sum_{k=1}^{K}w_k\frac{1}{\sqrt{2\pi}\sigma_k}exp\left\{-\frac{1}{2\sigma_k^2}(x-\mu_k)^2 \right\}$ : Location-scale mixture of normals
$f(x) = w_1\frac{1}{\sqrt{2\pi}\sigma}exp\left\{-\frac{1}{2\sigma^2}(x-\mu)^2 \right\} + (1-w_1)\frac{1}{\lambda}exp\left\{ -\frac{x}{\lambda} \right\}$ : Different families for each one of the kernels of distribution
$F(x)=\sum_{k=1}^Kw_kG(x|\theta_k)$. $\theta$ : parameter of the family(noraml $\mu$ and $\sigma^2$)
Expectation of X when $f(x)=\sum_{k=1}^Kw_kg_k(x)$
Expected value of Mixture : $E_f(x)=\int_{-\infin}^{\infin} xf(x)dx$