The expectation of Random variable $X$ under distribution $\mathbb{P}$ is defined as

$\mathbf{E}[X] = \sum\limits_x x\mathbb{P}[X=x]$

If the random variable $X$ is a function of other random variables $z_1,...,z_n$, and these have Joint distribution $\mathcal{D}$, then we sometimes specify this in the subscript of the big $\mathbf{E}$,

$\mathbf{E}_{z_1,...,z_n \sim \mathcal{D}}\left[X\right] = \sum\limits_{z_1,...,z_n} X(z_1,...,z_n)\mathcal{D}(z_1,...,z_n)$