Suppose that $X$ denotes a discrete random variable. Let $f(\cdot)$ be the PDF (which is in fact the probability mass function since $X$ is discrete) of $X$ with finite support $[\underline{X}, \overline{X}]$ such that $X\sim f_X(X)$. A multivariate signal of dimension $k$ such that $S = (S_1,\dots,S_k) \sim f_S(S|X)$ with finite support as well,that is $[\underline{S}, \overline{S}]$. The signal is correlated with the random variable $\theta$. My question is how do I calculate the following conditional moments?
$$\mathbb{E}[X|(S_1,\dots,S_k)], \quad \operatorname{\mathbb{V}ar}[X|(S_1,\dots,S_k)]$$