What does $\mathbb{E}_{Y \sim p}[f(Y)]$ mean?

Question

Suppose $Y$ is a random variable that takes values in $\{1 \ldots n\}$, and $p \in \mathbb{R}^n$ is a vector such that $p_i$ is the probability that $Y$ takes value $i$ in an experiment. Further, suppose we have a function $f:\{1 \ldots n\} \to \mathbb{R}^k$. Then what does this notation mean? $$\mathbb{E}_{Y \sim p}[f(Y)]$$

Answer 1

If the "$Y \sim p$" is confusing you, just ignore it. It's just the usual expectation. $$\mathbb{E}_{Y \sim p}[f(Y)] = \sum_{i=1}^n [p_i \cdot f(i)]$$