Probability Theory

Linearity of Expectation

For random variables $X$ and $Y$ (which may be dependent), $$ \mathbb{E}[X + Y] = \mathbb{E}[X] + \mathbb{E}[Y] $$ more generally, for random variable $X_{1}, X_{2}, \cdots, X_{n}$ and constants $c_{1}, c_{2}, \cdots, c_{n}$, $$ \mathbb{E}\bigg[\sum_{i = 0}^{N-1}c_{i}X_{i} \bigg] = \sum_{i = 0}^{N-1} (c_{i} \cdot X_{i}) $$

Also, $$ \mathbb{E}[af(X) + b] = a\mathbb{E}[f(X)] + b $$ this regardless of the distribution of $X$, and regardless of whether $f$ is linear.

reference

1. [1] Linearity of Expectation, https://brilliant.org/wiki/linearity-of-expectation/