Linearity of expectation

Statement

Let $X$ and $Y$ be integrable real random variables on one probability space, and let $a, b \in \mathbb{R}$. Then $aX + bY$ is integrable and $$E[aX + bY] = a \, E[X] + b \, E[Y].$$

Remarks

The statement is the linearity of the Lebesgue integral. It holds for all integrable $X$ and $Y$, also when they are not independent. By induction, $E[X_1 + \dots + X_n] = E[X_1] + \dots + E[X_n]$. A standard use is the method of indicators. If $N$ counts how many of the events $A_1, \dots, A_n$ occur, then $N = \sum_i 1_{A_i}$ and $E[N] = \sum_i P(A_i)$. For nonnegative random variables, additivity $E[X + Y] = E[X] + E[Y]$ holds without integrability, in $[0, \infty]$.

Explanations

L1 human

The average of a sum is the sum of the averages, always. If a random permutation of $n$ letters goes into $n$ envelopes, each letter is in its own envelope with probability $1/n$. So the expected number of correct letters is $n \cdot 1/n = 1$.