Expected Values of Sums and Products of Random Variables

For any two random variables $X$ and $Y$ and any constants $a, b, c \in \mathbb{R}$, the expectation operator is linear:

$$E[aX + bY + c] = a\,E[X] + b\,E[Y] + c.$$

This identity holds regardless of whether $X$ and $Y$ are independent. It also extends to any finite number of random variables: for $X_1, X_2, \ldots, X_n$ and constants $a_1, \ldots, a_n, b$,

$$E\!\left[\sum_{i=1}^n a_i X_i + b\right] = \sum_{i=1}^n a_i\, E[X_i] + b.$$

For example, if $E[X] = 10$ and $E[Y] = 4$, then

$$\begin{align*}E[3X - 2Y + 7] &= 3\,E[X] - 2\,E[Y] + 7 \\[3pt] &= 3(10) - 2(4) + 7 \\[3pt] &= 30 - 8 + 7 \\[3pt] &= 29.\end{align*}$$

Notice that constants pass straight through $E[\,\cdot\,]$, and the constant term $7$ is its own expectation since $E[c] = c$ for any constant $c$.