Probability

Linearity of expectation

Turns the expectation of a complicated sum into a sum of simple expectations, whatever the dependence between the terms.

\mathbb{E}\!\left[\sum_i a_i X_i + b\right] = \sum_i a_i\,\mathbb{E}[X_i] + b

Where

$X_i$: Any random variables at all, on the same probability space.
$a_i, b$: Constants.

Assumptions

None. Independence is not required and is the most common thing people wrongly look for.

Sanity check. If your answer needed independence, you were not using linearity.

Where this is taught

Linearity of expectation · PROB · Expectation, variance and the big tricks