Linearity of Expectation

E[aX + bY] = a E[X] + b E[Y] — no independence needed.

The idea

Theorem (Linearity of expectation).

For any random variables $X$ and $Y$ and any constants $a, b, c$, $\mathbb{E}[aX + bY + c] = a\,\mathbb{E}[X] + b\,\mathbb{E}[Y] + c.$

The identity assumes nothing about how $X$ and $Y$ relate: they may be independent, or one may determine the other, and it holds either way.

The corresponding statement for products, $\mathbb{E}[XY] = \mathbb{E}[X]\,\mathbb{E}[Y]$, is false in general; it does require independence.

In practice, linearity computes the mean of a complicated random quantity by cutting the quantity into simple pieces, taking each piece's mean, and adding — even when the pieces depend on one another.

Ways to work on it

• Walkthrough. Compute the expectation of a linear combination of two random variables.
• Proof. See why — derive E[X+Y] = E[X] + E[Y] from the definition, with no independence.
• Practice. One step. Random integer means E[X], E[Y] and small coefficients.
• Hardest. One step. Three random variables (possibly dependent) and negative coefficients.

Not sure where to start? Take the ten-question placement test.