Expected Value & Variance Cheat Sheet

Linearity, indicators, tail sums, conditional expectation and variance identities.

Expectation

E [a X + bY] = a E [X] + b E [Y]

No independence needed.

E [1_{A}] = P (A)

E [X] = k \geq 1 \sum P (X \geq k)

E [X] = E [E [X ∣ Y]]

Var (X) = E [X^{2}] - E [X]^{2}

Var (X + Y) = Var X + Var Y + 2 Cov (X, Y)

Var X = E [Var (X ∣ Y)] + Var (E [X ∣ Y])