Expectation, Covariance, & Variance

Let $X, Y$ be random variables and $a, b, c \in R$ .

E [c] = c

E [a X] = a E [X]

E [X + Y] = E [X] + E [Y]

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

E [X Y] has no simplification in general

If $X, Y$ are independent:

E [X Y] = E [X] E [Y]

Cov (X, Y) = E [X Y] - E [X] E [Y]

Cov (X, Y) = Cov (Y, X)

Cov (X, c) = 0

Cov (a X + b, Y) = a Cov (X, Y)

Cov (X, aY + b) = a Cov (X, Y)

If $X, Y$ are independent:

Cov (X, Y) = 0

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

Var (X) = Cov (X, X)

Var (c) = 0

Var (X + c) = Var (X)

Var (a X) = a^{2} Var (X)

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

If $X, Y$ are independent:

Var (X + Y) = Var (X) + Var (Y)

Var (a X + bY + c) = a^{2} Var (X) + b^{2} Var (Y) + 2 ab Cov (X, Y)

If $X, Y$ are independent:

Var (a X + bY + c) = a^{2} Var (X) + b^{2} Var (Y)

E [(X + Y)^{2}] = E [X^{2}] + E [Y^{2}] + 2 E [X Y]

Var (X) \geq 0

Quartz 4