common-expectation-and-variance-operations

Notation

$X, Y, X_{i}$ : Random variables
$a, b, c, c_{i}, a_{i}$ : Constants/scalars
$X ⊥ Y$ : $X$ and $Y$ are independent
$Cov (X, Y)$ : Covariance between $X$ and $Y$ TODO: Create def note on covariance

E [c] E [X \pm c] E [c X] Var (X \pm c) Var (c X) = c = E [X] \pm c = c E [X] = Var (X) = c^{2} Var (X)

E [X \pm Y] E [a X + bY] Var (X \pm Y) Var (X + Y) Var (a X + bY) = E [X] \pm E [Y] = a E [X] + b E [Y] = Var (X) + Var (Y) \pm 2 Cov (X, Y) = Var (X) + Var (Y) when X ⊥ Y = a^{2} Var (X) + b^{2} Var (Y) + 2 ab Cov (X, Y)

E [i = 1 \sum n X_{i}] E [i = 1 \sum n c_{i} X_{i}] Var (i = 1 \sum n a_{i} X_{i}) Var (i = 1 \sum n c_{i} X_{i}) = i = 1 \sum n E [X_{i}] = i = 1 \sum n c_{i} E [X_{i}] = i = 1 \sum n a_{i}^{2} Var (X_{i}) + 2 i < j \sum a_{i} a_{j} Cov (X_{i}, X_{j}) = i = 1 \sum n c_{i}^{2} Var (X_{i}) when all X_{i} are independent

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