Proof | Property of Variance (5)

$V [a X + b Y] = a^{2} V [X] + b^{2} V [Y] + 2 a b C o v [X, Y]$
where $C o v [X, Y]$ is the covariance of X and Y.

Proof : Let X, Y be random variables and a, b be any constants.

$\begin{array}{rcl} V [a X + b Y] & = & E [{(a X + b Y) - E [(a X + b Y)]}^{2}] \\ = & E [{a (X - E [X]) + b (Y - E [Y])}^{2}] (∵ E [a X + b Y] = a E [X] + b E [Y]) \\ = & E [a^{2} (X - E [X])^{2} + b^{2} (Y - E [Y])^{2} + 2 a b (X - E [X]) (Y - E [Y])] \\ = & a^{2} V [X] + b^{2} V [X] + 2 a b C o v [X, Y] \end{array}$

href="https://mrs-mathpedia.com/">Mrs.Mathpedia