Question

if X and Y are random variables such that Var(X)<∞ and Var(Y)<∞,show that

Var(aX+bY)=a²Var(X)+b²Var(Y)+2ab cov(X,Y). Where a and b are real constants.

Answer 1

$E{[(aX+by)-(aμ_X+bμ_Y )]^2 }=E{[a(X-μ_X )+b(Y-μ_Y )]^2 } \\=E{[a(X-μ_X )]^2 }+E{[2ab(X-μ_X )(Y-μ_Y )]}+E{[b(Y-μ_Y )]^2 } \\=a^2 \text{Var⁡(X)}+2ab\text{Cov⁡(X,Y)}+b^2 \text{Var⁡(Y)} \\=a^2 \text{Var⁡(X)}+b^2 \text{Var⁡(Y)}+2ab\text{Cov⁡(X,Y)}$