Question

Let the joint probability density function of X and Y be bivariate normal. For what values of a is the variance of X + Y minimum ?

Answer 1

$Var(\alpha X+Y)=Var(\alpha X)+Var(Y) + 2cov(\alpha X,Y)$

$\Rightarrow Var(\alpha X+Y)=\alpha^2Var(X)+Var(Y) + 2\alpha cov(X,Y)$

$\Rightarrow Var(\alpha X+Y)=\alpha^2\sigma_X^2+ \sigma_Y^2+2\alpha *\sigma_X*\sigma_Y$

To find $\alpha$ such that $Var(\alpha X+Y)$ is minimum.

Now,

$Var(\alpha X+Y)\geq 0$

$\Rightarrow \alpha^2\sigma_X^2+ \sigma_Y^2+ 2\alpha *\sigma_X*\sigma_Y \geq 0$

$\Rightarrow (\alpha \sigma_X+\sigma_Y)^2 \geq 0$

$\Rightarrow (\alpha \sigma_X+\sigma_Y) \geq 0$

$\Rightarrow \alpha \geq - \frac{\sigma_Y}{\sigma_X}$