Let X and Y be two random variables that have the same varia

Let X and Y be two random variables that have the same variance. Define

Z = X + Y and W = X - Y . Show that Z and W are uncorrelated.

Solution

Variance of Z = var x+ var T

and variance of W = var x - var y

Var z.-var W = sigmax^2-sigmay^2 =0

i.e. Z-W has variance =0

So Z and W are uncorrelated