7 Show that if X and Y are uncorrelated then varx y sigma

7. Show that if X and Y are uncorrelated , then var[x + y] = sigmax^2 + sigmaY^2

Solution

Let us first solve Var(X + Y ) = E[(X + Y )(X + Y )] E[X + Y ]²
= E[X² + 2XY + Y ²] (µx + µy)²
= E[X² + 2XY + Y ²] µ²x 2µxµy µ²y
= (E[X²] µ²x) + (E[Y ²] µ²y) + 2(E[XY ] µxµy)
= Var(X) + Var(Y ) + 2Cov(X, Y ).

Now X and Y are uncorrelated if Cor(X, Y ) = 0.

Hence putting this condition we get var(X +Y)=var(X)+var(Y)