covariance and correlation of x and y

Question

Given that the independent random variables $X$ and $Y$ have variance $36$ and $16$ respectively. Find (i) $Var(X + Y)$ (ii) $Var(X – Y)$ (iii) the correlation coefficient between $(X + Y)$ and $(X – Y)$ I found the answers for the first $2$ parts which is basically the addition of the variances and that is $52$. But for the third part $\text{correlation}= Cov(x+y,x-y)/(stdx.stdy)$ how do i find $Cov(x+y,x-y)$ because $cov(x,y)=E(XY)-E(X)E(Y)$. how do i find the expectations as they are not given. Your help will be much appreciated

Answer 1

The covariance of two independent r.v is equal to zero, hence $\mbox{cov}[X,Y] = 0$. To find the answer to your question use: $$\mbox{cov}[X+Y,X-Y] = \mbox{cov}[X,X]+\mbox{cov}[Y,X]-\mbox{cov}[X,Y]-\mbox{cov}[Y,Y].$$