covariance matrix of X+Y and X-Y

Question

This question comes up in almost every past paper i do and is worth 10 marks and just can't work it out...

Let $X$ and $Y$ have the joint pdf $$f(x,y)= \begin{cases} e^{-y}, \text{if} \ 0 < x < y < \infty \\ 0, \texttt{otherwise} \end{cases}$$

It first asks to find the marginal probability density functions of X and Y which i found to be

for x

$e^{-x}$

and for y

$y\cdot e^{-y}$

and i found these to not be independent. It then asks to compute the covariance matrix of $X+Y$ and $X-Y$ and this i have no idea how to do.

Any help or hints will really be appreciated.

Answer 1

Let $A=X+Y,B=X-Y$. You have to calculate 5 things: $E[A],E[B],E[A^2],E[B^2],E[AB]$. Each of these is an integral calculation; for example

$$E[AB]=\int_0^\infty \int_0^y (x+y)(x-y) e^{-y} dx dy.$$

Once you have these six quantities, calculate $\text{Var}(A)=E[A^2]-(E[A])^2,\text{Var}(B)=E[B^2]-(E[B])^2$, and $\text{Cov}(A,B)=E[AB]-E[A]E[B]$. Then the covariance matrix is

$$\begin{bmatrix} \text{Var}(A) & \text{Cov}(A,B) \\ \text{Cov}(A,B) & \text{Var}(B) . \end{bmatrix}.$$