How to compute joint pdf of f(x,y) =128e^(-8x-8y)?

Question

I am asked to find the joint pdf of $f(x,y)=128e^{(-8x-8y)}$ where $0 < x < y< \infty$ or $0$ otherwise. find the following: $$E(X)= ?$$

$$E(Y)= ? $$

$$E[X(Y-X)]= ? $$

Answer 1

Here if you see, the marginal of X is a Exponential Distribution with mean 1/16.

Thus E(X) = 1/16

And for the marginal of Y it is coming like this,

f(y) = 16e^(-8y)(1 - e^(-8y)) ; y>0

= 0 ; otherwise.

Thus by integrating you can find E(Y) = 3/16.

And for E [ X( Y - X )] = E(XY) - E(X^2)

By calculating you can see E(XY) = 1/128

And E(X^2) = Var( X ) + [ E( X ) ]^2

Thus X ~ Exponential Distribution with mean 1/16

Var( X ) = 1/256 and [ E(X) ]^2 = 1/256

E ( X^2) = 1/128

Thus E[ X(Y - X )] = 1/128 - 1/128 = 0

That's your answer.

And since I don't know the use of latex I have write in this manner. And if any help regarding Statistics is required, you can message me. I hope it helps you.