# How to find E(XY) when X and Y are NOT indepdant?

laura_a

## Homework Statement

I have a joint pdf f_{XY}(x,y) = (2+x+y)/8 for -1<x<1 and -1<y<1

## Homework Equations

I have to work out E(XY) but I have previously worked out that X and Y are NOT independant (that is f_{XY}(0,1) doesn't equal f_X{0}*f_Y{1}). I am using maxima so I don't need help with any integration, I just need to know what formula because I've read that E(XY) = E(X)E(Y) only when they're indepdant... so what happens when they're not?

zhentil
You integrate xy against the pdf. Do you not have the textbook?

laura_a
No, there is no textbook for this, I have bought some books, but none of them are written for people who aren't the best at statistics. I have no idea what you mean, isn't there an easier way using E(X) and E(Y) which I already have?

zhentil
No, there's not. Is this for a class?

DavidWhitbeck
Oh goody double posting! Laura you now have two people telling you the same thing-- integrate. I don't know why you had to start two threads on the same topic instead of just being patient.

wisling
You can use the definition of an expectation.
E(XY) = $$\oint\oint$$x*y*f(x,y) dy dx
Or you could argue that since the function is symmetric about 0 and the intervals [-1, 1] are centred about 0 that E(XY) = 0

Homework Helper
The density isn't symmetric about zero.

Laura, for any joint continuous distribution, whether or not $$X, Y$$ are independent, you can find $$E[XY]$$ as

$$\iint xy f(x,y) \, dxdy$$