How to Calculate E(XY) for Random Variables X and Y?

[bioman]

I have two random variables X and Y, and I need to calculate E(XY). The expectation of X, E(X) = aZ, and the expectation of Y, E(Y) = bZ, where a and b are known constants and Z is a random variable.

So the question is how would I calculate E(XY)?
I was thinking that I could do the following:
E(XY) = E(aZ,bZ)
=> E(XY) = ab*E(ZZ)
=> E(XY) = ab*E(Z^2)

Is it correct to do this?? or how would I do it?

 

[Hurkyl]

E(X) can't be equal to aZ: E(X) is a number, and aZ is a random variable. Are you sure you stated the problem right?


Anyways, there generally aren't short-cuts to computing the expectation of a product of random variables.

 

[bioman]

Yes, you're right I've stated the problem wrong! I can restate in another, much easier way.

So basically I need to calculate E(XY), where E(X) = aE(Y), where the constant a is less than 1.

So any ideas on how to go about calculating E(XY)??
Any help or directions would be great!

 

[Hurkyl]

So basically I need to calculate E(XY), where E(X) = aE(Y), where the constant a is less than 1.

So any ideas on how to go about calculating E(XY)??

Can't be done with the information given.

 

[bioman]

What more information would I need to calculate this?
For example, I know what the E(Y) and Var(Y) is going to be, I also know what the constant a is going to be.
So I know what the mean and variance of X and Y are going to be and the constant a, so what more information do I need to get E(XY)?

Overall I'm trying to calculate the Cov(XY) = E(XY) - E(X)E(Y), and seeming as X and Y are dependent, shouldn't I be able to work out the covariance between them??
I think I have all the information necessary to get this expression, I'm probably just not supplying it to you here?

 

[Hurkyl]

Do you have their distribution? You can compute E(XY) directly, rather than looking for a shortcut involving other things you can compute.

 

[bioman]

No unfortunately I'm unable to get the distribution of XY (if that's what you were talking about).
I just have the mean and variance of X and Y to play with and the constant a.

So when you say

You can compute E(XY) directly

Is there a general formulae for calculating E(XY) for dependent variables??
I could only find a formulae for independent variables.

 

[Hurkyl]

$$E[XY] = \sum_{a, b} a b \mathop{\mathrm{P}}(X = a \mathrm{\ and\ } Y = b)$$
(Or an integral, if appropriate)

 

[bioman]

Ok thanks for that, I'll have a look into it.

Also I was thinking maybe I could do it the following way, but I'm not sure my
"random variable algebra" is correct:

So again suppose I need to calculate E(XY), where E(X) = aE(Y), where the constant a <= 1.
We have E(X|Y) = aY
=> E(YX|Y) = aYY = aY^2
=> E(XY) = E(E(YX|Y)) = E(aY^2)
=> E(XY) = aE(Y^2)

Would this be correct??

 

[Jason Swanson]

Ok thanks for that, I'll have a look into it.

Also I was thinking maybe I could do it the following way, but I'm not sure my
"random variable algebra" is correct:

So again suppose I need to calculate E(XY), where E(X) = aE(Y), where the constant a <= 1.
We have E(X|Y) = aY
=> E(YX|Y) = aYY = aY^2
=> E(XY) = E(E(YX|Y)) = E(aY^2)
=> E(XY) = aE(Y^2)

Would this be correct??

You say you are given that $$E[X]=aE[Y]$$. This does not imply that $$E[X|Y]=aY$$. As an example, suppose $$X\sim N(a,1)$$, $$Y\sim N(1,1)$$, and they are independent. Then $$E[X]=a=a\cdot 1=aE[Y]$$, but $$E[X|Y]=E[X]=a$$. Clearly, $$a\ne aY$$.

If you are given that $$E[X|Y]=aY$$, then your calculations are correct.