Chp 4: How to calculate E[XY] and cov[X,Y] for dependent variables?

[keepcalmandstudyon]

Hi,

I'm on chapter 4 and have a question about how to calculate E[XY] and cov[X,Y] for continuous dependent variables.

I understand that this formula below is only true when X and Y are independent random variables (because cov(X,Y)=E[XY]−E[X]E[Y]=0 only if X and Y are independent random variables).

E[X,Y]=∫∫xy f(x,y) dx dy​

(This formula is based on page 18 of chapter 4. I understand that E[X,Y] and E[XY] are the same thing and that g(x,y) is xy. Please correct me if I'm wrong here!)

So my question is - when X and Y are dependent, or when we don’t know whether or not X and Y are independent, how can we calculate E(XY)?

Page 22 of Chapter 4 mentions that we can find E[XY] for dependent random variables by using E[XY] = E[X]E[Y] + cov[X,Y]. But this seems to assume that we already know what cov[X,Y] is.

If we don’t know what cov[X,Y] is, how can we calculate cov[X,Y] when X and Y are dependent, or when we don’t know whether or not X and Y are independent?

I understand that cov[X,Y] = E[XY] – E[X]E[Y], but this formula wouldn’t help to find cov[X,Y] when we don’t know what E[XY] is.

Thanks in advance!

 

[Andrea Goude]

E[XY]=∫∫xy f(x,y) dx dy can be calculated by integrating.

See the question and solution on pages 25-26 for an example with U and V, you calculate E[UV] and cov(U,V).

Hope this helps.