First I would like to clear up some misunderstanding I have about independence. In my text I have:

If $\displaystyle F(x,y)=F_X(x)F_Y(y)$, then X and Y are indep.

However, in my notes from class I have:

If $\displaystyle f(x,y)=f_X(x)f_Y(y)$, then X and Y are indep.

Which is correct?

Is this the only way to show independence? Does it have anything to do with expectation values? I had a question in which I had:

$\displaystyle X=Z, Y=Z^2; Z \sim N(\mu,\sigma^2)$

I was first asked to find the followings: $\displaystyle <X>,<Y>,<XY>$ and here is what I did:

$\displaystyle <X>=<Z>=\mu$

$\displaystyle <Y>=<Z^2>=Var(Z)+<Z>^2=\sigma^2+\mu^2$

$\displaystyle <XY>=<Z^3>$, for this I took the 3rd derivative of the moment generating function and got this expression: $\displaystyle 2\sigma^2\mu+\mu^3$

I was then asked about whether X and Y are independent. Naturally I attempted to use what I just did: $\displaystyle <XY>\neq<X><Y>$ therefore dependent. Is this correct?