For my exam preparation I'm currently stuck at this task. Let X,Y be i.i.d and standard normal distributed variables. Calculate $E\left(\frac{(X+Y)^4}{X-Y}\right)$. So I have
- $E(X)=E(Y)=0$ and $Var(X)=Var(Y)=1$
- $E(X+Y)=E(X)+E(Y)=0$ and $E(XY)=0$
- $E(X^2)=1$ since $E(X^2)=Var(X)+(E(X))^2 = 1 + 0^2 = 1$
- Product of i.i.d is again i.i.d
Now I try to break the initial expression down, so I can put the values above into the expression: $$E\left(\frac{(X+Y)^4}{X-Y}\right) = E((X+Y)^4)E\left(\frac{1}{X-Y}\right)=(E(X+Y))^4E\left(\frac{1}{X-Y}\right)=0$$ which does not seems to be correct. Thansk for tips/adivece.