I have the following problem: $f_{(X,Y)}(x, y)$ as a PDF of a continuous random variable. I had to calculate the marginal PDF/CDFs, and the means and vars of these ($E(X), E(Y), Var(X), Var(Y)$).
Now I have another random variable $Z = (X-2)^2 + Y$, and I need to calculate $E(Z)$ and $Var(Z)$.
I had no problems with $E(Z)$, but I do have some trouble with $Var(Z)$.
$Var(Z) = Var(X^2 – 2X + 3Y)$. How do I need to proceed from here? Even if $X$ and $Y$ were not correlated, I still wouldn't know how to calculate $Var(X^2)$. (I did google this, but there were many confusing solutions). Is it $E(X^4) – [E(X^2)]^2$?
And if it is, could you still help me figure out how to calculate the whole $Var(X^2 – 2X + 3Y)$ part?
Best Answer
Use the following formula:
$$V(X+Y+Z)=V(X)+V(Y)+V(Z)+2Cov(X,Y)+2Cov(X,Z)+2Cov(Y,Z)$$
and treat $X^2$ as a RV, so
$$V(X^2)=E((X^2)^2)-(E(X^2))^2$$