I tried to reason out the LHS as:
$V(XY) = E(X^2Y^2)-E(XY)^2$
$V(XY) = E(X^2)E(Y^2)-E(X)^2E(Y^2)$
Now I'm not sure how to go about showing that the RHS should be equal to this. Anyone know how to do this?
I tried to reason out the LHS as:
$V(XY) = E(X^2Y^2)-E(XY)^2$
$V(XY) = E(X^2)E(Y^2)-E(X)^2E(Y^2)$
Now I'm not sure how to go about showing that the RHS should be equal to this. Anyone know how to do this?
Since $X$ and $Y$ are independent (And thus $X^2$ and $Y^2$ are independent), we have:
\begin{align} V(XY) &= E(X^2Y^2) - E(XY)^2 \\ &= E(X^2)E(Y^2) - V(X)V(Y) - E(X)^2E(Y)^2 + V(X)V(Y) \\ &= E(X^2) E(Y^2) - \big(E(X^2) - E(X)^2 \big)\big(E(Y^2) - E(Y)^2 \big) - E(X)^2 E(Y)^2 + V(X) V(Y) \\ &= E(X)^2 E(Y^2) + E(X^2) E(Y)^2 - 2E(X)^2 E(Y)^2 + V(X) V(Y) \\ &= E(X)^2 E(Y^2) - E(X)^2 E(Y)^2 + E(X^2) E(Y)^2 - E(X)^2 E(Y)^2 + V(X) V(Y) \\ &= E(X)^2 V(Y) + E(Y)^2 V(X) + V(X) V(Y) \end{align} as desired.