Let : $X_1$ and $X_2$ two independent variables, square-integrable et non constant.
At which necessary and sufficient condition, do we have $ Var(X_1 X_2)= Var(X_1) Var(X_2)$ ?
Attempt :
$V(Z)= V( E(Z|X_1) ) + E(V(Z|X_1))$ if $Z$ is a random variable.
Best Answer
$$var(X_1 X_2) = E(var(X_1 X_2|X_2))+var(E(X_1 X_2 | X_2)\\ =E(X_2^2 var(X_1)) + var(X_2 E(X_1))\\ = var(X_1) (E(X_2)^2 + var(X_2)) + E(X_1)^2 var(X_2)\\ = var(X_1) var(X_2) + E(X_2)^2 var(X_1) + E(X_1)^2 var(X_2)$$
Since $X_1, X_2$ are not constants then we must have $E(X_1)=E(X_2)=0$ which is necessary and sufficient.