Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$. Find:
(a) $E(X_1 X_2)$, and
(b) $Var(X_1 X_2)$.
Isn't (a) = zero, since this are independent?
How do I go about (b)

(a) By symmetry $E(X_i)=\frac{1}{2}$. By independence $E(X_1X_2)=E(X_1)E(X_2)$.
(b) The variance of $X_1X_2$  is $E((X_1X_2)^2)-(E(X_1X_2))^2$.  It remains to calculate $E((X_1X_2)^2)$. By independence, this is $E(X_1^2)E(X_2^2)$.
To calculate $E(X_i^2)$, note this is $\int_0^1 x_i^2 \,dx_i$. 
