Variance of sum of multiplication of independent random variables

Question

Suppose that we have $Z=\sum_{i=1}^n (a_i+b_iX_i)(c_i+d_iY_i)$. Where $a_i,b_i,c_i$ and $d_i$ are real numbers and $X_i$s and $Y_i$s are all independent random variables. How can I find the variance of the random variable Z?

Answer 1

$\newcommand{\E}{\operatorname{\mathbb{E}}} \newcommand{\Var}{\operatorname{\mathbb{Var}}}$If the random variables are mutually independent: $$\begin{align}A\bot B &\iff \E(A B)=\E(A)\E(B) \\ A \bot B &\iff (n+A) \bot (m+B) \\ A \bot B &\iff nA\bot mB \\ A \bot C, B\bot C &\implies AB \bot C \end{align}$$

So: $\begin{align} \text{Let } P_i & = a_i+b_iX_i \\ \E(P_i) & = a_i+b_i\E(X_i) \\ \E(P_i^2) & = a_i^2+2a_ib_i\E(X_i)+b_i^2\E(X_i^2) \\ \Var(P_i) & = \E(P_i^2)-\E(P_i)^2 \\ & = a_i^2+2a_ib_i\E(X_i)+b_i^2\E(X_i^2) - a_i^2 - 2 a_ib_i\E(X_i)- b_i^2\E(X_i)^2 \\ & = b_i^2\Var(X_i) \\[2ex]\text{by symmetry} \\ \text{Let } Q_i & = c_i+d_iY_i \\ \E(Q_i) & = c_i+d_i\E(Y_i) \\ \Var(Q_i) & =d_i^2\Var(Y_i) \\[2ex]\text{Let } R_i & = P_iQ_i \\ \E(R_i) & = \E(P_i)\E(Q_i) & \ni P_i \bot Q_i \\ \E(R_i^2) & = \E(P_i^2)\E(Q_i^2) \\ \Var(R_i) & = \E(P_i^2)\E(Q_i^2)-\E(P_i)^2\E(Q_i)^2 \\ &= \Var(P_i)\Var(Q_i) + \Var(P_i)\E(Q_i)^2 + \Var(Q_i)\E(P_i)^2 \\ & = b_i^2d_i^2\Var(X_i)\Var(Y_i) \\ & \quad + b_i^2\Var(X_i)(c_i+d_i\E(Y_i))^2 \\ & \quad + d_i^2\Var(Y_i)(a_i+b_i\E(X_i))^2 \\[2ex] \text{Let } Z & = \sum_i R_i \\ \Var(Z) & = \sum_i \Var(R_i) & \ni R_i\bot R_j, \forall i\neq j \\ & = \sum_i b_i^2d_i^2\Var(X_i)\Var(Y_i) \\ & \quad + \sum_i b_i^2\Var(X_i)(c_i+d_i\E(Y_i))^2 \\ & \quad + \sum_i d_i^2\Var(Y_i)(a_i+b_i\E(X_i))^2 \end{align}$