Let $\{X_i\}$ be iid random variables where $X_i \sim N(\mu_x,\sigma^2)$ and $\{Y_i\}$-iid random variables where $Y_i \sim N(\mu_y,\sigma^2)$. Every $X$ is also independent from $Y$. I need to find such variance:
$$Var\Big(\sum_{\substack{i=1,…,n\\ j=1,…,n}} X_i Y_j \Big)$$
I tried to transform it into sums of variances and covariances but then i couldn't manage to calculate the covariance. I did something like this but i feel like its wrong.
$$\sum_{\substack{1\leq i < k \leq n\\ 1\leq j < l \leq n \\ i \neq k \lor j\neq l}} Cov(X_iY_j,X_k Y_l) = \sum_{\substack{i=1,…,n\\ 1\leq j < l \leq n}} Cov(X_iY_j,X_i Y_l) + \sum_{\substack{1\leq i < k \leq n\\ j=1,…, n}}Cov(X_iY_j,X_k Y_j)$$
$$+ \sum_{i\neq j \neq k \neq l} Cov(X_iY_j,X_k Y_l)+ \sum_{\substack{ i \neq j \neq k\\ j=1,…, n}}Cov(X_iY_j,X_k Y_k) + \sum_{\substack{ i \neq k \neq l\\ j=1,…, n}}Cov(X_iY_i,X_k Y_j)$$
I would appreciate any advice on how to calculate this.
Best Answer
Notice that $$S = \sum_{1 \le i, j \le n} X_i Y_j = \sum_{i=1}^n X_i \sum_{j=1}^n Y_j.$$ Then since $$S_X = \sum_{i=1}^n X_i \sim \operatorname{Normal}(n \mu_x, n \sigma^2), \\ S_Y = \sum_{j=1}^n Y_j \sim \operatorname{Normal}(n \mu_y, n \sigma^2),$$ the variance is computed as $$\begin{align} \operatorname{Var}[S] &= \operatorname{Var}[S_X S_Y] \\ &= \operatorname{E}[(S_X S_Y)^2] - (n^2 \mu_x \mu_y)^2 \\ &\overset{\text{ind}}{=} \operatorname{E}[S_X^2]\operatorname{E}[S_Y^2] - n^4 \mu_x^2 \mu_y^2 \\ &= (\operatorname{Var}[S_X] + n^2 \mu_x^2) (\operatorname{Var}[S_Y] + n^2 \mu_y^2) - n^4 \mu_x^2 \mu_y^2 \\ &= n^2 (n (\mu_x^2 + \mu_y^2) + \sigma^2 )\sigma^2. \end{align}$$