Question
Let $X_1,X_2,\dots$ be IID random variables with finite mean $\mu_X$ and variance $\sigma_X^2$ and $Y_1,Y_2,\dots$ be another sequence of IID random variables with finite mean $\mu_Y$ and variance $\sigma_Y^2$. Note that the $X_i$ and $Y_i$ sequences are independent of each other. Consider the sequence $Z_i = X_iY_i$. Does the sequence $$\dfrac{Z_1 + Z_2 + \dots + Z_n}{n}$$ converge to some distribution as $n \rightarrow \infty$? If it does, state the sum of the mean and variance of the distribution. If it doesn't, answer $-100$. Answer this when $\mu_X = 5, \mu_Y = 6, \sigma_X^2 = 25,$ and $\sigma_Y^2 = 36$.