How to model this distribution using Central Limit theorem?

Question

I have distribution that can be defined as below,

$S=a_0\cdot b_0 + a_1\cdot b_1 + a_2\cdot b_2 + \cdots +a_{n-1}\cdot b_{n-1}$

Now, I want find the distribution of $S$ when, $a_i$'s are selected from a certain distribution with standard deviation $\sigma$ (for simplicity we can assume it a Gaussian distribution). And $b_i$'s can be $+5$ with probability $p$ and $-5$ with probability $1-p$. How, as far as I know $S$ will be distributed normally too, but what will be the standard deviation of such distribution.

Answer 1

$S$ will not be normally distributed, but no matter

If everything is independent then

• $a_i$ has mean $\mu$ and variance $\sigma^2$ so $E[a_i^2]=\sigma^2+\mu^2$
• $b_i$ has mean $5-10p$ and $E[b_i^2]=25$
• $a_i b_i$ has mean $\mu(5-10p)$ and $E[a_i^2 b_i^2]=E[a_i^2]E[b_i^2]= 25(\sigma^2+\mu^2)$ so has variance $25\sigma^2 +100p \mu^2-100p^2 \mu^2$
• $S=\sum a_i b_i $ has mean $n\mu(5-10p)$ and variance $25n(\sigma^2 +4p(1-p)\mu^2)$ and so a standard deviation which is the square-root of that
• if $p=\frac12$ then the mean of $S$ is $0$ and the standard deviation of $S$ is $5\sqrt{n(\sigma^2+\mu^2)}$