# obtaining a distribution from the sum of normal distributions

We are given that $Y_i$ ($i=1,2,3...10$) is an independent random sample from a $N\sim(-1,4)$ distribution, and that Q is given by:

$$Q=\frac{1}{400}\Bigl(\sum_{i=1}^{10}(Y_i +1)\Bigl)^2$$

The question is to name the distribution of Q, and state the parameter(s) of the distribution.

My working was: $\sum_{i=1}^{10}(Y_i +1)$ = $N\sim(-10, 400)$

$\Bigl(\sum_{i=1}^{10}(Y_i +1)\Bigl)^2=\chi^2(1)$

My specific question is what is the distribution of Q. But more generally, what is the general approach and intuition to dealing with ratios in these sort of problems. To take another example, suppose $X_i$ a random variable with a $N\sim(0, \sigma^2)$, ($i=1,2,..5$). If $U= X_1 +X_2 +X_3 +X_4$, then $N\sim(0,4\sigma^2)$. I can accept this. But $\frac{U}{4}$ is given as $N\sim(0,\frac{4\sigma^2}{16})$

I am not clear what $\frac{U}{4}$ represents and why we would want to divide U by 4 (or in the first example, why divide by 400) and why the variance changes to $\frac{4\sigma^2}{16}$ (and also, for the normal distribution, whether the divisor would affect the mean also). Is there any intuitive explanation for including a divisor and is there any general rule, when trying to solve these problems, how to deal with the divisor. Hope this makes sense. Thanks

• Hints: Work the problem from the inside out--that is, in the order in which the operations are performed. Given you know the distribution of $Y_i$, (1) What is the distribution of $Y_i+1$? (2) What is the distribution of the sum of the $Y_i+1$? (3) What is the distribution of the square of that sum? To solve (3) it helps to express the result of (2) as a rescaled version of a simple, well-known distribution. Although all three parts can be tackled from basic definitions, at this stage of learning you likely already know the answers (and the point of this problem is to exercise that knowledge). – whuber Jul 29 '18 at 16:03
• It might help to realize the Chi-squared distribution family is a particular subset of the Gamma distribution and to look at scaling Gamma random variables – deasmhumnha Jul 30 '18 at 1:44