I am studying for a preliminary examination in the fall over probability and have come across this question:
Let $X_1, X_2, ... ,X_n$ be independent standard normal random variables. Find the distribution of the following random variables:
(a) $X_1^2$
(b) $\sum_{k=1}^{n} X_k^2$
I know the answer to part (a), $X_1^2$ is chi-squared with one degree of freedom, my question is on part (b). I know that this sum is chi-squared with n degrees of freedom, I found this using the method of characteristic functions. However, in our course we never mentioned what the characteristic function of a chi-squared random variable is, I had to look that up online and of course I won't have that luxury during the actual exam. So, my question here is just: Is there another simple way to answer part (b), in particular is there a way that uses hardly any knowledge of the chi-squared random variable? Thanks in advance.