The sum of squares of $k$ independent standard normal random variables $\sim\chi^2_k$
I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then $X_1^2+X_2^2+\dots+X_k^2\sim\sigma^2\chi^2_k$. How do I go about obtaining the pdf?
If I have $k$ independent normal random variables where $X_i\sim\mathcal{N}(0,\sigma_i^2)$ then what is the distribution of $X_1^2+X_2^2+\dots+X_k^2$?
Best Answer
Let's answer the first one. If you know the PDF for $Z$, say $f_{Z}\left(z\right)$, then $f_{c\cdot Z}\left(c\cdot z\right)$ is found from the probability definition:
\begin{equation} \begin{split} \text{Pr}\left\{c\cdot Z < z \right\} &= \text{Pr}\left\{Z < \cfrac{z}{c} \right\} = F_{Z}\left(\cfrac{z}{c}\right) \quad \text{so} \\ \cfrac{d}{dz}\left[F_{Z}\left(\cfrac{z}{c}\right)\right] &= \cfrac{1}{c} f_{Z}\left(\cfrac{z}{c}\right) \end{split} \end{equation}
So, applied to a chi-square, just scaled the PDF for $\chi_{N}^{2}$ by $\cfrac{1}{\sigma^{2}}$ and scale it's argument by the same $\cfrac{1}{\sigma^{2}}$ and plot it.