How do I go about finding the the pdf of the statisitc $\sum_i x_i ^2$ such that each $x_i$ is iid from a $N(\sigma , \sigma)$ distribution? I've searched, but cannot find a straightforward answer. I've looked at charts such as this (which I highly recommend reviewing if you have not seen one like it). I've also reviewed this answer about the noncentral chi-squared distribution, but it seems to be specifically for iid variables drawn from the distribution $N(0, \sigma^2)$.
[Math] Distribution of the sum of squared independent normal random variables
normal distributionprobability distributionsprobability theory
Best Answer
If $X_i$ are iid normal with mean $\mu$ and variance $\sigma^2$, then $\sigma^{-2} \sum_{i=1}^n X_i^2$ has noncentral chi-squared distribution with $n$ degrees of freedom and noncentrality parameter $\lambda = n (\mu/\sigma)^2$ (as defined e.g. in Wikipedia; conventions may sometimes differ). So what you have is a multiple of a noncentral chi-squared random variable. If $Y$ has pdf $f_Y$, then $Z = c Y$ (for $c > 0$) has pdf $f_Z(z) = c^{-1} f_Y(z/c)$.