Distribution of sum of squares of normals that have mean zero but not variance one?

Question

I am trying to find the distribution of a random variable that is calculated according to $Y:=\sum_{i=1}^n X_i^2$ where $X_i $ is distributed as $ \mathcal{N}(0,\sigma^2_i)$. Does there exist a particular of calculating this?

Thank you so much!

Answer 1

$X_i \sim \mathcal{N}(0,\sigma^2_i) \Rightarrow \frac{X_i}{\sigma_i}\sim \mathcal{N}(0,1) $

$\therefore$ $\frac{X_i^2}{\sigma_i^2} \sim \chi^2(1)=\Gamma(1/2,2)$

$X_i^2 \sim \sigma_i^2\Gamma(1/2,2)=\Gamma(1/2,2\sigma_i^2)$

If your $\sigma_i$s are fixed (i.e all the same) then

$\sum_{i=1}^n X_i^2 \sim \sum_{i=1}^n\Gamma(1/2, 2\sigma^2)=\Gamma(n/2,2\sigma^2)$ suppose $\sigma_i$s are equal to $\sigma$

i.e $\sum_{i=1}^n X_i^2$ has a gamma distribution with $k=n/2,\theta=2\sigma^2$

If your $\sigma_i$s are not fixed then

ref this