Find distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$

Question

Let $X_{1},...,X_{n}\sim f(x)=Kx^{2}e^{\frac{−x^{2}}{2σ^{2}}}$ i.i.d. I need find the distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$. To do this, calculate the normalizing constant, K, given by: $$K=\frac{1}{\sigma^{3} \sqrt{2\pi}}$$ with $$f(x)=\frac{1}{\sigma^{3} \sqrt{2\pi}}x^{2}e^{\frac{−x^{2}}{2σ^{2}}}$$ Later, use tranformation density to find distribution of $Y = X_{i}^2$, with $f(y)=\frac{1}{2\sigma^{3} \sqrt{2\pi}}\sqrt{y}e^{\frac{−|y|}{2σ^{2}}}$. However, I did not reach any clear result, could someone help me?

Answer 1

The moment generating function of $\frac{X^2}{\sigma^2}$ where $X\sim f$ is $$M_{X^2 /\sigma^2}(t)=\int_{-\infty}^{\infty}Kx^2 \exp\Big\{-\frac{x^2}{\sigma^2}\Big(\frac{1}{2}-t\Big)\Big\}\mathrm{d}x=(1-2t)^{-3/2}$$

This happens to be the moment generating function for a $\chi^2_3$ distribution.

Hence $\sum_{i=1}^n\frac{X_i^2}{\sigma^2}\sim \chi^2_{3n}$