Consider a vector $\mathbf{x}\in\mathbb{R}^n$, where each element in $\mathbf{x}$ is sampled independently from a normal distribution $\mathcal{N}(0,\sigma^2)$.
What is the probability density function of $||\mathbf{x}||_2^2$?
Consider a vector $\mathbf{x}\in\mathbb{R}^n$, where each element in $\mathbf{x}$ is sampled independently from a normal distribution $\mathcal{N}(0,\sigma^2)$.
What is the probability density function of $||\mathbf{x}||_2^2$?
Note that $ \frac{1}{\sigma^2}\|\mathbf{x}\|_2^2=\frac{1}{\sigma^2}(x_1^2+\cdots+x_n^2)=z_1^2+\cdots+z_n^2, $ where $z_i\sim \mathcal{N}(0,1)$ and hence $\frac{1}{\sigma^2}\|x\|_2^2$ follows a $\chi^2(n)$-distribution.