Distribution of the step size of diffusion in 3-dimensional space

Question

I need to find the distribution of the random variable $$Y=\sqrt{X_1^2+X_2^2+X_3^2}$$ where $X_i\sim{\cal{N}}(0,\sigma^2)$ and $\sigma^2$ is related to diffusion coefficient. All $X_i$s are independent. I wonder whether there is a general form for $Y$.

Answer 1

Write $Z_i = \dfrac{X_i}{\sigma} \sim \mathcal{N}(0, 1)$.

Then, $$Y = \sigma\sqrt{Z_1^2+Z_2^2+Z_3^2}\text{.}$$ Using a standard result, $Z_1^2+Z_2^2+Z_3^2 \sim \chi^2_3$.

The square root of a $\chi^2_3$ distribution is a chi distribution with three degrees of freedom. Since we are multiplying this by $\sigma$, we obtain a random variable proportional to a chi distribution with 3 degrees of freedom. One name for this is the Maxwell-Boltzmann distribution.