Rayleigh Distribution from Independent Normals

Let $X$ and $Y$ be independent $N(0, \sigma^2)$ random variables. Define $R = \sqrt{X^2 + Y^2}$.

(a) Using the CDF method, derive the PDF of $R$ for $r \geq 0$.

(b) Identify the distribution by name and compute $E[R]$.

(c) Evaluate $E[R]$ and $\text{Var}(R)$ numerically for $\sigma = 1$.