Take two i.i.d. Gaussian R.V.s $X$ are $Y$ both of which are $~N(0,a\sigma)$.
Define a new R.V. $D = \sqrt{X^2 + Y^2}$.
What's the expected value $E(D)$?
In researching this I'm seeing references to chi-squared distributions and chi distributions (of different degrees of freedom?), but I haven't learned about those yet. Do they apply here? What are their means and how does the original mean/variance carry through?
If all you want is the expected value of $D$, then write the expectation as an integral and switch to polar coordinates: $$ \begin{align} ED=E\sqrt{X^2+Y^2}&=\iint\sqrt{x^2+y^2}f(x)f(y)\,dy\,dx\\ &=\int_{x=0}^\infty\int_{y=0}^\infty\sqrt{x^2+y^2} \frac1{2\pi (a\sigma)^2}\exp[-(x^2+y^2)/2(a\sigma)^2]\,dy\,dx\\ &=\int_{\theta=0}^{2\pi}\int_{r=0}^\infty r\frac1{2\pi(a\sigma)^2}\exp[-r^2/2(a\sigma)^2]\,r\,dr\,d\theta\\ &=\frac1{(a\sigma)^2}\int_{r=0}^\infty r^2\exp[-r^2/2(a\sigma)^2]\,dr\\ &=\sqrt{\frac\pi2}a\sigma \end{align} $$