I'm stuck on a question with a weird twist on the usual Rayleigh distribution.
Instead of assuming that the components of the distribution are independent, we are given alternative conditions and asked to show independence. The setup is as follows:
- Let $R \geq 0$ be a random variable with density $f_R(r) = re^-\frac{r^2}{2}$
- Let $\Theta$ be uniformly distirbuted on $[-\pi,\pi]$ so that $f_\Theta(\theta) = \frac{1}{2\pi}$
- Show that the variables $X = R\cos(\theta)$ and $Y = R\sin(\theta)$ are independent and have density $f_X(t) = f_Y(t) = \frac{e^{-\frac{t^2}{2}}}{\sqrt{2\pi}}$
My current approach is to show that $f_{XY}(x,y) = f_X(x)f_y(y)$. I have used the multivariate change of density formula to obtain the correct distribution for $f_{XY}(x,y)$, which is $\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}$. However I cannot figure out how to obtain expressions for $f_X(x)$ and $f_Y(x)$.
Please note The question is actually designed to show that $\int_{-\infty}^\infty e^{-\frac{t^2}{2}} = \sqrt{2\pi}$, so we cannot assume that this is true. This effectively means that we cannot integrate $f_{XY}$ over the range of Y to get $f_X$.