I am struggling with the following question:
- Let $\left(X, Y\right)$ be a pair of random variables with joint density function $\mathrm{g}\left(x,y\right) = \frac{1}{2}xy$ if $\left(x,y\right)\in D$, $0$ else. Here, $D$ denotes the first quarter of the disk of radius $2$ centered at $\left(0,0\right)$.
- First the problem asks if $X$ and $Y$ are independent. This is easy, we can simply integrate.
My issue is with the second question:
- Find the density of the random variable $R = \,\sqrt{\, X^{2} + Y^{2}\, }\,$ and then that of $R^{2}$.
I imagine this is related to a change of variables as it looks like polar coordinates but I don't understand what I am supposed to do here.
Hint
The cumulative distribution function of $\ R\ $ is given by \begin{align} P(R\le \rho)&=\iint_{D\cap \cal{D}_\rho}\frac{xy}{2}dxdy\\ &=\frac{1}{2}\int_0^\rho\int_0^\frac{\pi}{2}r^3\cos\theta\sin\theta\, d\theta dr\ , \end{align} where $\ \cal{D}_\rho=\left\{(x,y)\left|\sqrt{x^2+y^2}\le \rho\right.\right\}\ $, and its density function can be obtained by differentiating this. And $\ P(R^2\le r)=$$P\left(R\le \sqrt{r}\right)\ $.