(Solved): can i get help on E and F. use R for E. Thank you [40 pts total] Being able to sample from the unif ...

can i get help on E and F. use R for E. Thank you

[40 pts total] Being able to sample from the uniform distribution on a disk (the region on the plane bounded by a circle) can be useful for conducting Monte Carlo studies. Two Statistics graduate students are trying to write down the probability density function (PDF) for a uniform distribution on a disk of radius \( \rho \). Let \( R \) and \( \Theta \) denote the polar coordinates of a point on the disk (you should have learned about polar coordinates in previous math courses). One of the students suggests the following PDFs for \( R \) and \( \Theta \), respectively, \[ f_{R}(r)=\left\{\begin{array}{ll} \frac{2 \pi r}{\pi \rho^{2}} & 0 \leq r \leq \rho, \\ 0 & \text { otherwise }, \end{array} \quad f_{\Theta}(\theta)=\left\{\begin{array}{ll} \frac{1}{2 \pi} & 0 \leq \theta \leq 2 \pi, \\ 0 & \text { otherwise, } \end{array}\right.\right. \] Answer the following questions. (a) \( [6 \mathrm{pts}] \) Show that the PDFs for \( R \) and \( \Theta \) are both valid PDFs. (b) \( [6 \mathrm{pts}] \) Find the CDF of \( R \) and the CDF of \( \Theta \). (c) \( [6 \mathrm{pts}] \) Find the means of \( R \) and \( \Theta \). (d) \( [4 \mathrm{pts}] \) Find \( E\left[\Theta^{3}\right] \). (e) [10 pts] One student suggested the following set of PDFs: \[ R \sim U(0, \rho), \quad \text { and } \quad \Theta \sim U(0,2 \pi), \] Therefore, sampling simply involves generating both \( R \) and \( \Theta \) from uniform distributions separately. However, the other student argue that this scheme will not simulate points uniformly from the disk. We will verify this claim via simulation for \( \rho=1 \). Generate 3000 samples each from Uniform \( [0,1] \) and Uniform \( [0,2 \pi] \) distributions. Call the samples (vectors) \( r \) and \( \theta \) respectively. Obtain the two vectors \( y=r \sin (\theta) \) and \( x=r \cos (\theta) \) from the vectors \( r \) and \( \theta \). Plot \( y \) against \( x \) to visualize the points on a disk of unit radius. Do you see the problem? Comment. (f) [8 pts] Find the probability density function of the circumference \( (2 \pi R) \) and area \( \left(\pi R^{2}\right) \) for the density given in the main problem description - i.e., assuming \[ f_{R}(r)=\left\{\begin{array}{ll} \frac{2 \pi r}{\pi \rho^{2}} & 0 \leq r \leq \rho \\ 0 & \text { otherwise } \end{array}\right. \]