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 p. Let R and 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 O, respectively,
fR(r) = { 2πτ/πr², 0≤ r ≤p
{ 0 otherwise
fe(θ) = { 1/2π, 0 ≤ θ ≤ 2π,
{ 0 otherwise
Show that the PDFs for R and are both valid PDFs.