Let (x, y ) be chosen uniformly from inside the circle of radius one centered at the origin. let r denote the distance of the chosen point from the origin. determine the density of r. from there, determine the density of the random variable r2 = x2 + y2

Denote by the unit disk centered at the origin,

which has area . Then have joint PDF

Transform the pair of random variables to a new pair such that

In this new coordinate system, the support is

Using the method of transformations, the joint PDF of is

where is the Jacobian matrix for the transformation. We have

Then

Since the joint density doesn't depend on (it varies uniformly over the interval with probability ), it follows that and are independent of one another and

Then letting , by the method of transformations we have

step-by-step explanation: