Let V = X+ Y and W = X/Y. Show that the joint density of V and W is given by 1 fv,w(v, w) = 27 /w(w +1) e¯v/2 V, W > 0.

For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a)  Find the marginal density fX(x) for the total amount of time spent at the clinic. (b)  Find the conditional density for waiting time, given the total time. (c)  Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)