The StatDem party contests elections in two countries, Statland and Statopia, the proportion of support the party has in each country being modelled by the random variables X and Y respectively. It is thought that the joint density function for the two variables is of the form f(x, y) = (k(x + 2y) for 0 < x < 1,0 < y < 1 O elsewhere for some constant k. Give answers to the following to 2 decimal places. Part a) Find the value of k. Part b) Find the probability that the StatDem party has the support of more than half the population in each country. Part c) Find the marginal density of the support in Statland evaluated at the point X=1/2. Part d) Find the marginal density of the support in Statopia evaluated at the point Y=1/2.

Let X and Y be two random variables with joint density function f(x,y) = (3 − x + 2y) / 60, for 1 < x < 3, 0 < y < 5. Is P(X > 2, Y < 3) equal to P(X > 2) × P(Y < 3)?

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.)

Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise  c=1/2 P(X < 1), Determine whether X and Y are independent