3. [16 marks] A pair of random variables (X, Y) has a joint probability distribution in which Y has marginal probability density function fr(y) = My (1 - 4), yE ( 0, 1 ) 10, else and the conditional probability density function of X given {Y = w} is uniform on the interval (0, y). (a) Write down the joint probability density function of (X, Y), clearly specifying the support of the distribution. [2 marks] (b) Determine the marginal probability density function of X. [3 marks]Prob. 1. Let X and Y be statistically independent random variables with uniform distribution on [-2, 2] and [-1, I]. respectively. (a) Sketch the probability density functions, Jy (x) and f(y)- (b) Let Z - X - Y. Sketch the probability density function, f (=). (c) Obtain P[Z 21]. (d) Obtain E[Z]. (c) Obtain of- Prob. 2. Any point (X, Y) in the shaded area below is selected equally likely. (a) Determine the joint probability density function fry (x.y). Clearly state the domain of the joint density function. (b) Sketch the probability density function /, (x). (c) Sketch the probability density function f,(y). (d) Are X and Y independent? (e) Obtain P (-0.5