a) The joint probability distribution of two discrete random variables X and Y is given below. X = 1 X = 2 X = 3 X = 4 Y = 1 0.06 0.02 0.2 0.18 Y = 2 0.12 0 P 0.06 Y = 3 0 0.08 0.16 0 i. ii. Find the probability p. [2 marks] Find the marginal distributions fx (x) and fy (y) iii. Find the conditional distribution fx| x=2 [4 marks] iv. Find P(XY). b) The probability density function of a continuous random variable T is given by f(t) = {kt sin(t), 0t1 Find k. 0, otherwise [4 marks] [3 marks] [7 marks] c) The cumulative distribution function of a random variable is given by F(x) = Szexp (), 1- exp(--), x b Where b is a positive constant and the median of the distribution is . Find the upper quartile of this distribution. [7 marks]