a) Let X and Y have the joint density function given by Find the: f(x, y) =kxy, 0 x 1,0 y 1; = 0 elsewhere i) value of k ii) p(X 0.5, Y 0.75) iii) marginal density functions for X and Y iv)p(X 0.5/Y 0.75) v) conditional density function of X given Y = y (12 Marks) b) When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. Let X denote the number of headlights that need adjustment, and let Y denote the number of defective tires. i) If X and Y are independent with px (0) = 0.5, px (1) = 0.3, px (2)= = 0.2 and py (0) = 0.6, py (1) = 0.1, py (2) = py (3) = 0.5, py (4) = 0.2; display the joint probability mass function of (X,Y) in a joint probability table. ii) Compute p(X 1 and Y < 1) (5 Marks) (1 Mark) iii) Determine the probability of no violations? (1 Mark)