2. Let (, F, P) be a probability space. Let X, Y be two independent ran- dom variables such that 1 3 P(X = 0) P(X = 1) P(X = 2) P(X = 3) P(X = 4) 10 10 5' 3 P(Y = 0) P(Y = 1) 10 1 10 7 1 P(Y2) = P(Y = 3) , P(Y = 4) 10 20 20 (i) Find the probability generation function Gx(s) and Gy(s) of X and Y, respectively. [12 Marks] (ii) Using Gx(s) and Gy(s), find the mean and variance of X and Y. [12 Marks] (iii) Find the probability generating function Gx+y(s) of X + Y. [8 Marks]

4. Let (, F, P) be a probability space and let X and Y be random variables on it. Assume that the random variable X takes on value 0 and 1, and the distribution of X is given by Px (0)=0.3, Px(1) = 0.7. Further assume that the random variable Y takes on value 0 and 1, and the distribution of Y is given by Py (0)=0.8, Py(1) = 0.2. (i) Assume that the random variables X and Y are independent. Find the joint distribution of X and Y, Px,y. [5 Marks] (ii) Give an example of a possible joint distribution Px,y such that the random variables X, Y are not independent. [15 Marks] Hint. Denote Px,y (0, 0) = = a, Px,y (0, 1) = b, Px,y(1, 0) = = C, Px,y (1, 1) = = d. We must have (why?) a, b, c, d = [0, 1], a+b+c+d= 1, a+b= 0.3, a+c=0.8. Choose a, b, c, d satisfying the above conditions and such that a 0.24.

Q2) Suppose that X is a continuous random variable with probability distribution fx(x)= 0 ≤x≤4 Find the probability distribution of Y = (X - 2)².

B) Let X and Y be discrete random variables with joint probability function 2-y+1 9 for x = 1,2 and y = 1, 2 p(x, y) otherwise Calculate E(Y/X).