MATH 2131 3.00 MS2 Assignment 3 Total marks = 55 Question 1: Let (X, Y ) have the joint pdf ( 8xy, 0 < x < y < 1, fX,Y (x, y) = 0, otherwise. (a) (4 marks). Find E(XY ). (b) (6 marks). Find Cov(X, Y ). (c) (4 marks). Find X,Y . Question 2: Let X1 , . . . , Xn P be a random sample on mean is Pnr.v. X whose n 2 2 2 and variance is . Let X = i=1 Xi /n and S = i=1 (Xi X) /(n 1). Show that = . (a) (2 marks). E(X) = 2 /n. (b) (2 marks). V ar(X) (c) (5 marks). E(S 2 ) = 2 . Question 3: (6 marks). A bowl contains 10 balls numbered 1, 2, . . . , 10. How many balls selected at random with replacement will be needed to ensure that the proportion of 2's drawn is between 0.05 and 0.15 with probability at least 0.9? Use Chebyshev's Inequality. Question 4: (6 marks). A radio show conducts a game according to the following rules. A number between 1 and 100 is selected at random, and the disc jockey announces what number is chosen. Call it n. The announcer then randomly selects a number between 1 and n and asks that the first caller guess the value of the second number chosen. Assuming that the caller's guess is equally likely to be any number between 1 and n, determine the expected value of the number guessed by the caller. 1 Question 5: (6 marks). If U = a + bX and V = c + dY with b, d 6= 0, show that |U,V | = |X,Y |. Question 6: (3 marks). If X and Y are independent r.v.'s, show that E(X|Y = y) = E(X). Question 7: Let (X,Y) have the joint pdf ( ey , 0 < x < y < , fX,Y (x, y) = 0, otherwise. (a) (7 marks). Find E(X|Y = y) and V ar(X|Y = y). (b) (4 marks). Find the pdf of r.v. E(X|Y ). 2