# Homework 7

1. Walpole 3.52 plus part (e) below. Problem from the textbook is copied here: A coin is tossed twice. Let Z denote the number of heads on the ﬁrst toss, and let W be the total number of heads on the two tosses. If the coin is unbalanced and a head has a 40% chance of occurring, ﬁnd(a) The joint pmf of W and Z. (b) The marginal pmf of W. (c) The marginal pmf of Z. (d) The probability that at least one head occurs (in the two tosses). Additionally, (e) Are Z and W independent? 2. Consider the joint discrete random variables K and L. You are given that the joint pmf is given as: fK,L(k,l) = akl, k ∈ {1,2,3},l ∈ {1,2,3} 0, o.w. for some constant a. (a) Plot the joint pmf or create a table listing its values. (By hand is ﬁne.) (b) Find the value of a that makes fK,L(k,l) a valid joint pmf. (c) Find the marginal pmfs fK(k) and fL(l). (d) Find the conditional pmf fL|K(l|k = 1). (e) Are K and L independent? (f) Find the mean and covariance of K and L. (g) Find the correlation coeﬃcient, ρKL. 3. (a) Walpole 3.42. Copied here: Let X and Y denote the lengths in life, in years, of two components in an electrical system. If the joint pdf of these variables is: fX,Y (x,y) = e−(x+y), x 0,y 0 0, o.w. ﬁnd P [0 < X < 1|Y = 2]. (b) For the same joint pdf, are X and Y independent? 4. Walpole 3.45. Copied here: Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic model that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint pdf, fX,Y (x,y) = 1/y, 0 < x < y < 1 0, o.w.   Find P [X + Y 0.5]. Neal’s note: 0 < x < y < 1 is three inequalities: x 0, y x, and y < 1. I would start by drawing on the x-y plane these three inequalities. You draw an inequality by a) drawing the equality, for example, x = 0, and then b) shading the side where the inequality is true. 5. Walpole 3.40 plus parts (d) and (e) below. An abbreviated version of the problem statement in the book is: Given a joint pdf for X and Y : fX,Y (x,y) = 2 3(x + 2y), 0 ≤ x ≤ 1,0 ≤ y ≤ 1 0, o.w. (a) Find the marginal pdf of X. (b) Find the marginal pdf of Y . (c) Find P [X < 0.5]. Additionally, (d) Find the covariance of X and Y . (e) Find the correlation coeﬃcient, ρXY .