# Homework 5_SOLUTION

1. For problem 3 in Homework 4, ﬁnd the mean (expected value) of X. 2. Walpole 4.8. Copied here: Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.22, 0.36, 0.28, and 0.14, respectively, that she will be able to sell it for a proﬁt of \$250, sell it for a proﬁt of \$150, break even, or sell it for a loss of \$150. What is her expected proﬁt? 3. For problem 4 in Homework 4: (a) Find the mean of X. (b) Assume that this is a game in a casino: the payoﬀ is \$X and it costs \$2 to play the game each time. Compute the expected net gain/loss per game. 4. For problem 5 in Homework 4, ﬁnd the mean of X. 5. Small batch board manufacturers have a frustratingly low probability of successfully delivering a working board to you, and they charge you for the board regardless of if it works or not. Assume that the charge per manufacture is \$500, and the probability of it working is p. Assume that the number of times until you need to send the board for manufacture until a working board comes back, random variable K, has the geometric pmf (which happens to be a good assumption): fK(k) = P [K = k] = p(1 − p)k−1, k = 1,2,... 0, o.w. We need to budget for the cost of board manufacturing: What is the expected value of the money we will spend on the board manufacturing? Note: you might need that P∞ n=1 nxn−1 = 1 (1−x)2 . 6. (From Yates and Goodman, Probability and Stochastic Processes): The pdf of random variable Y is given by fY (y) = 3y2/2, −1 ≤ y ≤ 1 0, o.w. (a) Plot the pdf (by hand is ﬁne). (b) Find the expected value µY = E [Y ].XC