I. A simple random sample of three items is selected from a shipment of 20 items of which four are defective. Let X be the number of defective items in the sample. (a) Find the PMF of X. (b) Find the mean value and variance of X. 2. Let X have PMF X 1 2 3 4 p(x) 0.4 0.3 0.1 0.2 (a) Calculate E(X ) and E(1/X).(b) In a win-win game, the player will win a monetary prize, but has to decide between the fixed price of $1000/E (X) and the random price of $1000/X, where the random variable X has the PMF given above. Which choice would you recommend the player make? 3. A customer entering an electronics store will buy a at screen TV with probability 0.3. Sixty percent of the cus- tomers buying a flat screen TV will spend $750.00 and 40% will spend $400.00. Let X denote the amount spent on flat screen TVs by two random customers entering the store. (a) Find the PMF of X. (Him. 5X 2 {0, 400, 750, 800, 1150, 1500} and P(X = 400) = 2 x 0.7 x 0.3 x 0.4.) (b) Find the mean value and variance of X. 4. A metal fabricating plant currently has five major pieces under contract, each with a deadline for comple- tion. Let X be the number of pieces completed by their deadlines. Suppose that X is a random variable with PMF p(x) given by X 0 1 2 3 4 5 p(x) 0.05 0.10 0.15 0.25 0.35 0.10 (a) Compute the expected value and variance of X. (b) For each piece completed by the deadline, the plant receives a bonus of $15,000. Find the expected value and variance of the total bonus amount