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Refer to Exercise 3. a. Calculate the covariance between X, = the number of customers in the express checkout and X- = the number of customers in the superexpress checkout. b. Calculate V(X, +X,). How does this compare to V(X,) + V(X2)? Reference Exercise 3. A certain market has both an express checkout line and a superexpress checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X1 and X2 is as given in the accompanying table. X a. What is P(X, = 1, X, = 1), that is, the probability that there is exactly one customer in each line? b. What is P(X, = X,), that is, the probability that the numbers of customers in the two lines are identical? c. Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X, and X2, and calculate the probability of this event. d. What is the probability that the total number of customers in the two lines is exactly four? At least four?A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly selected couple dining at this restaurant, let X = the cost of the man's dinner and Y = the cost of the woman's dinner. The joint pmf of X and Y is given in the following table: p(x, y) 12 15 20 12 05 05 10 15 05 10 .35 20 0 20 10 a. Compute the marginal pmf's of X and Y. b. What is the probability that the man's and the woman's dinner cost at most $15 each? c. Are X and Y independent? Justify your answer. d. What is the expected total cost of the dinner for the two people? e. Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message "You will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen." How much would the restaurant expect to refund?a. Let X, have a chi-squared distribution with parameter v1 (see Section 4.4), and let X, be independent of X, and have a chi-squared distribution with parameter v2- Use the technique of Example 5.21 to show that X, + X, has a chi-squared distribution with parameter V1 _ V2- b. In Exercise 71 of Chapter 4, you were asked to show that if Z is a standard normal rv, then Zz has a chi-squared distribution with v = 1. Let Z,, Z 2. .. .. Z, ben independent standard normal ry's. What is the distribution of X? Justify your answer. c. Let X1, . . . . X, be a random sample from a normal distribution with mean m and variance of. What is the distribution of the sum 1 ? Justify your answer. Reference example 5.21 Service time for a certain type of bank transaction is a random variable having an exponential distribution with parameter I. Suppose X1 and X2 are service times for two different customers, assumed independent of each other. Consider the total service time To = X1 + X2 for the two customers, also a statistic. The cdf of 7, is, for f 2 0, X2 Reference exercise 7 in chapter 4 The time X (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution with A = 25 and B =35. a. Determine the pdf of X and sketch the corresponding density curve. b. What is the probability that preparation time exceeds 33 min? c. What is the probability that preparation time is within 2 min of the mean time? [Hint. Identify u from the graph of f(x).] d. For any a such that 25 150? . In the context of this situation, describe type I and type II errors. Which type of error would you consider more serious? Explain