57. The warranty on a machine specifies that it will be replaced at failure or age 4, whichever occurs first. The machine’s age at failure, X, has density function

 Let Y be the age of the machine at the time of replacement.
Calculate the variance of Y.
(A) 1.3
(B) 1.4
(C) 1.7
(D) 2.1
(E) 7.5

92. A mattress store sells only king, queen and twin-size mattresses. Sales records at the store indicate that the number of queen-size mattresses sold is one-fourth the number of king and twin-size mattresses combined. Records also indicate that three times as many king-size mattresses are sold as twin-size mattresses. Calculate the probability that the next mattress sold is either king or queen-size. (A) 0.12 (B) 0.15 (C) 0.80 (D) 0.85 (E) 0.95

93. The number of workplace injuries, N, occurring in a factory on any given day is Poisson distributed with mean λ. The parameter λ is a random variable that is determined by the level of activity in the factory, and is uniformly distributed on the interval [0, 3]. Calculate Var(N). (A) λ (B) 2 λ (C) 0.75 (D) 1.50 (E) 2.25

94. A fair die is rolled repeatedly. Let X be the number of rolls needed to obtain a 5 and Y the number of rolls needed to obtain a 6. Calculate E(X | Y = 2). (A) 5.0 (B) 5.2 (C) 6.0 (D) 6.6 (E) 6.8

95. A driver and a passenger are in a car accident. Each of them independently has probability 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0, 1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1. (A) 0.510 (B) 0.534 (C) 0.600 (D) 0.628 (E) 0.800

96. Each time a hurricane arrives, a new home has a 0.4 probability of experiencing damage. The occurrences of damage in different hurricanes are mutually independent. Calculate the mode of the number of hurricanes it takes for the home to experience damage from two hurricanes. (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

97. Thirty items are arranged in a 6-by-5 array as shown.  Calculate the number of ways to form a set of three distinct items such that no two of the selected items are in the same row or same column. (A) 200 (B) 760 (C) 1200 (D) 4560 (E) 7200

98. An auto insurance company is implementing a new bonus system. In each month, if a policyholder does not have an accident, he or she will receive a cash-back bonus of 5 from the insurer. Among the 1,000 policyholders of the auto insurance company, 400 are classified as lowrisk drivers and 600 are classified as high-risk drivers. In each month, the probability of zero accidents for high-risk drivers is 0.80 and the probability of zero accidents for low-risk drivers is 0.90. Calculate the expected bonus payment from the insurer to the 1000 policyholders in one year. (A) 48,000 (B) 50,400 (C) 51,000 (D) 54,000 (E) 60,000

99. The probability that a member of a certain class of homeowners with liability and property coverage will file a liability claim is 0.04, and the probability that a member of this class will file a property claim is 0.10. The probability that a member of this class will file a liability claim but not a property claim is 0.01. Calculate the probability that a randomly selected member of this class of homeowners will not file a claim of either type. (A) 0.850 (B) 0.860 (C) 0.864 (D) 0.870 (E) 0.890

100. A survey of 100 TV viewers revealed that over the last year:i) 34 watched CBS.ii) 15 watched NBC.iii) 10 watched ABC.iv) 7 watched CBS and NBC.v) 6 watched CBS and ABC.vi) 5 watched NBC and ABC.vii) 4 watched CBS, NBC, and ABC.viii) 18 watched HGTV, and of these, none watched CBS, NBC, or ABC.Calculate how many of the 100 TV viewers did not watch any of the four channels (CBS,NBC, ABC or HGTV).(A) 1(B) 37(C) 45(D) 55(E) 82

51. An auto insurance company insures an automobile worth 15,000 for one year under a policy with a 1,000 deductible. During the policy year there is a 0.04 chance of partial damage to the car and a 0.02 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function Calculate the expected claim payment. (A) 320 (B) 328 (C) 352 (D) 380 (E) 540

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