77. The profit for a new product is given by Z = 3X – Y – 5. X and Y are independent random variables with Var(X) = 1 and Var(Y) = 2.
Calculate Var(Z).
(A) 1
(B) 5
(C) 7
(D) 11
(E) 16

78. A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 10. The company will begin using the second generator immediately after the first one fails. Calculate the variance of the total time that the generators produce electricity. (A) 10 (B) 20 (C) 50 (D) 100 (E) 200

79. In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be mutually independent, exponentially distributed random variables with respective means1.0, 1.5, and 2.4. Calculate the probability that the maximum of these losses exceeds 3. (A) 0.002 (B) 0.050 (C) 0.159 (D) 0.287 (E) 0.414

80. Let X denote the size of a surgical claim and let Y denote the size of the associated hospital claim. An actuary is using a model in which E(X)=5, E(X2)=27.4,E(Y)=7,E(Y2)=51.4,Var(X+Y)=8. Let C1=X+Y denote the size of the combined claims before the application of a 20% surcharge on the hospital portion of the claim, and let C2 denote the size of the combined claims after the application of that surcharge. Calculate Cov(C1,C2) . (A) 8.80 (B) 9.60 (C) 9.76 (D) 11.52 (E) 12.32

81. Two life insurance policies, each with a death benefit of 10,000 and a one-time premium of 500, are sold to a married couple, one for each person. The policies will expire at the end of the tenth year. The probability that only the wife will survive at least ten years is0.025, the probability that only the husband will survive at least ten years is 0.01, and the probability that both of them will survive at least ten years is 0.96. Calculate the expected excess of premiums over claims, given that the husband survives at least ten years. (A) 350 (B) 385 (C) 397 (D) 870 (E) 897

82. A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease present and 0 for disease not present. Let X denote the disease state (0 or 1) of a patient, and let Y denote the outcome of the diagnostic test. The joint probability function of X and Y is given by: P[X = 0, Y = 0] = 0.800 P[X = 1, Y = 0] = 0.050 P[X = 0, Y = 1] = 0.025 P[X = 1, Y = 1] = 0.125 Calculate Var(Y|X =1) (A) 0.13 (B) 0.15 (C) 0.20 (D) 0.51 (E) 0.71

83. An actuary determines that the annual number of tornadoes in counties P and Q are jointly distributed as follows:   Calculate the conditional variance of the annual number of tornadoes in county Q, given that there are no tornadoes in county P. (A) 0.51 (B) 0.84 (C) 0.88 (D) 0.99 (E) 1.76

84. You are given the following information about N, the annual number of claims for a randomly selected insured:  Let S denote the total annual claim amount for an insured. When N = 1, S is exponentially distributed with mean 5. When N ＞ 1, S is exponentially distributed with mean 8.  Calculate P(4 ＜ S ＜ 8). (A) 0.04 (B) 0.08 (C) 0.12 (D) 0.24 (E) 0.25

85. Under an insurance policy, a maximum of five claims may be filed per year by a policyholder. Let p n( ) be the probability that a policyholder files n claims during a given year, where n = 0,1,2,3,4,5. An actuary makes the following observations: i) p(n)≥p(n+1) for n=0, 1, 2, 3, 4 .ii) The difference between p (n) and p (n+1) is the same for n = 0,1,2,3,4.iii) Exactly 40% of policyholders file fewer than two claims during a given year.Calculate the probability that a random policyholder will file more than three claims during a given year.(A) 0.14(B) 0.16(C) 0.27(D) 0.29(E) 0.33

86. The amounts of automobile losses reported to an insurance company are mutually independent, and each loss is uniformly distributed between 0 and 20,000. The company covers each such loss subject to a deductible of 5,000. Calculate the probability that the total payout on 200 reported losses is between 1,000,000 and 1,200,000. (A) 0.0803 (B) 0.1051 (C) 0.1799 (D) 0.8201 (E) 0.8575

87. An insurance agent offers his clients auto insurance, homeowners insurance and renters insurance. The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive. The profile of the agent’s clients is as follows:i) 17% of the clients have none of these three products.ii) 64% of the clients have auto insurance.iii) Twice as many of the clients have homeowners insurance as have renters insurance.iv) 35% of the clients have two of these three products.v) 11% of the clients have homeowners insurance, but not auto insurance.Calculate the percentage of the agent’s clients that have both auto and renters insurance.(A) 7%(B) 10%(C) 16%(D) 25%(E) 28%

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