111. Let X be a continuous random variable with density function

 Calculate the value of p such that E(X) = 2.
(A) 1
(B) 2.5
(C) 3
(D) 5
(E) There is no such p.

112. The figure below shows the cumulative distribution function of a random variable, X. Calculate E(X). (A) 0.00 (B) 0.50 (C) 1.00 (D) 1.25 (E) 2.50

113. Two fair dice are rolled. Let X be the absolute value of the difference between the two numbers on the dice. Calculate the probability that X ＜ 3. (A) 2/9 (B) 1/3 (C) 4/9 (D) 5/9 (E) 2/3

114. An actuary analyzes a company’s annual personal auto claims, M, and annual commercial auto claims, N. The analysis reveals that Var(M) = 1600, Var(N) = 900, and the correlation between M and N is 0.64. Calculate Var(M + N). (A) 768 (B) 2500 (C) 3268 (D) 4036 (E) 4420

115. An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2. Calculate the expected claim payment made for an auto loss. (A)(B)(C)(D)(E)

116. A student takes a multiple-choice test with 40 questions. The probability that the student answers a given question correctly is 0.5, independent of all other questions. The probability that the student answers more than N questions correctly is greater than 0.10. The probability that the student answers more than N + 1 questions correctly is less than 0.10. Calculate N using a normal approximation with the continuity correction. (A) 23 (B) 25 (C) 32 (D) 33 (E) 35

117. In each of the months June, July, and August, the number of accidents occurring in that month is modeled by a Poisson random variable with mean 1. In each of the other 9 months of the year, the number of accidents occurring is modeled by a Poisson random variable with mean 0.5. Assume that these 12 random variables are mutually independent. Calculate the probability that exactly two accidents occur in July through November. (A) 0.084 (B) 0.185 (C) 0.251 (D) 0.257 (E) 0.271

118. An airport purchases an insurance policy to offset costs associated with excessive amounts of snowfall. For every full ten inches of snow in excess of 40 inches during the winter season, the insurer pays the airport 300 up to a policy maximum of 700. The following table shows the probability function for the random variable X of annual (winter season) snowfall, in inches, at the airport. Calculate the standard deviation of the amount paid under the policy. (A) 134 (B) 235 (C) 271 (D) 313 (E) 352

119. Damages to a car in a crash are modeled by a random variable with density function where c is a constant. A particular car is insured with a deductible of 2. This car was involved in a crash with resulting damages in excess of the deductible. Calculate the probability that the damages exceeded 10. (A) 0.12 (B) 0.16 (C) 0.20 (D) 0.26 (E) 0.78

120. Two fair dice, one red and one blue, are rolled. Let A be the event that the number rolled on the red die is odd. Let B be the event that the number rolled on the blue die is odd. Let C be the event that the sum of the numbers rolled on the two dice is odd. Determine which of the following is true. (A) A, B, and C are not mutually independent, but each pair is independent. (B) A, B, and C are mutually independent. (C) Exactly one pair of the three events is independent. (D) Exactly two of the three pairs are independent. (E) No pair of the three events is independent.

121. An urn contains four fair dice. Two have faces numbered 1, 2, 3, 4, 5, and 6； one has faces numbered 2, 2, 4, 4, 6, and 6； and one has all six faces numbered 6. One of the dice is randomly selected from the urn and rolled. The same die is rolled a second time. Calculate the probability that a 6 is rolled both times. (A) 0.174 (B) 0.250 (C) 0.292 (D) 0.380 (E) 0.417

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