232. The number of claims X on a health insurance policy is a random variable with
 E[X²]=61 and E[X-1]²=47.
Calculate the standard deviation of the number of claims.
(A) 2.18
(B) 2.40
(C) 7.31
(D) 7.50
(E) 7.81

233. Ten cards from a deck of playing cards are in a box: two diamonds, three spades, and five hearts. Two cards are randomly selected without replacement. Calculate the variance of the number of diamonds selected, given that no spade is selected. (A) 0.24 (B) 0.28 (C) 0.32 (D) 0.34 (E) 0.41

237. A car and a bus arrive at a railroad crossing at times independently and uniformly distributed between 7:15 and 7:30. A train arrives at the crossing at 7:20 and halts traffic at the crossing for five minutes. Calculate the probability that the waiting time of the car or the bus at the crossing exceeds three minutes. (A) 0.25 (B) 0.27 (C) 0.36 (D) 0.40 (E) 0.56

238. Skateboarders A and B practice one difficult stunt until becoming injured while attempting the stunt. On each attempt, the probability of becoming injured is p, independent of the outcomes of all previous attempts. Let F(x, y) represent the probability that skateboarders A and B make no more than x and y attempts, respectively, where x and y are positive integers. It is given that F(2, 2) = 0.0441. Calculate F(1, 5). (A) 0.0093 (B) 0.0216 (C) 0.0495 (D) 0.0551 (E) 0.1112

239. The number of minor surgeries, X, and the number of major surgeries, Y, for a policyholder, this decade, has joint cumulative distribution functionfor nonnegative integers x and y. Calculate the probability that the policyholder experiences exactly three minor surgeries and exactly three major surgeries this decade. (A) 0.00004 (B) 0.00040 (C) 0.03244 (D) 0.06800 (E) 0.12440

240. A company provides a death benefit of 50,000 for each of its 1000 employees. There is a 1.4% chance that any one employee will die next year, independent of all other employees. The company establishes a fund such that the probability is at least 0.99 that the fund will cover next year’s death benefits. Calculate, using the Central Limit Theorem, the smallest amount of money, rounded to the nearest 50 thousand, that the company must put into the fund. (A) 750,000 (B) 850,000 (C) 1,050,000 (D) 1,150,000 (E) 1,400,000

241. An investor invests 100 dollars in a stock. Each month, the investment has probability 0.5 of increasing by 1.10 dollars and probability 0.5 of decreasing by 0.90 dollars. The changes in price in different months are mutually independent. Calculate the probability that the investment has a value greater than 91 dollars at the end of month 100. (A) 0.63 (B) 0.75 (C) 0.82 (D) 0.94 (E) 0.97

242. Let X denote the loss amount sustained by an insurance company’s policyholder in an auto collision. Let Z denote the portion of X that the insurance company will have to pay. An actuary determines that X and Z are independent with respective density and probability functions  Calculate the variance of the insurance company’s claim payment ZX. (A) 13.0 (B) 15.8 (C) 28.8 (D) 35.2 (E) 44.6

243. A couple takes out a medical insurance policy that reimburses them for days of work missed due to illness. Let X and Y denote the number of days missed during a given month by the wife and husband, respectively. The policy pays a monthly benefit of 50 times the maximum of X and Y, subject to a benefit limit of 100. X and Y are independent, each with a discrete uniform distribution on the set {0,1,2,3,4}. Calculate the expected monthly benefit for missed days of work that is paid to the couple. (A) 70 (B) 90 (C) 92 (D) 95 (E) 140

244. The table below shows the joint probability function of a sailor’s number of boating accidents and number of hospitalizations from these accidents this year.   Calculate the sailor’s expected number of hospitalizations from boating accidents this year. (A) 0.085 (B) 0.099 (C) 0.410 (D) 1.000 (E) 1.500

245. On Main Street, a driver’s speed just before an accident is uniformly distributed on [5, 20]. Given the speed, the resulting loss from the accident is exponentially distributed with mean equal to three times the speed. Calculate the variance of a loss due to an accident on Main Street. (A) 525 (B) 1463 (C) 1575 (D) 1632 (E) 1744

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