247. Random variables X and Y have joint distribution
  
 Let a be the value that minimizes the variance of X.
Calculate the variance of Y.
(A) 2/5
(B) 8/15
(C) 16/25
(D) 2/3
(E) 7/10

201. A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive for more than one of these three risk factors. For each risk factor, a patient tests positive with probability p, where 0 ＜ p ＜ 1. The outcomes of the tests across all patients and all risk factors are mutually independent. Determine an expression for the probability that exactly n patients are tested, where n is a positive integer less than 100. (A)(B)(C)(D)(E)

202. A representative of a market research firm contacts consumers by phone in order to conduct surveys. The specific consumer contacted by each phone call is randomly determined. The probability that a phone call produces a completed survey is 0.25. Calculate the probability that more than three phone calls are required to produce one completed survey. (A) 0.32 (B) 0.42 (C) 0.44 (D) 0.56 (E) 0.58

203. Four distinct integers are chosen randomly and without replacement from the first twelve positive integers. X is the random variable representing the second smallest of the four selected integers, and p is the probability function of X. Determine p(x) for x = 2,3,…,10. (A)(B)(C)(D)(E)

204. Losses due to burglary are exponentially distributed with mean 100. The probability that a loss is between 40 and 50 equals the probability that a loss is between 60 and r, with r ＞ 60. Calculate r. (A) 68.26 (B) 70.00 (C) 70.51 (D) 72.36 (E) 75.00

205. The time until the next car accident for a particular driver is exponentially distributed with a mean of 200 days. Calculate the probability that the driver has no accidents in the next 365 days, but then has at least one accident in the 365-day period that follows this initial 365-day period. (A) 0.026 (B) 0.135 (C) 0.161 (D) 0.704 (E) 0.839

206. The annual profit of a life insurance company is normally distributed. The probability that the annual profit does not exceed 2000 is 0.7642. The probability that the annual profit does not exceed 3000 is 0.9066. Calculate the probability that the annual profit does not exceed 1000. (A) 0.1424 (B) 0.3022 (C) 0.5478 (D) 0.6218 (E) 0.7257

207. Individuals purchase both collision and liability insurance on their automobiles. The value of the insured’s automobile is V. Assume the loss L on an automobile claim is a random variable with cumulative distribution function  Calculate the probability that the loss on a randomly selected claim is greater than the value of the automobile. (A) 0.00 (B) 0.10 (C) 0.25 (D) 0.75 (E) 0.90

208. The lifetime of a machine part is exponentially distributed with a mean of five years. Calculate the mean lifetime of the part, given that it survives less than ten years. (A) 0.865 (B) 1.157 (C) 2.568 (D) 2.970 (E) 3.435

209. Let X be a random variable with density function Calculate P [X≤0.5|X≤1.0]. (A) 0.433 (B) 0.547 (C) 0.632 (D) 0.731 (E) 0.865

210. Events E and F are independent. P[E] = 0.84 and P[F] = 0.65. Calculate the probability that exactly one of the two events occurs. (A) 0.056 (B) 0.398 (C) 0.546 (D) 0.650 (E) 0.944

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