课程名称︰机率导论 课程性质︰数学系大二必修 课程教师︰陈宏 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2015/04/23 考试时限（分钟）：120 试题 : 1. (40 points) An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder's loss, Y, follows a distribution with density function -2 ╭ y , for y > 1 f(y) = ╯ ╰ 0, otherwise (a) (25 points) What is the expected value of the benefit, X, paid under the insurance policy? (b) (20 points) How do you write X in terms of a function of Y and 10? 2. (45 points) A company prices its hurricane insurance under the following assumptions: (i) In any calendar year, there can be at most one hurricane. (ii) In any calendar year, the probability of a hurricane is 0.05. (iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year. Let N denote the number of hurricanes in a 20-year period. (a) (15 points) Give possible values of N can take. (b) (15 points) Give the probability mass function of N. (c) (15 points) Calculate the probability that there are fewer than 3 hurricanes. 3. (40 points) Let X and Y be continuous random variables with joint density function ╭ 24xy, for 0 < x < 1 and 0 < y < 1-x f (x, y) = ╯ X, Y ╰ 0, otherwise Calculate P(Y < X|X = 1/3). 4. (45 points) The joint density of (X, Y) is given by ╭ 3x, if 0 ≦ y ≦ x ≦ 1, f(x, y) = ╯ ╰ 0, otherwise (a) (30 points) Compute the conditional density of Y given X = x. (b) (15 points) Are X and Y independent? Justify your claim. 5. (30 points) Suppose X is uniform on (0, 1), Y is exponential with parameter 1, and X and Y are independent. Compute the PDF of X/Y. 6. (40 points) Suppose N is the number of flips of a fair coin until the first 2 head. Suppose Y is uniform on (0, N ). Suppose X is exponential with ＿ parameter √Y. Compute the expectation of X. 7. (40 points) A company agrees to accept the highest of four sealed bids on a property. The four bids, X , X , X , and X , are regarded as four 1 2 3 4 independent random variables with common cumulative distribution function 1 3 5 F(x) =─(1 + sinπx) for─ ≦ x ≦─. 2 2 2 Calculate the expected value of the accepted bid. 8. (40 points) An insurance policy pays a total medical benefit consisting of two parts for each claim. Let X represent the part of the benefit that is paid to the surgeon, and let Y represent the part that is paid to the hospital. The variance of X is 5,000, the variance of Y is 10,000, and the variance of the total benefit, X+Y, is 17,000. Due to increasing medical costs, the company that issues the policy decides to increase X by a flat amount of 100 per claim and to increase Y by 10% per claim. Calculate the variance of the total benefit after these revisions have been made. 9. (40 points) Assume a crime has been committed. It is known that the perpetrator has certain characteristics, which occur with a small frequency -8 8 p (say, 10 ) in a population of size n (say, 10 ). A person who matches these characteristics has been found at random (e.g., at a routine traffic stop or by airport security) and, since p is so small, charged with the crime. There is no other evidence. What should the defense be? (a) Let N be the number of people with given characteristics. Describe the distribution of random variable N. (b) Choose a person from among these N, label that person by C, the criminal. Then, choose at random another person, A, who is arrested. The question is whether C = A, that is, whether the arrested person is guilty. Please determine P(C = A|N ≧ 1). 10. (40 points) Let X and Y be independent random variables. Express Var(XY) in 2 2 terms of [E(X)] , [E(Y)] , Var(X), and Var(Y). -- 2 2 1 ψxavier13540 给定一个二次元(|R )上的开集 G，设 f: G →|R ∈ C 。考虑一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，则 ╰x(0) = x° t ω(x°) 只能是一定点、一周期轨道或连接有限个 critical point 的连通路径，不会像三 次元一样可能出现混沌(chaos)。此即为 ODE 动力系统中的 Poincaré–Bendixson 定理。 --

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