课程名称︰机率导论 课程性质︰系必修 课程教师︰张志中 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰100/4/13 考试时限（分钟）：120 是否需发放奖励金：Y （如未明确表示，则不予发放） 试题 : 1.(20%) Let S = {1,2,...,n} and 2^S represent the collection of all the subsets of S. The ecperiment of randomly choosing an element of 2^S is performed in such a way that each element can be chosen equally likely. This experiment is carried out twice independently. Denote the outcome of the j-th experiment, j = 1,2, by X_j 属於 2^S, i.e. X_j is the (random) subset of S chosen at the j-th experiment. (a) Construct a probability model to describe this experiment, which includes constructing a sample space Ω, a probability measure Ρ, and two independent random varables X_1 and X_2. You must verify that X_1 and X_2 are indeed independent. (b) Evaluste Ρ(X_1 属於 X_2) and Ρ(X_1∩X_2 = ψ). 2.(The Borel-Cantelli lemma)(24%) Given a probability model (Ω,F,Ρ) and a ∞ sequence of events {H_n} , prove the ollowing two statements. n=1 ∞ (a) If ΣΡ(H_n) ﹤∞, then Ρ(limsup H_n) = 0. n=1 n→∞ ∞ (b) If H_1, H_2,... are independent and ΣΡ(H_n) = ∞, n=1 then Ρ(limsup H_n) = 1. n→∞ (c) In the probability model of tossing a fair coin indinitely many times, let H_n be the event that the n-th tossint is a head. Describe the event limsup H_n verbally. n→∞ 3.(16%) Answer True or False to the following two statements and then explain or prove your answers. (a) Let Y be a discrete random variable with probability mass function p (χ_k) = Ρ(Y = χ_k) = p_k, k = 1,2,3,... Y If the infinite sum Σ(χ_k)(p_k) converges, then Ε[Y] exits. (b) Let Z be a continuous random variable with probability density function f. M If lim ∫ zf(z)dz exists, then Ε[Z] exists. M→∞ -M 4.(30%) We start with a stick of unit length. We break it at a point which is chosen according to a uniform distribution over [0,1]. For the piece, of length Y, that contains the left end of the stick, we then repeat the same process. After the second breaking, let X be the length of the resulting piece that contains the left end. Now we gave three pieces of stick with lengths X, Y-X, and 1-Y, respectively. (a) Find the joint probability density function Y and X. (b) Evaluate Ε[X]. (c) fing the probability that the three pieces can form a triangle. 5.(10%) A game is played as follows: A random number X is chosen uniformly from [0,1]. Then a sequene Y1, Y2,... of random numbers is chosen independently and uniformly from [0,1]. The game ends the first time that Yj > X. If j=1, you lose c dollars. If j≧2, you receive √j dollars. (a) Find c so that the game is fair theoretically. (b) With th c found above, is it really a fair game? Why or why not? 6.(Bonus problem)(10%) There are two envelope and two numbers, say a and b. Here a and b, a ﹤b, are tqo fixed real numbers. Each envelope contains one number. To win the game, one need to choose the envelope containing the larger correctly. Moreover, one is allowed to choose an envelope at random and look at the number in it first. Then the opportunity to swop is offered before making the final decision. Either develop a strategy to win the game with probability grater than a half, or show that such strategy does not exist. --

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