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）︰2015/06/23 考试时限（分钟）：110 试题 : 1. (40 points) Suppose that X is uniformly distributed on the interval [0, 1] and that, given X = x, Y is uniformly distributed on the interval [1-x, 1]. (a) (15 points) Determine the joint density f(x, y). (Be sure to specify the range.) (b) (10 points) Find the probability P(X ≧ 1/2, Y ≧ 1/2). (As usual, you can leave your answer in raw form, such as 1 - 1/e.) (c) (15 points) Find the conditional density, f (x|1/3), of X given Y = X|Y 1/3. Be sure to specify the range. 2. (40 points) Suppose X and X are independent random variables, each having 1 2 density -4 ╭ 3x , for 1 < x < ∞ f(x) = ╯ ╰ 0, otherwise (a) (20 points) Let Z = max(X , X ) denote the larger (maximum) of the two 1 2 random variables. Find the p.d.f (density) of Z. (b) (20 points) Let Y = X /X . For y ≧ 1, find P(Y ≧ y). 2 1 3. (40 points) Suppose that X and Y are independent random variables with moment -2 -3 generating function M (t) = (1 - 2t) and M (t) = (1 - 2t) . Find Var(X + X Y Y). 4. (40 points) Show that if N (t) and N (t) are independent Poisson processes 1 2 with rate λ and λ , then N(t) = N (t) + N (t) is a Poisson process with 1 2 1 2 rate λ + λ . (Hint: There are two possible approaches to this problem. The 1 2 first approach is to find P[N(t) = n], the probability mass function of the Poisson process, and the second approach is to check on the requirements of a Poisson process for N(t).) 5. (40 points) Suppose that in a certain very volatile industry, failures among companies occur at the rate of 2.5 per year. Assuming the occurrence of such business failures to be a Poisson process. (a) (20 points) Find the probability that 4 or more failures will occur next year. (b) (20 points) Find the probability that the next two failures will occur within three months of each other. 6. (45 points) A non-negative random variable X has mean 100 and variance 100. (a) (15 points) What does Markov's inequality say about P(X ≧ 400)? (b) (15 points) What does Chebyshev's inequality say about P(X ≧ 400)? (c) (15 points) Let S be the sum of n independent variables, each with the n same distribution as X. Find a sequence x so that P(S /n > 100 + x ) n n n converges to 1/4 as n → ∞. 7. (60 points) (coupon collector problem) Sample from n cards, with replacement, indefinitely, and let N be the number of cards you need to get each of n different cards are represented. (a) (30 points) Let N be the number of coupons needed to get i different i coupons after having i-1 different ones. Determine E(N ) and Var(N ). i i (b) (30 points) Write N in terms of N , ..., N . Find a sequence a such that 1 n n as n → ∞, N/a converges to 1 in probability. n 8. (55 points) (Random Incidence Paradox) Consider observing successive buses on an urban bus line as they arrive at and depart from the same bus stop. Suppose that 8 of 9 waits between successive buses are 1 minute (short gap), but that 1 in 9 each intervals equals 10 minutes (long gap). (a) (10 points) What is the probability that this passenger faces a long wait of 10 minutes for the next bus? Suppose that a second would-be passenger arrives at the same bus stop, but at a time that is random with respect to bus arrivals. (b) (15 points) It states that is the probability that this passenger faces a long wait of 10 minutes for the next bus is calculated as follows 1 ─×10 9 5 ─────── =─. 1 8 9 ─×10 + ─×1 9 9 Please explain why the above calculation is correct. (c) (10 points) Consider the first passenger who just missed the bus. Calculate his or her expected waiting time. (d) (20 points) Determine the average waiting time of the second passenger. /* 第7题是说，假设你现在在收集某个东西(e.g. 以前7-11的小磁铁)，总共有n种，可是你在 拆开包装以前不知道里面是哪一种。而你的目标，就是把每个种类的东西都收集到。 */ -- 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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