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）︰2015/06/09 考试时限（分钟）：60 试题 : 2 1. (25 points) Let X be a normal random variable with mean 0 and variance σ . Let S be a random variable which is independent of X and such that P(S = 1) = P(S = -1) = 1/2. (a) (15 points) Show that SX is a normal random variable and determine its mean and variance. (You can use moment generating function to show it.) (b) (10 points) Show that X and SX are uncorrelated, but X and SX are not independent. 2. (25 points) Let N, X , X be three independent random variables. Let Y = 1 2 N Σ X where N is a random variable taking values 1 and 2 with probability 1/2 i=1 i 2 and 1/2 while E(X ) = μ and Var(X ) = σ . Determine E(Y) and Var(Y). i i 3. (25 points) A basketball team scores baskets according to a Poisson process with rate λ = 2 baskets per minute. (a) (10 points) What is the expected amount of time until the team scores its first basket? (b) (5 points) Given that at the five minute mark of the game the team has scored exactly one basket, what is the probability that the team scored the basket in the first minute? (c) (10 points) What is the probability that the team scores exactly three baskets in the first five minute of the game? (Computation of the numerical value is not necessary.) 4. (25 points) Show that a Poisson process is continuous in probability by arguing P(N - N > η) = P(N - N ≧ 1) < ε t+h t t+h t Hint: Associate the distribution of N - N and the distribution of N . t+h t h 5. (25 points) Distribute n balls independently at random into n boxes. Let N n be the number of empty boxes. Denote N by the sum of I , ..., I , where I = n 1 n i 1 if the i-th box is empty and 0, otherwise. (a) (4 points) Write N in terms of I , ..., I . n 1 n (b) (6 points) Derive E(I ) and Var(I ). 1 1 (c) (10 points) Derive E(I I ) and Cov(I , I ). 1 2 1 2 -1 (d) (5 points) Show that N /n converges to e in probability as n goes to n infinity. /* 1. 我修改了第4题的题叙，让题意比较清楚。 2. 5(d)原本没有"in probability"，这是考试中老师加上去的。 */ -- 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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