课程名称︰统计学导论 课程性质：数学系选修 课程教师︰郑明燕 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2017/4/27 考试时限（分钟）：110 试题 : β 1. Let X be a random variable with cdf F(x) = 1 - exp(-αx ) for x ≧ 0 and F(x) = 0 for x < 0, where α > 0 and β > 0. (a) (3 pts.) Show that F is a cdf. (b) (3 pts.) Find the density of X. (c) (4 pts.) Find the probabilities P(1 ≦ X < 2) and P(X = 3). 2. Let X and Y have the joint density 6 2 f(x,y) = ---(x+y) , 0 ≦ x ≦ 1, 0 ≦ y ≦ 1. 7 (a) (8 pts.) Find the two conditional densities. (b) (12 pts.) Find E(Y|X = x) and Var(Y|X = x), 0 ≦ x ≦ 1, and vertify that E[Var(Y|X)] < Var(Y). 3. A six-sided die is rolled 240 times. (a) (6 pts.) Approximate the probability that the face showing a three turns up between 30 and 60 times. (b) (6 pts.) Approximate the probability that the sum of the face values is greater than 400. 4. (8 pts.) If X and Y are independent exponential random variables with λ = 1, find the distribution of X/Y. 5. (10 pts.) Let X , ..., X be a simple random sample without replacement 1 n _ from a finite population {x , ..., x }. Is X an unbiased estimator of 1 n 2 μ , where μ is the population mean? If not, what is the bias? 6. Let X , ..., X be i.i.d. uniform on [0,θ]. 1 n (a) (6 pts.) Find the method of moments estimate of θ and its mean and variance. (b) (8 pts.) Find the MLE of θ and its mean and variance. (c) (6 pts.) Compare the bias, variance and mean squared error of the MLE to those of the method of moments estimate. 7. Let X , ..., X be i.i.d. random variables with the density function 1 n θ f(x|θ) = (θ+1)x , 0 ≦ x ≦ 1. (a) (5 pts.) Find the MLE of θ. (b) (5 pts.) Find the asymptotic variance of the MLE. (c) (5 pts.) Find a sufficient statistic for θ. (d) (5 pts.) Is the MLE a UMVUE? Why or why not? --

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