课程名称︰统计导论 课程性质︰数学系选修 课程教师︰江金仓 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2016.12.7 考试时限（分钟）：50 试题 : Introduction to Statistics (Test3) 1.(10%) Are two independent events mutually exclusive? Explain your answer. 2. In the AIDS study, let D and T denote the events of HIV+ patients and patients who are diagnosed as HIV+ patients, respectively. Suppose that the sensitivity P(T|D) and the specificity P(T'|D') are known. (2a)(10%) Compute the odds ratio of D versus D'. (2b)(5%) What additonal condition is required to compute the positive predictivity P(D|T) and the negative predictivity P(D'|T')? 3.(10%) Suppose that P(D)～0 in a case-control study. Show that the relative risk of E versus of E' can approximated by the odds ratio of E versus E'. 4.(7%)(8%) An experiment consists of a sequence of independent coin tosses. Let X denote the number of heads occurring within n tosses and Y be the number of tails occurring before the rth head. Write the probability density function of X and Y. 5. Let T be the waiting time until an event occurs of the Poisson process {N_t:t>0} with P(N_t=m)={(λt)^m exp(-λt)/(m!)}1_{0,1,...,n,...}(m). (5a)(8%) Derive the density function of T. (5b)(7%) Show that T has a memoryless property. 6.(10%) Let X_1,...,X_n be a random sample from a Bernoulli(p), 0<p<1. Assume that the sample size is large enough. Find an approximated probability of P(Σ_{i=1}^{n} X_i<x) with continuity correction, wherex=1,2,...,n. 7.(5%)(10%) Let X_1,X_2,...,X_n be a random sample from a finite population {x^(1),...,x^(N)}. Find an unbiased estimator of mean μ=Σ_{i=1}^{N} x^(i)/N and variance σ^2=Σ_{i=1}^{N} (x^(i)-μ)^2/N. 8.(4%)(6%) Let X and Y be random variables with E[X]=μ_x, E[Y]=μ_y, Var(X)= σ_x^2, Var(Y)=σ_y^2, and Cov(X,Y)=σ_xy^2. What are the mean and variance of aX-bY for any constants a and b? --

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