课程名称︰统计学 课程性质︰数学系选修、系统生资学程选修、神经认知学程必修 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︰2012年06月18日 考试时限：15:30 - 18:20 是否需发放奖励金：是 试题 : Statistic Final Examination June 18, 2012 1. Consider the simple sampling with replacement from a finite population. Let X1,...,Xn denote the sample. (a) (8 pts.) Show that n　　　_ s^2 = 1/(n-1) * Σ(Xi -X)^2 　　　　　　　　　　　　　　　　　　　　　 i=1 　　　is an unbiased estimate of the population variance σ^2. 　 (b) (8pts.) Is s an unbiased estimate of σ? 2. (16 pts.) True or false? 　 (a) A 95% confidence interval for the population mean μ contains μ with 　　　 probability 0.95. 　 (b) A 95% confidence interval contains 95% of the population. 　 (c) A 90% confidence interval for the average number of children per 　　　 household based on a sample is found to be (0.7,2.1). So we conclude 　　　 that 90% of the households have between 0.7 and 2.1 children. 　 (d) The significance level of a test is the probability that the null 　　　 hypothesis is true. 　 (e) The p-value of a test is the smallest significance level at which the 　　　 null hypothesis would be rejected. 　 (f) If a test is rejected at the significance level α, the probability 　　　 that the null hypothesis is true equals α. 　 (g) The likelihood function of a parameter is random. 　 (h) The maximum likelihood principle finds the most plausible model based on 　　　 the observed data. 3. Suppose that X1,...,Xn are i.i.d with density function f(x|θ) = e^[-(x-θ)], x≧θ 　 and f(x|θ)=0 otherwise. 　 (a) (8 pts.) Find the method of moments estimate of θ. 　 (b) (8 pts.) Find the mle of θ.(Hint: For what values of θ is the 　　　 likelihood positive?) 4. (10 pts.) Suppose that X1,...,Xn is a i.i.d. sample from Normal(μ,σ^2) and 　 μ has a Normal( μ0,(σ0)^2 ) prior distribution. Show that the posterior 　 distribution of μ is Normal( μ1,(σ1)^2 ), where 　　　　　　　　 　　　_ 　　　　　　　n[σ^(-2)]x + [(σ0)^(-2)]μ0 μ1 = --------------------------------- , n[σ^(-2)] + (σ0)^(-2) 1 　　 (σ1)^2 = -------------------------- . 　　　　　　　　n[σ^(-2)] + (σ0)^(-2) PLEASE TURN OVER 5. Let X1,...,Xn be i.i.d with Poisson distribution 　　　　　　　　　　　　　　　e^(-λ)*λ^x 　　　　　　　　P(X1 = x) = -------------- , x=0,1,2,... 　　　　　　　　　　　　　　　　　 x! 　 Let θ= e^(-λ) = P(X1=0). 　 (a) (8 pts.) Find a Uniformly Minimum Variance Unbiased Estmate of θ. 　 (b) (8 pts.) Compute the Cramer-Rao lower bound for θ. Is it attained by 　　　 some estimate? 6. Let Xi~Binomial(ni,pi), i=1,...,m, be independent random variables. 　 (a) (10 pts.) Derive a likelihood ratio test statistic for the hypothesis 　　　 H0: p1=p2=...=pm against the alternative that the pi are not all equal. 　 (b) (6 pts.) What is the large-sample distribution of the test statistic in 　　　 part (a)? 7. Let X1,...,Xn be a sample from a Poisson(λ) distribution. 　 (a) (10 pts.) Find the likelihood ratio test for testing H0:λ=λ0 versus 　　　 HA: λ=λ1, where λ1>λ0. Explain how to determine the critical point 　　　 at sighificance level α. 　 (b) (10 pts.) Show that the test in part(a) is uniformly most powerful for 　　　 testing H0:λ=λ0 versus HA:λ>λ0. --

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