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）︰2017/6/22 考试时限（分钟）：110 试题 : 1.(15 pts.) Let X ,..., X be a sample from an Exponential(λ) distribution. 1 n (a) Find the likelihood ratio test for testing H : λ = λ versus 0 0 H : λ = λ , where λ > λ . Explain how to determine the critical A 1 1 0 point at significance level α. (b) Show that the test in part (a) is uniformly most powerful for testing H : λ = λ versus H : λ > λ . 0 0 A 0 2.(15 pts.) Let X ~ Binomial(n , p ), i = 1,..., m, be independent random i i i variables. (a) Derive a likelihood ratio test statistic for the hypothesis H : p = p = ... = p against the alternative that the p are not 0 1 2 m i all equal. (b) What is the large-sample distribution of the test statistic in part(a)? 3.(15 pts.) Let X ,..., X be a sample from a cdf F, and let F denote 1 n n the ecdf. (a) Determine the distribution of F (x) and find E[F (x)], Var[F (x)]. n n n (b) Find Cov[F (u),F (v)]. n n 4.(15 pts.) Suppose a nonnegative random variable X has hazard funciton h(t). (a) Find the hazard function of aX where a is a positive constant. (b) Find the hazard function h(t) when X is uniform(0,1). 1 (c) Find the density function of X when h(t) = -----, t > 0. 1+t 5.(15 pts.) Let X ,..., X be a sample from an Exponential(λ ) distribution 1 n 1 and Y ,..., Y be an indepedent sample from an Exponential(λ ) distribution. 1 n (a) Determine the expected rank sum of the X's. (b) Determine the variance of the rank sum of the X's. 2 6.(15 pts.) Let X ,..., X be i.i.d. N(μ , σ ), and let Y ,..., Y be i.i.d. 1 n X X 1 m 2 N(μ , σ ), where the two samples are independent. Y Y 2 2 2 2 (a) Find the distribution of s /s , where s and s are sample variance. X Y X Y 2 2 (b) Construct a level 100(1-α)% confidence interval for the ratio σ /σ . X Y (c) Test the hypothesis H : σ = σ versus H : σ ≠ σ 0 X Y 1 X Y at significance level α. 7.(30 pts.) Consider the one-way analysis of variance model Y = μ+α +ε , ij i ij i = 1,..., I, j = 1,..., J, where Y is the jth observation for the ith ij treatment level, μ is the overall mean, α is the effect of the ith i I 2 treatment with Σ α = 0, and ε 's are independent N(0, σ ) errors. i=1 i ij (a) Find the mle's of the parameters μ and α , i = 1,..., J. i (b) Find the expectation of the sum of squares within and between groups respectively. (c) Write down the One-Way Analysis of Variance table. (d) Test the null hypothesis H : α = α = ... = α = 0 at significant 0 1 2 I level α. --

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