課程名稱︰統計學導論
課程性質︰數學系選修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/6/22
考試時限（分鐘）：110
試題 :
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) =