課程名稱︰統計學
課程性質︰數學系選修、系統生資學程選修、神經認知學程必修
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︰2012年06月18日
考試時限：15:30 - 18:20
是否需發放獎勵金：是
試題 :
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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