課程名稱︰統計學
課程性質︰數學系大學部選修
課程教師︰邱政民
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2007/05/04
考試時限（分鐘）：60分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Statistics Quiz 3 (05/04/2007)
1. Suppose that X is a discrete random variable with P(X=1)=θ and P(X=2) =
1-θ. Three independent observations of X are made: x_1=1, x_2=2, x_3=2.
(a)(7%) Find the method of moment estimate of θ.
(b)(6%) What is the maximum likelihood estimate of θ.
(c)(7%) If Θ has a prior distribution that is uniform on [0,1], what is its
posterior distribution?
2.(20%) Suppose that a random sample of size 20 is taken from a normal distri-
bution with unknown mean and known variance equal to 1, and the sample mean
_
is found to be x = 10. A normal distribution was used as the prior for the
mean, and it was found that posterior mean was 15 and the posterior standard
deviation was 0.1. What were the mean and standard deviation of the prior?
-1
Hint: You may use the fact that the conjugate prior of N(μ,ξ ) for a
-1
random sample of size n is N(μ_0,ξ_prior) with posterior precision
ξ_post = nξ_0 + ξ_uprior, and posterior mean as the weighted sum of
sample mean and prior mean.
3. Let X_1, X_2,..., X_n be an i.i.d. sample from an exponential distribution
with the density function
1
f(x|τ) = --- exp(-x/τ) , 0<= x < ∞
τ
(a) (10%) What is the exact sampling distribution of the mle of τ?
(b) (10%) Use the central limit theorem to find a normal approximation to
the sampling distribution of the mle of τ.
(c) (5%) Find the form of an approximate confidence interval for τ.
(d) (5%) Find the form of an exact confidence interval for τ.
(e) (10%) Is there any other unbiased estimate with smaller variance?
Explain.
4. Let X_1, X_2,...,X_n be an i.i.d. sample from a Poisson distribution with
n
mean λ, and let T = Σ X_i.
i=1
(a) (7%) Show that the distribution of X_1, X_2,...,X_n given T is indepen-
dent of λ and conclude that T is sufficient for λ.
(b) (6%) Show that X_1 is not sufficient for λ.
(c) (7%) Use the factorization theorem to show that T is sufficient by
indentifying the factorized function of the theorem.
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