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課程名稱︰統計學∕數理統計
課程性質︰選修 (非數學系需有統計基礎)
課程教師︰鄭少為 教授
開課系所︰數學系
考試時間︰2005/05/05 18:00-21:00
試題：
(可攜帶textbook, Lecture Notes/Lecture Notes with Hand-Written Notices,
homework solution, 計算機或電子字典)
(所有 Hint 的結果都可以直接引用而不需證明)
1. (18pts, 2pts for each)
For the following statements, please answer true or false. If false,
please explain why.
c c
(a) For two events A and B, if P(A|B)＝P(A), then P(A |B)＝P(A ).
(b) For a random variable X with cdf F, P(a≦X≦b)＝F(b)-F(a) always.
(c) A random variable X with possible values 0 and 1 will have
E(X^k)＝E(X), for k＝2,3,4,...
(d) If the correlation between two random variables is 0, then there is
no association between the two random variable.
2
(e) χ_2 is an Exponential distribution E(1/2).
[Hint. Homework 4, Ch6, #8]
(f) Let X_1,...,X_n be i.i.d. from a distribution with finite variance.
_
As the sample size n increases, the standard error of X increases.
(g) A point estimate, i.e. an estimated value, of a parameter θ is a
random variable with an associated probability distribution, called
sampling distribution.
(h) The maximum likelihood principle finds the most plausible model based
on the observed data.
(i) Because sufficient statistics contain all information about the
unknown parameters, we can always throw away all original data and
keep only the sufficient statistics.
2. Let X be a random variable with Exponential distribution E(1). Define
Y＝[X]＋1, where [X] is the largest integer not exceeding X. Then, Y is
a discrete random variable.
(a) (4pts) Compute the probability mass function of Y.
(b) (2pts) Can you identify the distribution of Y as something you have
seen in lecture before?
(c) (2pts) What is E(Y)?
3. Let X_1, X_2 be i.i.d. from Normal distribution N(0,1). Define
_ _
W_1＝√3 X_1＋X_2 and W_2＝X_1－√3 X_2.
(a) (2pts) Write down the joint pdf of (X_1, X_2).
(b) (4pts) Compute the joint pdf of (W_1, W_2).
(c) (2pts) Find out the marginal distributions of W_1, W_2.
(d) (2pts) Examine whether or not W_1 and W_2 are independent.
2 2
(e) (2pts) What is the probability that W_1＋W_2＜8 ?
[Hint. problem 1(e)]
4. Let Y_1,...,Y_n be i.i.d. from the Uniform distribution U(θ,θ+1).
︿
(a) (4pts) Find the method of moments estimator for θ (denoted as θ_1)
︿
and show that θ_1 is an unbiased estimator of θ.
︿
(b) (2pts) Find the standard error of θ_1.
︿
(c) (6pts) Let θ_2＝ Y_(n)－(n/(n+1)), where Y_(n)＝max{Y_1,...,Y_n}.
︿
Show θ_2 is an unbiased estimator of θ.
[Hint. (i) Let T_i＝Y_i－θ, then T_i has a Uniform distribition
U(0,1).
(ii) Let T_(n)＝max{T_1,...,T_n}, then T_(n)＝Y_(n)－θ.
Therefore E(T_(n))＝E(Y_(n)－θ) and Var(T_(n))＝Var(Y_(n)).]
︿
(d) (4pts) Find the standard error of θ_2.
︿ ︿
(e) (2pts) Find the efficiency of θ_1 relative to θ_2, denoted as
︿ ︿
eff_n(θ_1,θ_2), and find their asymptotic relative efficiency.
(f) (4pts) When the sample size n is greater than 7, explain which of
︿ ︿
θ_1 and θ_2 is better in terms of mean square error.
︿ ︿
[Hint. eff_n(θ_1,θ_2)＜1, when n＞7.]
5. Let Y_1,...,Y_n be i.i.d. from pdf:
f(y|θ)＝┌ (1/θ)‧ry^(r－1)‧e^(－(y^r)/θ), y＞0
└ 0 , otherwise
where r is a known positive constant and θ＞0.
(a) (4pts) Use the factorization theorem to find a sufficient statistic.
(b) (4pts) Find the MLE of θ.
r
(c) (4pts) Show that Y_1 has an Exponential distribution E(1/θ).
(d) (5pts) Calculate Fisher information and from which find the asymptotic
variance of the MLE. [Hint. use (c).]
6. Let Y_1,...,Y_n be i.i.d. from pdf:
f(y|θ)＝┌ θ‧y^(θ－1), 0＜y＜1
└ 0 , otherwise
where θ＞0.
(a) (4pts) Show that the pdfs form a one-parameter exponential family and
find a sufficient and complete statistic for θ.
(b) (4pts) Let W_i＝－log(Y_i). Show that W_i has an Exponential distri-
bution E(θ).
n 2
(c) (5pts) Show that 2θΣ W_i has a χ_2n distribution and show that
i=1
n
E(1/(2θΣ W_i))＝ 1/(2(n－1)) (1)
i=1
[Hint. (i) If X～E(λ), then aX～E(λ/a), where a is a constant.
(ii) problem 1(e).
2 2
(iii) If U～χ_n, then E(1/U)＝1/(n－2) and E(1/U )＝1/((n－2)(n－4)).]
(d) (2pts) Use (1) to find an unbiased estimator of θ.
(e) (4pts) Find the Cramer-Rao lower bound for all unbiased estimators of
θ.
(f) (4pts) Examine whether the variance of the unbiased estimator in (d)
achieves the Cramer-Rao lower bound. [Hint. use hint (iii) in (c).]
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