→ senyek9527:囧 錢好少 果然全部都是英文的比較虧XD 04/26 11:26
課程名稱︰Mathematical Statistics II
課程性質︰數學系 選修 (學分數學系本系不承認)
課程教師︰陳宏
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰96.04.26
考試時限（分鐘）：110 min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Mathematical Statistics II
Midterm: Chapters 9 and 10
April 26th 8:10 to 10 AM
1.(50%) Let X1, ...., Xn be i.i.d. N(μ,σ). Let C = Σn j=1 (Xj - Xbar)^2
(a) For what constants c1(n) depending on n is c1(n)V an unbiased estimator
of σ^2?
(b) For what constants c2(n) depending on n is the mean-square error
Eσ^2[(c2(n)V-σ^2)^2] mininmized for all σ>0?
Hints: V/σ^2 has a chi-square(n-1) distribution. A chi-square(d)
distribution is Γ(α,λ) with α = d/2 and λ = 1/2. (The density function
is (λ^α)[x^(α-1)]e^(-λx)/Γ(α).) If Y has Γ(α,λ) distribution then
EY = α/λ and Y has variance α/(λ^2).
2.(70%) Suppose that X1, X2, ..., Xn1, Y1, Y2,..., Yn2,and W1, W2,..., Wn3 are
independent random samples from normal distributions with respective unknown
means μ1,μ2,and μ3 and variances σ1^2, σ2^2, σ3^2. Suppose that we use
θcap = a1Xbar + a2Ybar + a3Wbar to estimate θ = a1μ1 + a2μ2 + a3μ3.
Let S1^2, S2^2, and S3^2 denote the respective sample variances of X1,X2,...
Xn1, Y1, Y2,..., Yn2, and W1, W2,..., Wn3.
(a) What is the distribution of the estimator θcap?
(b) What is the distribution of (n1-1)S1^2 + (n2-1)S2^2 + (n3-1)S3^2 when
σ1^2 = σ2^2 = σ3^2 = σ^2?
(c) What is the approximated distribution of
θcap - θ
T = -----------------------------------
(S1^2/n1 + S2^2/n2 + S3^2/n3)^(1/2)
3.(50%) Suppose that Y1,Y2,...,Yn denote a random sample from a Poisson
distribution with mean λ. Find the MVUE of P(Yi = 0) = exp(-λ).
4.(80%) Suppose that Y1, Y2,..., Yn constitute a random sample from the density
function
{exp(-(y-θ)), y > θ
f(y|θ) = {
{0, elsewhere
where θ is an unknown, positive constant.
(a) Find and estimator θ1cap for θ by the method of moments.
(b) Find an estimator θ2cap for θ by the method of maximum likelihood.
(c) Derive the asymptotic distribution of θ2cap.
5.(60%) Under H0 a random variable has the probability density function
f0(x) = 2x, where 0 <= x <= 1 and under H1 it has the density function
f1(x) = 3x^2 , 0<= x <= 1. A single observation, X is made.
(a) What is the rejection region of a level α = 0.10 most powerful test?
(b) What is the poer of the test?
(c) If X = 0.8, what is the p-value?
6.(40%)In an experiment to evaluate the efficacy of a new technique to increase
the strength of a manufactured item, four independent measurements are taken
under the standared procedure and another four are taken with the new
technique. All eight observations are independent of each other. It is found
that among the eight measurement, the four strongest are all from the new
technique.
(a) Formulate H0 and H1.
(b) Is this difference significant at the α=0.05 level? (You can answer this
without a table)
完
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1. 總分350, 除以1000 為學期成績.
2. 以灰色字體的是下標.
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機會就像老二 握緊就會變大
時間就像乳溝 擠擠還是有的
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