私立中原大學八十九學年度博士班招生考試命題紙

所組別：

數學學系

科目：

統計學

考試時間：

6月14日第2節

1.	(20 points)
	Let X, Y, and Z be i.i.d. random variables (a)Find P(X>Y, X>Z) (b)Find P(X>Y>Z)

2.

(40 points)

The contaminated normal distribution is represented by pN(μ1, σ12)+(1-p)N(μ2, σ2 2), here the addition refers to "mixing" --with probability p the process is realized from N(μ1, σ12)) ;with probability(1-p)the process is realized from N(μ2, σ22) Express Y in terms of U, X1, and X2 (a) Let U~U(0, 1), X1~N(μ1, σ12), X2~N(μ2, σ22) Let U, X1, X2 be independent. Let Y denote the contaminated normal random variable. (b) Let f1 and f2 be the p.d.f.of X1 and X2 , respectively.Find the p.d.f.of Y. (c) Let Z=pX1+(1-p)X2.Investigate the behaviors of the distributions of Y and Z.

3.

(40 points)

Observations(xi, Yi), i=1, ...., n, are made according to the model Yi=α+βxi+εi , where x1, ....., xn are fixed constants and ε1, ...., εn are i.i.d.N(0, σ2). The model is then reparameterized as Yi=α'+β'(xi-)+εi Let and denote the MLEs of α and β, respectively, and and denote the MLEs of α' and β', respectively. (a)Show that '= (b)Show that '≠.In fact, show that '=.Find the distribution of ' (c)Show that ' and ' are uncorrelated and, hence, independent under normality.

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