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經貿組(7-7)

國立高雄大學九十五學年度研究所碩士班招生考試試題

科目：統計學

系所：經濟管理研究所全球經貿組

可使用計算機

考試時間：100分鐘

本科原始成績：滿分100分

否

4. Let $X\_1,\; X\_2,\; \backslash dots,\; X\_n$ be independent, uniformly distributed random variables on the interval $[0,\; \backslash theta]$ . (20%)

(a) Find the maximum likelihood estimator $\backslash hat\{\backslash theta\}$ of $\backslash theta$ .

(b) Is $\backslash hat\{\backslash theta\}$ an unbiased estimator of $\backslash theta$ ?

(c) Is $\backslash hat\{\backslash theta\}$ a consistent estimator of $\backslash theta$ ?

5. For a given experiment, $S$ denotes the sample space and $A\_1,\; A\_2,\; A\_3,\; \backslash dots$ represent possible events. Assume that a number $P(A)$ , called the probability of $A$ , satisfies the following three axioms:

(Axiom 1) $P(A)\; \backslash ge\; 0$ for every $A$ ;

(Axiom 2) $P(S)\; =\; 1$ ;

(Axiom 3) $P\backslash left(\backslash bigcup\_\{i=1\}^\{\backslash infty\}\; A\_i\backslash right)\; =\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; P(A\_i)$ if $A\_i\; \backslash cap\; A\_j\; =\; \backslash emptyset$ for $i\; \backslash ne\; j$ .

Consider the experiment of rolling a fair die, where the experimental outcome is defined as the number of dots appearing on the upward face of the die. Show that the probability that one dot appears on the upward face of the die is $1/6$ .

(15%)

Area in Upper Tail

$t$ Distribution

Degrees of Freedom	Upper 5% point	Upper 2.5% point
6	2.447	3.143
7	2.365	2.998
8	2.306	2.896
9	2.262	2.821
10	2.228	2.821
$\backslash vdots$	$\backslash vdots$	$\backslash vdots$
$\backslash infty$	1.645	1.96

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