臺灣大學數學系
八十七學年度博士班入學考試題 機率與統計
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如您決定攻讀機率, 機率考題成績佔成績之70%, 機率部份必須作答四題, 統計部份選答兩題 (請註明選取題目); 如您決定攻讀統計, 統計考題成績佔成績之70%, 統計部份必須作答四題, 機 率部份選答兩題(請註明選取題目).
機率考題：
以下五題每題20分, 以 代表一機率空間.
(1)
,
(a)
描述一個使 zer-one law 成立的條件.
(b)
若已知 , 證: ( 表 的 complement).
(c)
若已知 , zero-one law 成立, 且有一常數 , , 證:
(2)
令 為定義於 Ω 之兩個實值隨機變數, 令 表其機率分佈, 若 為獨立, 證
明: Borel , . 將此結果推至 個隨機變數
之和的情況.
(3)
令 , , 為有界函數, 且 , , 為分佈函數. 設 ,
, 討論函數方程 之解的存在與唯一 (確實寫下其解), 以
機率觀點說明解的含義. (* 表 convolution) (4)
令 , , 表一實值 Markov 過程, 令 表其遷移機率 (transition
probability). 何謂過程為時間齊性 (temporal homogeneous)? 何謂過程為空間齊性 (spatial homogeneous)? 以 描述之, 且予證明.
(5)
就你所瞭解的, 描述數值 之獨立增分 (independent increments) 隨機過程的 Levy-Ito 理 論.
統計考題：
(1)
and are two random variables.
(a)
Find a function of , such that
(b)
Find a function of , such that
Note: You can assume that the minimizers of and are and the median of , respectively.
(2)
Let be independent variables.
(a)
Find the M. L. E.'s of , , and . (b)
Suppose is fixed but . Show that (obtained in (a)) is not a consistent estimate of .
(3)
Let be a sample from a population. Find the U. M. V. U. estimate of .
(4)
Suppose that the observations are directions around a circle, i.e. of angles between and . Consider the simple null hypothesis that the random variables are i.i.d. in the uniform density over , that is, that the angles are completely random.
Suppose that we wish to test for symmetrical clustering around the zero direction.
(a)
Suggest a test statistic when clustering may be around and π.
(b)
Propose a normal approximation to the distribution to the distribution of proposed test statistic under the null hypothesis.
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