臺灣大學數學系

 八十七學年度第二學期碩博士班資格考試試題

統計與機率

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Probability

1. Let $X_1,\cdots,X_n$ be a random sample from logistic distribution with cdf $F(x) = 1 / (1+e^{-x})$. Let $V_n = \max(X_1,\cdots,X_n)$.
   (11) Show $V_n \buildrel P \over \longrightarrow \infty$.
   (12) Show $V_n -\log n$ converge to a limiting distribution.
   (13) Find $\lim_{n \rightarrow \infty} P(V_n - \log n \leq 0)$.
2. Let $\{N_k\}$ be a sequence of positive, integral-valued random variables such that $k^{-1}N_k \buildrel P \over \longrightarrow c$ as $k \rightarrow \infty$, where $0 < c < \infty$. Let $\{X_n\}$ be a sequence of independent, identically distributed random variables with $EX_n = 0$ and $EX_n^2 = 1$, $n \geq 1$. Find the asymptotic distribution of $\sum_{j=1}^{N_n} X_j/\sqrt{N_n}$ as $n \rightarrow \infty$. Justify your answer.
3. Let $\{X_n\}$ be a sequence of random variables satisfying
   
   $$X_1 > X_2 > \cdots > 0 \,\,\,\hbox{almost surely.}$$
   
   Show that $X_n \buildrel {a.s.} \over \longrightarrow 0$ if $X_n \buildrel {P} \over \longrightarrow 0$
4. Suppose that $X$ and $Y$ are independent random variables with a common distribution function $F$ that is positive and continuous. What is the conditional probability of $[X \leq x]$ given the random variable $M = max(X,Y)$?

Statistics

1. Let $X_1,\ldots,X_n$ be a random sample from a population with density $f(x,\theta) = \theta(\theta +1) x^{\theta -1}(1 -x)$, $0 < x < 1$, $\theta > 0$.
   1. Show that $T_n = {{2\bar{X}}\over {1- \bar{X}}}$ is a method of moment estimate of θ.
   2. Show that
      
      $${{\sqrt{n}(T_n - \theta)}\over{\theta(\theta +2)^2/2(\theta +3)}} \rightarrow N(0,1)$$
      
      

      in law of large number.

2. Let $Y_{ij} = \beta_i + \epsilon_{ij}$, $1 \leq j \leq n_i$, $i=1, \cdots,p$, where the $\epsilon_{ij}$ are independent $N(0,\sigma_i^2)$ variables, $i=1, \cdots,p$. Derive the likelihood ratio test of $H_0: \sigma_1^2= \cdots = \sigma_p^2$ versus $H_a: \sigma_i^2 \ne \sigma_j^2$ for some $i$, $j$ is used.

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