臺灣大學數學系

 八十九學年度第一學期碩博士班資格考試試題

統計與機率

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§機率與統計擇一科作答, 如兩科都作答, 皆以零分計算.
§機率 Probability (70/100)

1.
(15/100)(1.1) State and prove the weak law of large numbers.
(1.2) (Weierstrass Approximation Theorem.) Given a continuous function $f:[0,1] \rightarrow \mathbb{R}$ construct first the Bernstein polynomials $\{ B_n(x;f), n \in \mathbb{N}\}$ associate with $f$. Then apply the weak law of large numbers to prove that $B_n(x;f) \rightarrow f(x)$ uniformly on $[0,1]$.

2.
(20/100) (2.1) Show that for any random variables $X$ one has
\begin{displaymath} \sum_{n=1}^{\infty} P(\vert X\vert \ge n) \le E(X) \le 1+\sum_{n=1}^{\infty} P(\vert X\vert \ge n). \end{displaymath}


(2.2) Let $(Y_n)$ be an iid sequence of random variables. Show that $E(\vert Y_1\vert) < \infty$ if and only if $\lim\limits_{n \rightarrow \infty} \vert Y_n\vert/n = 0$ almost surely (a.s.). (Hint. Apply Borel-Cantelli lemma and the result in (2.1) with $X=Y_1/\varepsilon$ for $\varepsilon > 0.$)

3.
(20/100) (3.1) Let $(X_n)$ be a sequence of random variables. Assume it is known that $\displaystyle \sum_{n=1}^{\infty} X_n$ converges in probability (i.p.) if and only if $\displaystyle \sum_{n=1}^{\infty} X_n$ converges a.s. Apply this result to prove that if $\displaystyle \sum_{n=1}^{\infty} \mbox{Var}(X_n) < \infty$, then $\displaystyle \sum_{n=1}^{\infty} (X_n - E(X_n))$ converges a.s.
(3.2) Let $(Z_n)$ be an iid sequence of random variables with $E(Z_n) = 0$, Var$(Z_n) = 1$. Denote $\displaystyle S_n = \sum_{j=1}^n Z_j$. Apply the result in (3.1) and Kronecker's lemma to find the range of δ, as large as you can, for which $\displaystyle \frac{S_n}{n^{\frac12} (\log n)^{\delta}} \longrightarrow 0$ a.s.

4.
(15/100) (Solve only one of the following two questions.)
(4.1)
Suppose $X_n, n \ge 0; Y_n, n \ge 1$ are random variables such that $X_n$ converges to $X_0$ in distribution and $Y_n$ converges to 0 i.p. Show that $X_n + Y_n$ converges to $X_0$ in distribution.
(4.2)
Let probability space $(\Omega, \cal F, \cal P)$ and sub-field $\cal G \subset \cal F$ be given. Let $X,Y$ be two random variables such that $X, XY \in L_1(\Omega, \cal F, \cal P)$ and $Y$ is $\cal G$-measurable. Show that $E[XY\vert{\cal G}] = YE[X{\cal G}]$ a.s.

§統計
Let $\chi^2_{\alpha,df}$ denote the α-th quantile of Chi-square distribution with degree of freedom df.

df 21 22 23 24 25 26 27 28 29 30
$\chi^2_{0.05,df}$ 11.59 12.34 13.09 13.85 14.61 15.38 16.15 16.93 17.71 18.49
$\chi^2_{0.95,df}$ 32.67 33.92 35.17 36.42 37.65 38.89 40.11 41.34 42.56 43.77
df 31 32 33 34 35 36 37 38 39 40
$\chi^2_{0.05,df}$ 19.28 20.07 20.87 21.66 22.47 23.27 24.07 24.88 25.7 26.51
$\chi^2_{0.95,df}$ 44.99 46.19 47.4 48.6 49.8 51 52.19 53.38 54.57 55.76
df 41 42 43 44 45 46 47 48 49 50
$\chi^2_{0.05,df}$ 27.33 28.14 28.96 29.79 30.61 31.44 32.27 33.1 33.93 34.76
$\chi^2_{0.95,df}$ 56.94 58.12 59.3 60.48 61.66 62.83 64 65.17 66.34 67.5
df 51 52 53 54 55 56 57 58 59 60
$\chi^2_{0.05,df}$ 35.6 36.44 37.28 38.12 38.96 39.8 40.65 41.49 42.34 43.19
$\chi^2_{0.95,df}$ 68.67 69.83 70.99 72.15 73.31 74.47 75.62 76.78 77.93 79.08

  1. 　
    1. (A bio-assay problem) Suppose that the probability of death $p(x)$ is related to the dose $x$ of a certain drug in the following manner
      \begin{displaymath}p(x)=\frac{1}{1+e^{-(\alpha+\beta x)}},\end{displaymath}

      where $\alpha>0,\beta\in R$ are unknown parameters. In an experiment, $k$ different(given) doses $x_1,x_2,\ldots,x_k$ of the drug are considered, dose level $x_i$ is applied to $n_i$(given) animals and the number $Y_i$ of deaths among them are recorded. Derive sufficient statistics for $(\alpha,\beta)$. (8 points)

    2. Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables with p.d.f. $f(\cdot;\theta),\theta\in\Omega\subset R$,and let $\overrightarrow{T}= (T_1,T_2,\ldots,T_m),T_j=T_j(X_1,X_2,\ldots,X_n),j=1,2,\ldots,m.$ be any statistic. Let $U=U(X_1,X_2,\ldots,X_n)$ be an unbiased statistic for θ. Prove or disporve that
      1. $E_\theta(U\vert\overrightarrow{T})$ is an unbiased statistic for θ. (4 points)
      2. $\sigma^2_\theta[E_\theta(U\vert\overrightarrow{T})]\le \sigma^2_\theta[\overrightarrow{U}]$. (3 points)
  2. Let $X_1,X_2,\ldots,X_n$ be independent random variables with p.d.f. $f$ given by
    \begin{displaymath}f(x;\theta)=\frac{1}{\theta}e^{-x/\theta},\quad x>0,\theta>0. \end{displaymath}

    1. Derive(base on Neyman-Pearson Lemma) the uniformly most powerful test for testing the hypothesis $H:\theta=\theta_0$ against the altenative $A:\theta<\theta_0$ at level of significance α. (8 points)
    2. Determine the minimum sample size n required to obtain power at least $0.95$ against the alternative $\theta=500$ when $\theta_0=1000$ and $\alpha=0.05$.(7 points)


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