臺灣大學數學系

 八十八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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機率(機率組)

1. (25/105) For any random variable $X$ on $(\Omega ,{\cal F}, P)$ denote the characteristic function of $X$ by $\varphi_X(t)=E[e^{itX}]$.

   (1.1)

   Show that $\varphi_X(t)$ is uniformly continuous for $t\in\mathbb{R}$.

   (1.2)

   Let $X$ be a Normal random variable with mean μ and variance $\sigma^2$. Find $\varphi_X(t)$.

   (1.3)

   Apply (1.2) to evaluate $E[Y^n], n\in\mathbb{N}$, where $Y$ is a Normal random variable with mean $0$ and variance $\sigma^2$. State clearly which properties or facts you are using to solve this question.

2. (30/105) Let $(\Omega ,{\cal F}, P)$ be a fixed probability space.

   (2.1)

   Let $A_n\in {\cal F}, n\in\mathbb{N}$. Prove Borel-Cantelli lemma: If $\sum_{n=1}^{\infty} P(A_n)<\infty$, then $P(A_n \;i.o.)=0$.

   (2.2)

   Let $(X_n)$ be a sequence of random variables. Apply (2.1) to prove that as $n\to\infty, X_n$ converges to $X$ in probability, denoted as $X_n\buildrel P \over \longrightarrow X$, if and only if each subsequence $(X_{n_k})$ contains a further subsequence $(X_{n_{k(j)}})$ which converges to $X$ almost surely.

   (2.3)

   Let $(Y_n), (Z_n)$ be two sequences of random variables such that $Y_n\buildrel P \over \longrightarrow Y, Z_n\buildrel P \over \longrightarrow Z$. Show that $Y_n+Z_n\buildrel P \over \longrightarrow Y+Z$ and $Y_nZ_n\buildrel P \over \longrightarrow YZ$. (Note that if you cannot prove (2.2), you can still apply it to establish (2.3).)

3. (20/105) Let $P_N$ be the product of $N$ Bernoulli measures of $(\eta_1,\cdots,\eta_N)$ on $\Omega_N=\{0,1\}^N$ such that $P(\eta_k=1)=p, P(\eta_k=0)=1-p, 1\le k\le N, 0<p<1$. Put $S_N=\eta_1+\cdots+\eta_N$.

   (3.1)

   Denote by $P_{N,m}$ the conditional distribution of $P_N$ given $S_N=m, m\in \{0,1,...,N\}$. Find $P_{N,m}$.

   (3.2)

   Evaluate by using the limiting behavior of $S_N/N$.

機率(統計組)

1. (15/105) (1.1) Let $X$ be a Normal random variable with mean μ and variance $\sigma^2$. Find its moment generating function $\varphi_X(t)=E[e^{tX}], t\in\mathbb{R}$.
   (1.2) Apply (1.1) to evaluate $E[Y^n], n\in\mathbb{N}$, where $Y$ is a Normal random variable with mean $0$ and variance $\sigma^2$.
2. (15/105) Let $P_N$ be the product of $N$ Bernoulli measures of $(\eta_1,\cdots,\eta_N)$ on $\Omega_N=\{0,1\}^N$ such that $P(\eta_k=1)=p, P(\eta_k=0)=1-p, 1\le k\le N, 0<p<1$. Put $S_N=\eta_1+\cdots+\eta_N$.
   (2.1) Denote by $P_{N,m}$ the conditional distribution of $P_N$ given $S_N=m, m\in \{0,1,...,N\}$. Find $P_{N,m}$.
   (2.2) Evaluate $E[\,\eta_1\eta_2\,\mid\,S_N=m\, ]$.

統計(機率組做 1,2 題, 統計組全做)

1. (15 points) Assume $X_1,\ldots,X_n$ are i.i.d. according to $U(0,\theta)$, $\theta > 0$.
   
   

   (i)

   (5 points) Find the maximum likelihood estimator $\hat{\delta}_n$ of $(\theta-1)^2$.
   
   

   (ii)

   (10 points) It is known that the limit distribution of $n(\theta - X_{(n)})$ is exponential distributed with parameter θ (i.e., The density function $f(y) = \theta^{-1} \exp(-y/\theta)$.). Here $X_{(n)}$ is the largest order statistic. Use this result to determine the nondegenerate limit distribution of $\hat{\delta}_n - (\theta-1)^2$ under proper normalization.

2. (15 points) Let $X_1,\ldots,X_m$ and $Y_1,\ldots,Y_n$ be independent normal $N(\xi,\sigma^2)$ and $N(\eta,\tau^2)$, respectively, and consider the test of $H: \sigma^2 = \tau^2$ against $\sigma^2 < \tau^2$ with rejection region
   
   $$\sqrt{(m+n)\rho (1- \rho)/2}[\log S_Y^2 - \log S_X^2] \geq z_{\alpha},$$
   
   where $\rho = \lim_{n \rightarrow \infty} m/(m+n)$, $0 < \rho < 1$, $S_X^2 = \sum(X_i - \bar{X})^2/(m-1)$, $S_Y^2 = \sum(Y_j - \bar{Y})^2/(n-1)$, and $P(Z > z_{\alpha}) = \alpha$ where $Z$ is a standard normal random variable. Show that this test has asymptotic level α.
3. (10 points) The random variable $Y$ has a binomial distribution with an unknown number θ of trials, and known probability of success, $1/2$. (Namely, $Y \sim Bin(\theta, 1/2)$.) Find an approximated $95\%$ confidence interval of θ of the form $[0,c]$ where $c$ is to be determined.
4. (15 points) Suppose that the independent pairs of random variables $(Y_1,Z_1),\ldots,(Y_n,Z_n)$ are such that $Y_j$ and $Z_j$ are independent in $N(\xi_j,\sigma^2)$ and $N(\beta \xi_j,\tau^2)$, respectively.
   (i) (8 points) Use the method of moment to derive the estimate β.
   (ii) (7 points) Is the estimate obtained in (i) consistent?
5. (15 points) In life-testing experiments, it is quite often that the experiment is terminated whenever the first $r$ failures have occurred among $n$ tested units. Suppose the survival time of a particular units follows an exponential distribution $Exp(\theta)$. (i.e., $P(X > x) = \exp(-\theta x)$.)
   (i) Derive the maximum likelihood estimate of θ under the above setting.
   (ii) Discuss whether the resulting estimator is consistent when $r=2$.

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