臺灣大學數學系

八十七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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probability

1. In $(\Omega, {\cal F}, P)$, let $A_n \in {\cal F}$. The set $\{ A_n \quad i.o. \}$ is defined as $\{ w; w \in A_n$ for an infinite number of $n \}$. Show the following statements hold.
   1. $\sum_1^{\infty}P(A_n) < \infty$ implies $P(A_n \,\,\, i.o.)=0$.
   2. If $A_n$ are independent events, then $\sum_1^{\infty}P(A_n)=\infty$ implies $P(A_n \,\,\, i.o)=1$.
2. For any random variable $X$ and real number $r >0$,
   1. Show that $E {\vert X \vert} ^{r}=r \int_0^{\infty}t^{r-1} P(\vert X \vert \geq t)\ dt$.
   2. If $E \vert X \vert^r < \infty$, then $P(\vert X \vert \geq t)=o(t^{-r})$ as $t \rightarrow \infty$. (You can assume that (a) holds to solve this problem.)
   3. Comment on the connection of statement (b) with $r=2$ and the Chebyschev inequality.
3. Let the distribution functions $F, F_1, F_2, \cdots$ possess respective characteristic functions $\phi, \phi_1, \phi_2, \cdots$. Show that the following three statements are equivalent:
   1. $F_n$ converges in distribution to $F$;
   2. $\lim_n \phi_n (t) = \phi (t)$, each real $t$;
   3. $\lim_n \int gdF_n= \int gdF$, each bounded continuous function $g$.
4. Consider a two-state Markov chain. The variables $X_1, X_2, \cdots$ each take on the values 0 and 1, with the joint distribution determined by $P(X_1=1)=p_1$ and $P(X_{i+1} =1 \mid X_i =0)=\pi_0$ and $P(X_{i+1}=1 \mid X_i=1)=\pi_1$ of which we assume $0< \pi_0,\, \pi_1 <1$. Set $\bar{X}_n= \sum_{i=1}^n (X_i / n)$. Show that $\bar{X}_n$ is a consistent estimate of $p=\pi_0 / (\pi_0 +\pi_1)$.

Statistics

1. Find the asymptotic distribution of $\hat p_n^2$ where $\hat p_n$ is the proportion of success of a binomial distribution with $n$ trials and the probability of success $p$.
2. In life-testing experiments, it is quite often that the experiment is terminated whenever the first $r$ failures have occurred among $n$ tested units. This scheme is usually referred to as Type II censored sampling. Suppose the survival time of a particular unit follows an exponential distribution $Exp(\theta)$ (i.e., $F(x)=1-\exp(-\theta_x))$. Derive the maximum likehood estimate of θ under Type II censored sampling and discuss whether the resulting estimator is consistent when $r=1$.

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