臺灣大學數學系

九十一學年度第一學期碩博士班資格考試題

統計(Statistics)

Sept 11, 2002

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• (25 points) A random sample is drawn from a population that is normally distributed with mean θ and variance θ, where $\theta > 0$.
  (11) (10 points) Show that the MLE of θ, $\hat{\theta}$, is a root of the quadratic equation $\theta^2 + \theta - W = 0$, where $W = n^{-1} \sum_{i=1}^n X_i^2$, and determine which root equals the MLE.
  (12) (15 points) Find the approximate variance of $\hat{\theta}$.
• (20 points) For a random sample $X_1,\ldots,X_n$ of Bernoulli($p$) variables, it is desired to test
  $$H_0: p =0.49\;\;\;$$versus$$\;\;\;H_1: p = 0.51.$$
  The test being considered is to reject $H_0$ if $\sum_{i=1}^n X_i$ is large.
  Note that $X_1$ will take on the value of 0 or $1$ with probability $1-p$ and $p$, respectively.
• (20 points) Suppose that $U_1, U_2,\ldots,U_m$ are iid uniform(0,1) random variables, and let $S_n = \sum_{i=1}^n U_i$. Define the random variable $N$ by
  $$N = \min \{k: S_k > 1\}.$$
  (31) (5 points) Show that $P(S_k \leq t) = t^k/k!$.
  (32) (10 points) Show that $E(N) = e$.
  (33) (5 points) How large should $n$ be so that you are $95\%$ confident that you have the first four digits of $e$ correct?
• (20 points) $X$ and $Y$ are independent random variables with $X \sim exponential(\lambda)$ and $Y \sim exponential(\mu)$. (The density function of $X$ is $\lambda^{-1} \exp(-x/\lambda)$ where $x>0$.) It is impossible to obtain direct observations of $X$ and $Y$. Instead, we observe the random variables $Z$ and $W$, where
  $$Z = \min \{X, Y\}\;\;\;$$and$$\;\;\; W = \left\{\begin{array}{ll} 1 & \mbox{if $Z=Z$} \\ 0 & \mbox{if $Z=Y$.} \end{array} \right.$$
  (41) Find the joint distribution of $Z$ and $W$.
  (42) Prove that $Z$ and $W$ are independent.
• (15 points) Find a $(1- \alpha)$ confidence interval for θ, given $X_1,\ldots,X_n$ iid with pdf $f(x\vert\theta) = 2x/\theta^2$, where $0 < x < \theta$ and $\theta > 0$.

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