臺灣大學數學系

八十六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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1. (15 points) Let $Y$ be a number uniformly distributed over the unit interval$(0,1)$. Let $X_1 , X_2 , \cdots$ be the successive digits in the decimal expansion of $Y$ , that is,
   
   $$Y={X_1 \over 10} + {X_2 \over 10^2} + \cdots +{X_n \over 10^n}+\cdots.$$
   
   
   1. (7 points) Prove that $X_1 , X_2 , \cdots$ are independent discrete valued random variables uniformly distributed over the integers $0$ to $9$.
   2. (8 points) Assume (a) holds. Show that for any integer $k \,(0 \leq k \leq 9)$, the set of numbers $Y$ for which the relative frequency of $k$ in the decimal expansion of $Y$ is $1/10$ has probability $1$. Does this contradict the fact that only three occur in the decimal expansion of $1/3$ ?
2. (10 points) Give a proof to show that if $E(X_n) \rightarrow 0$ and $Var(X_n) \rightarrow 0$ , then $X_n \, {\buildrel P \over \rightarrow} \, 0$.
3. (10 points) Let $X_1 , X_2$ be independent, identically distributed random variables with common uniform distribution on $(0,1)$. Let $X_{(2)} = \max (X_1 , X_2)$. Find $P(X_1 \leq x \mid X_{(2)}=y)$ and $E(X_1 \mid X_{(2)}=y)$.
4. (15 points) Let $X_1 , X_2 , \cdots$be a sample from a $N(\mu , 1)$ population. Find the UMVU estimate of $P(X_1 \geq 0)=\Phi(\mu)$. Here Φ denotes the distribution function of a standard normal random variable.
5. (30 points) Let $X_1, \cdots , X_n$ be independent random variables, $X_i \sim Poisson(\lambda_i)$ , $i=1, \cdots , n$.
   1. (10 points) Deduce that the likelihood-ratio statistic for $H_0 : \lambda_1 = \cdots = \lambda_n$ versus $H_a : \lambda_i \ne \lambda_j$ for some $i,j$, is given by
      
      $$2\log \Lambda = 2 \sum_{i=1}^n X_i \log({X_i \over \bar X}),$$
      where $\bar X =\sum_{i=1}^n X_i/n$.
   2. (10 points) Explain why $-2 \log \Lambda$ can be approximated by
      
      $${1 \over {\bar X}} \sum_{i=1}^n (X_i - \bar X)^2 \,.$$
      
      
   3. (10 points) If $H_0$ is true and $\lambda_1 , \cdots , \lambda_n \rightarrow \infty$, it is claimed that $2\log \Lambda$ has approximately the chi-square distribution with $n-1$ degrees of freedom. Please show that the above claim is correct. (You can assume (b) holds to proceed your proof.)
6. (20 points) Given the general linear model
   
   $$E(Y \mid {\cal X}, {\cal Z})={\cal X} \beta + {\cal Z} \gamma \, ;\qquad \qquad Var(Y)={\cal I} \sigma^2$$
   
   where $Y$ is $n \times 1$, ${\cal X}$ is $n \times p, \beta$ is $p \times 1$, ${\cal Z}$ is $n \times p$, γ is $q \times 1$. Suppose that we believe that $Y$ is not related to ${\cal Z}$. (In fact, $Y$ is related to ${\cal Z}$ via the above model with $\gamma \ne 0$.) We fit the model
   
   $$E(Y \mid {\cal X})={\cal X} \beta\,; \qquad \qquad Var(Y)={\cal I} \sigma^2 \,.$$
   
   
   1. (10 points) What is the effect of the false model assumption on the estimation of β?
   2. (10 points) Suppose that $p=2, q=1$, the $i$th element of ${\cal Z}$ is $x_{ni}^2$, and the $i$th row of ${\cal X}$ is $(1, x_{ni})$. Re-do (a) as $n \rightarrow \infty$ under the assumptions that the unobserved noises are normally distributed with mean $0$ and variance $\sigma^2$, and that $x_{ni}=i / n$.

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