Stat 5102 First Midterm ExamMarch 2, 2001

Name Student ID

The exam is open book, including handouts. It is closed notes. You may use one $8 \frac{1}{2} \times 11$ sheet of paper with formulas, etc. You may use a calculator.

Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from.

The points for the questions total to 100. There are pages and 5 problems.

1. [20 pts.] Suppose $X_1$, $X_2$, $\ldots$ are i. i. d. $\NormalDis(0, 4)$ and $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ and $S_n^2$ are the sample mean and variance for a sample of size $n = 9$ (with $S_n^2$ defined as in equation (7.17) in the notes with $n - 1$ in the denominator).

   Find $P(X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n < S_n)$ exactly (not approximately).

2. [20 pts.] Suppose $X_1$, $X_2$, $\ldots$ are i. i. d. from the distribution with density
   $$f_\beta(x) = \frac{2 + 2 (\beta - 1) x}{1 + \beta}, \qquad 0 < x < 1$$
   where $\beta > 0$ is an unknown parameter.

   Find a method of moments estimator for $\beta$.

3. [20 pts.] Suppose $X_1$, $X_2$, $\ldots$ are i. i. d. $\PoissonDis(\mu)$. Because $E(X_i) = \var(X_i) = \mu$, both the sample mean $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ and sample variance $V_n$ are consistent estimators of $\mu$.

   The fourth central moment of the $\PoissonDis(\mu)$ distribution is

   $$E\{ (X - \mu)^4 \} = \mu + 3 \mu^2.$$
   1. Find the asymptotic distribution of $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$. (Express all parameters in terms of $\mu$.)
   2. Find the asymptotic distribution of $V_n$. (Express all parameters in terms of $\mu$.)
   3. Find the asymptotic relative efficiency of $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ with respect to $V_n$ considered as estimators of $\mu$.
   4. State which is the better estimator.

4. [20 pts.] Suppose $X_1$, $X_2$, $\ldots$ are i. i. d. $\ExpDis(\lambda)$ random variables, where $\lambda > 0$ is an unknown parameter. As usual, let $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ denote the sample mean. What is the asymptotic distribution of
   $$W_n = \frac{X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n}{1 + X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n}$$

5. [20 pts.] Suppose $X_1$, $X_2$, $\ldots$ are i. i. d. $\GeoDis(p)$ random variables, where $p$ is an unknown parameter such that $0 < p < 1$ (and by $\GeoDis(p)$ we mean the distribution described by Section B.1.8 of the Appendix on Brand Name Distributions in the notes). As usual, let $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ denote the sample mean. Find an asymptotic 95% confidence interval for $p$ corresponding to data $n = 100$ and $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n = 4.73$.