Stat 5102 First Midterm ExamMarch 1, 2000

Name Student ID

The exam is open book, including handouts. It is closed notes. 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.

[15 pts.] Suppose X₁, X₂, $\ldots$ are i. i. d. $\NormalDis(3, 4)$ and $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ is the sample mean for a sample of size n = 9. Find $P(X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n < 2)$.

[20 pts.] Suppose X₁, X₂, $\ldots$ are i. i. d. from the distribution with density


$$f(x) = (\theta - 1) x^{- \theta}, \qquad x > 1,$$


where $\theta > 2$ is an unknown parameter. Find a method of moments estimator for $\theta$.

[25 pts.] Suppose X₁, X₂, $\ldots$ are i. i. d. $\UniformDis(a, b)$, meaning they have p. d. f.


$$f_{a, b}(x) = \frac{1}{b - a}, \qquad a < x < b,$$


where a and b are unknown parameters.
Hint:

$$\var(X_i) = \frac{(b - a)^2}{12}$$


1. Find the asymptotic distribution of $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$.
2. Find the asymptotic distribution of $\widetilde{X}_n$.
3. Find the asymptotic relative efficiency of $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ with respect to $\widetilde{X}_n$ when both are considered estimators of the parameter $\theta = (a + b) / 2$. Also state which is the better estimator.

[20 pts.] Suppose X₁, X₂, $\ldots$ are i. i. d. $\NormalDis(\mu, \sigma^2)$distribution, where $\mu$ and $\sigma^2$ are unknown parameters, and suppose S²_n is the usual sample variance given by


$$S^2_n = \frac{1}{n - 1} \sum_{i = 1}^n (X_i - X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n)^2.$$


Find an exact 90% confidence interval for $\sigma^2$corresponding to data n = 10 and S² = 2.2.

[20 pts.] Suppose X₁, X₂, $\ldots$ are i. i. d. $\ExpDis(\lambda)$ random variables and, as usual, $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ denotes the sample mean. What is the asymptotic distribution of


$$Y_n = e^{- t / X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n},$$


where t is a known positive number?