Sample Questions

1.

The random variable X is $\sim N(\mu_1, \sigma_1)$ and the random variable $Y\sim N(\mu_2, \sigma_2)$. X and Y are independent. Identify by name the distribution of each of the following functions of these random variables:

(a)

$$\frac{(X-\mu_1)^2}{\sigma_1^2}$$

(b)

$$\frac{(X-\mu_1)^2/\sigma_1^2}{(Y-\mu_2)^2/\sigma_2^2}$$

(c)

$$\frac{(X-\mu_1)/\sigma_1}{\sqrt{(Y-\mu_2)^2/\sigma_2^2}}$$

2.

Define the following terms:

(a)

A consistent estimator

(b)

A uniformly most powerful test.

(c)

Pivotal quantity

(d)

F distribution (in terms of other distributions)

(e)

T distribution (in terms of other distributions)

3.

A study was done to compare the effectiveness of surgery compared to radiation therapy in controlling cancer. A researcher obtained historical records for patients who had been treated for a particular cancer, either by surgery or radiation therapy. For the n₁ patients who had surgery, f₁₁ showed improvement, f₁₂ did not change, and f₁₃showed improvement. For the n₂ patients who had radiation therapy f₂₁ showed improvement, f₂₂ did not change, and f₂₃showed improvement. The data can be summarized as follows:

Outcome

Treatment	No Improvement	No Change	Some Improvement	Totals
Surgery	f11	f12	f13	n1
Radiation Therapy	f21	f22	f23	n2
Totals	f+1	f+2	f+3

The researchers desire to determine if there is a difference in the distributions of outcomes between the two treatments, surgery and radiation therapy. Let p_ij denote the probability that a patient in treatment i (i=1,2) has outcome j=1,2,3. For example, p₁₂denotes the probability that a patient in group 1, the surgery group, shows no change, outcome category 2. The two samples, of sizes n₁ and n₂, are independent.

The null hypothesis of interest is:

$$H_o:\ p11=p21;\ p12 = p22;\ p13 = p23.$$

Derive the likelihood ratio test for this hypothesis. What is the name of a commonly used test procedure used to test this hypothesis? How is it related to the likelihood ratio test?

4.

Let X_i, $i=1,2,3,\ldots,n$, denote an iid random sample from

$$f(x) = \frac{e^{-\lambda}\lambda^x}{x!}$$

for x= 0, 1, 2 ,3...... Derive the maximum likelihood estimator for $\lambda$ and show that it is unbiased. State an approximate large sample confidence interval for $\lambda$.

5.

You have 10 iid observations from a Bernouilli distribution

f(x)=p^x(1 - p)^1-x,

where 0 < p < 1.0 and x = 0, 1. You want to test the simple hypotheses H₀: p = 0.7 vs. H_a: p = 0.9.

(a)

Give the critical region for the UMP test with level of significance $\le 0.15$.

(b)

Determine the power of the test for n=10.

(c)

Determine n so that the power is $\ge 0.80$.

6.

Determine the moment generating function for the random variable Xwith pdf f(x) = e^-x for x>0 and use the mgf to find the mean and variance of X.

7.

Let X and Y be iid standard normal random variables.

(a)

Show that the random variables X²+Y² and $X/\sqrt{X^2+Y^2}$are independent.

(b)

Show that X/Y has the Cauchy distribution.

8.

Let X_i, $i = 1,\ldots,n$, denote a sample from a Bernouilli distribution

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

where $0<\theta<1.0$ and x=0,1. Find the density $f(\theta\vert x)$, the posterior distribution of $\theta$ given the prior beta density $g(\theta)=6\theta(1-\theta)$ for $0<\theta<1.0$.

9.

Let X_i, $i = 1,\ldots,n$, denote a sample from the pdf $f(x) = e^{-(x-\theta)}$ for $x>\theta$.

(a)

Find the unique uniformly minimum variance unbiased estimate of $\theta$. (Hint: Consider X₍₁₎ the first order statistic from the sample.)

(b)

Show that $X_{(1)}-\theta$ is a pivotal quantity and construct a 95% upper confidence interval for $\theta$.

10.

Let X_i, $i = 1,\ldots,n$, denote a sample from the pdf

$$f(x)=\frac{xe^{-x/\theta}}{\theta^2},\ \mbox{for}\ x > 0.$$

(a)

Find the maximum likelihood estimator of $\theta$ and give its expected value and variance.

(b)

Determine the general form of a uniformly most powerful test for testing

$$H_0:\ \theta\le\theta_0\ \mbox{vs}\ H_a: \theta > \theta0.$$

(c)

Using the central limit theorem, approximate the power function of such a test and sketch its graph.

(d)

For $\theta_0 = 1$ and a 0.01 level of significance, approximate the smallest possible sample size so that the largest Type II error probability for $\theta>2$ is no greater than 0.03.