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Master of Applied / Industrial Statistics

Comprehensive Exam, Part I: Theory

May 2014

Directions: This is a closed book exam with a 3-hour time limit. Attached you will find

three pages of formulas as tables for the normal, t, χ2, and F distributions. You may use a

non-programmable, non-graphing calculator.

Answer Only Five of the Six Questions.

1. Two litters of a particular rodent species have been born, one with two brown-haired and

one gray-haired (litter 1), and the other with three brown-haired and two gray-haired (litter

2). We select a litter at random and then select an offspring at random from the selected

litter.

a) Find the probability that the animal chosen is gray-haired given that litter 2 is selected.

b) Find the probability that the animal chosen is brown-haired.

c) Find the probability that the animal is from litter 1 given that a brown-haired offspring

is selected.

2. Let r be a positive integer and 0 < p < 1. Let X1, . . . , Xn be independent random

variables, each following a negative Binomial distribution with the probability mass function

p(x) =

(x− 1

r − 1

)pr(1− p)x−r, x = r, r + 1, . . . .

a) Explain the meaning that X̄ =∑ni=1Xi/n converges in probability to r/p.

b) Argue why√n(X̄ − (r/p))/

√r(1− p)/p2 converges in distribution to N(0, 1).

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c) Use the result in (b) to approximate P (X̄ ≤ 19) when n = 30, r = 10, p = 0.5.

3. Suppose that X and Y have the joint density

fX,Y (x, y) =1

4(x+ 2y), 0 ≤ x < 2, 0 ≤ y < 1

a) Find the marginal density of X.

b) Find the conditional density function of Y given X. Are X and Y independent? Explain.

c) Find the density function of the transformed random variable Z = 9/(X + 1).

4. Let X1, X2, . . . , Xn independently follow a distribution with the density function

fX(x) = θxθ−1, 0 ≤ x ≤ 1,

where parameter θ > 0.

a) Find the method of moments estimator of θ.

b) Find the MLE of θ.

c) Find the asymptotic variance of the MLE.

5. Suppose that the number of printing errors per page printed by a particular newspaper

printer follows a Poisson distribution with mean θ (unknown). Assume the prior distribution

of θ is given by a Gamma(2, 1.5) distribution. Suppose six news pages were randomly

inspected and the numbers of printing errors per page, X1, X2, . . . , X6, were recorded.

a) Derive the posterior distribution of θ.

b) Find a Bayes estimator of θ (the mean of the posterior distribution).

c) Describe how you will obtain a 90% credible interval for θ.

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6. The waiting time for service in a popular deli shop is assumed to follow an Exponential(θ)

distribution. The consumers claim that the average waiting time at counter at lunch time is

more than 20 minutes. A consumer survey group is trying to test this claim with null and

alternative hypotheses as follows:

H0 : θ ≥ 120

vs. H1 : θ < 120

The survey group plans to randomly sample 5 regular customers and propose the following

test: Reject H0 and accept H1 if Y > 18 with Y = min{X1, X2, · · · , X5}.

a) Show that the sampling distribution of Y is Exponential(5θ).

b) Find the power function of the test.

c) Calculate the size (the significance level) of the test.