Math 152. Rumbos
Fall 2009
1
Lecture Examples
1. Let 1 , 2 , . . . , denote a random sample from a normal(, 2 ) distribution.
Show that the sample mean has a normal(, 2 /) distribution.
Suggestion: Use moment generating functions.
Solution: Compute the mg
Math 152. Rumbos
Fall 2009
1
Solutions to Review Problems for Exam #3
1. Let have a Gamma distribution with parameters = 4 and = > 0.
(a) Find the Fisher information ().
Solution: The pdf of is given by
( ) =
1
3 /
4
(4)
for 0 < <
and zero elsewhere, wh
Math 152. Rumbos
Fall 2009
Solutions to Exam #3
1. Dene the following terms:
(a) Likelihood ratio statistic
Answer: In general, suppose we want to test the hypothesis
H :
H1 :
1 ,
versus the alternative
based on a random, 1 , 2 , . . . , , sample from
Math 152. Rumbos
Fall 2009
Solutions to Exam #1
1. Dene the following terms:
(a) Random sample
Answer: A random sample of size from a given distribution
is a set of independent random variables, 1 , 2 , . . . , , which
have the same distribution as that f
Math 152. Rumbos
Fall 2009
1
Solutions to Review Problems for Exam #2
1. In the book Experimentation and Measurement, by W. J. Youden and published by the by the National Science Teachers Association in 1962, the author
reported an experiment, performed b
Math 152. Rumbos
Fall 2009
1
Solutions to Exam #2
1. Dene the following terms:
(a) Signicance level of a hypothesis test.
Answer: The signicance level, , of a hypothesis test is the
largest probability that the test will reject the null hypothesis
when th
Math 152. Rumbos
Fall 2009
1
Exam 2
October 30, 2009
Name:
This is a closed book exam. Show all signicant work and justify all your answers. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the following 5 pro
Math 152. Rumbos
Fall 2009
1
Exam 3
December 4, 2009
Name:
This is a closed book exam. Show all signicant work and justify all your answers. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the following 5 que
Math 152. Rumbos
Fall 2009
1
Solutions to Review Problems for Exam #1
1. Let and be independent normal(0, 1) random variables and dene
=
( )2
.
2
Give the distribution of .
Suggestion: First, determine the distribution of .
Solution: Since and are indepe
Math 152. Rumbos
Fall 2009
1
Solutions to Assignment #15
1. Suppose that when the radius of a disc in the plane is measured, an error is
made that has a normal(0, 2 ) distribution. If independent measurements
are made, nd an unbiased estimator for the are
Math 152. Rumbos
Fall 2009
1
Solutions to Assignment #13
1. Consider a test of the simple hypotheses
H : =
versus H1 : = 1
based on a random sample from a distribution with pmf ( ), for =
1, 2, . . . , 7. The values of the likelihood function at and 1 ar
Math 152. Rumbos
Fall 2009
1
Solutions to Assignment #14
1. Let 1 , 2 , . . . , denote a random sample from a Bernoulli() distribution
1
with 0 < < 1. We have seen that =
is the MLE for . Compute
=1
the mean squared error, MSE(), of .
Solution: Observe
Math 152. Rumbos
Fall 2009
1
Exam 1
September 30, 2009
Name:
This is a closed book exam. Show all signicant work and justify all your answers. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the following 4 p