Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Stat 342
Solution Homework 2
Fall 2016
Total Points: 45
1. [11 pts.] PITsimulation.R.
(a) Run the code as it is given line by line. X
(b) [3 pts.] I chose the following sample sizes: N=10, N=50, N=100, N=500 and N=1000. As the
sample size increases the e
Introduction to the Theory of Probability and Statistics II
STAT 342

Fall 2012
Stat 342 HomeWork 1 Solutions
Note: youre expected to follow, but not limited to, these rules for hw1 and all
your future hw.
1. Please list your team members and do not copy from each other.
2. Derivation steps are required. You cannot only write the nal
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Continuous rvs: the nonmonotone case
Note that unless g is strictly monotone (or at least there is a way to break up RY =
cfw_y : fY (y) > 0 into several intervals on each of which g is strictly increasing or decreasing), then Y being a continuous rv. do
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Stat 342
Solution Homework 1
Fall 2016
Total Points: 55
1. [12pts] Let the random variable Y have pdf
fY (y) =
(a) [6pts] For U = 3 Y find fU (u):
3 2
2y
if 1 y 1
otherwise
0
1 y 1 = 1 3 u 1 = 2 u 4
FU (u) = P (U u) = P (3 Y u) = P (Y u 3)
= P (Y 3 u) = 1
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Stat 342
Homework 1
Fall 2016
Due Friday, September 9th
at the beginning of class
Make sure to show all your work. Points are given for both, the correct answer and a detailed solution to
the correct answer.
1. Let Y be a random variable with pdf given by
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Stat 342
Homework 0
Fall 2016
Due Friday, September 2nd
at the beginning of class
The majority of the following problems are taken either from the notes, homework
assignments or the exams during the Spring 2014 and Spring 2015 semester. They are
a represe
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2016
Statistics 342
Exam #1 Solution
February 26, 2014
Instructor: Dr. Genschel
Name SOLUTION, 65 points
Please answer each question in the space provided, and show all your work. Credit cannot be
given if work is not shown. Points are tentative. Good luck.
1.
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
3. Method of Moment Generating Functions
Definition 10 The moment generating function (mgf ) of a random variable Y
is
MY (t) = E[etY ]
assuming that E[etY ] < 1 exists for all t 2 ( h, h) for some h > 0. If E[etY ] fails to
exist in some neighborhood of
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2016
Statistics 342
Exam #1
February 18, 2015
Instructor: Dr. Genschel
Name
Please answer each question in the space provided, and show all your work. Credit cannot be
given if work is not shown. Points are tentative. Good luck.
1. What is your favorite place
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2016
Statistics 342
Exam #1
February 26, 2014
Instructor: Dr. Genschel
Name
Please answer each question in the space provided, and show all your work. Credit cannot be
given if work is not shown. Points are tentative. Good luck.
1. (2 points) Outside of any st
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2016
Statistics 342
Exam #1
February 18, 2015
Instructor: Dr. Genschel
Name Total: 60 points
Please answer each question in the space provided, and show all your work. Credit cannot be
given if work is not shown. Points are tentative. Good luck.
1. [1pt.] What
Introduction to the Theory of Probability and Statistics II.
STAT 342

Fall 2015
Homework 1
[15 pts] 1. Rapid Fire.
(a) What is the value of
0
y 5 ey dy?
Solution: (6) = 5! = 120
(b) What is another name for the Gamma(1, 3) distribution?
Solution: Exponential(3)
(c) What is the constant c if f (x) = cx(1 x)2 is a pdf with support [0,
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Stat 342
Homework 2
Fall 2016
Due Friday, September 16th
at the beginning of class
Make sure to show all your work. Points are given for both, the correct answer and a detailed solution to
the correct answer.
1. Refer to the Rcode PITsimulation.R. I adde
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
2. Method of Transformations
We compute the pdf of U = g(Y ) directly through a transformation technique which
is only valid if the function g is either monotone increasing or monotone decreasing (or
piecewise monotone) for all y in the support of fY (y).
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
Stat 342
Homework 0 Solution
Fall 2016
Points: 40
The majority of the following problems are taken either from the notes, homework
assignments or the exams during the Spring 2014 and Spring 2015 semester. They are
a representative subset of calculus probl
Introduction to the Theory of Probability and Statistics II.
STAT 342

Spring 2014
1
Introduction
Connection between Stat 341 and Stat 342
Big Picture:
1
Some Definitions and Notational Conventions
Definition 1 (Population) The entire set of individuals or objects that are of interest in a study.
Definition 2 (Sample) A subset of the po
Negative Binomial Distribution
A discrete random variable Y is said to have a negative binomial distribution if The experiment involves independent and identical trials. Each trial has two possible outcomes, success and failure. The probability of success
Hypergeometric Distribution
A random variable Y has a hypergeometric distribution if A sample of size n is selected without replacement from a population of size N . Each member of the population belongs to one of two groups; success or failure. The numbe
STATISTICS 342  Homework Assignment #1
Due Friday, January 22, 2010
1. Let Y be a uniform random variable with 1 = 0 and 2 = 1. (a) Using R, generate 10,000 observations of Y . Find the following statistics for your 10,000 observations: minimum, maximum,
Geometric Distribution
A discrete random variable is said to have a geometric distribution if The experiment involves independent and identical trials. Each trial has two possible outcomes, success and failure. The probability of success on each trial is
Gamma Distribution
The Gamma Distribution is used to model continuous data whose values are positive and have a probability histogram that is skewed to the right. Most of the data are near the origin, but some of the observations trail o to the larger num
Binomial Distribution
A discrete random variable is said to have a binomial distribution if The experiment involves a xed number of independent and identical trials, n. Each of the n trials has two possible outcomes, success and failure. The probability o