1
Chapter 7: Functions of Random Variables
7.1 Introduction
As we observed in Chapter 6, many situations we wish to study produce a set of random
variables, X1, X2, , Xn, instead of a single random variable. Questions about the
average life of components,
1
Lecture 1. Transformation of Random Variables
Suppose we are given a random variable X with density fX (x). We apply a function g
to produce a random variable Y = g(X). We can think of X as the input to a black
box, and Y the output. We wish to nd the d
TRANSFORMATIONS OF RANDOM VARIABLES
1. I NTRODUCTION
1.1. Definition. We are often interested in the probability distributions or densities of functions of
one or more random variables. Suppose we have a set of random variables, X1, X2 , X3 , . . . Xn , w
2
Functions of random variables
There are three main methods to find the distribution of a function of
one or more random variables. These are to use the CDF, to transform the pdf directly or to use moment generating functions. We
shall study these in tur
11 TRANSFORMING DENSITY FUNCTIONS
It can be expedient to use a transformation function to transform one probability density
function into another. As an introduction to this topic, it is helpful to recapitulate the
method of integration by substitution of
Transforming a Random Variable
Our purpose is to show how to find the density function fY of the
transformation Y = g(X) of a random variable X with density
function fX.
Let X have probability density function (PDF) fX(x) and
let Y = g(X).
We want to find
Method of Transformations
Goal:
To find the probability distribution of U = h(X1, . . . , Xn).
Method 2: The Method of Transformations
This method applies to the situation in which the
function h is either increasing or decreasing.
Theorem: Let X have pr
1
Discrete Random Variables
For X a discrete random variable with probabiliity mass function fX , then the probability mass function fY
for Y = g(X) is easy to write.
X
fY (y) =
fX (x).
xg 1 (y)
Example 2. Let X be a uniform random variable on cfw_1, 2, .
Transformations
Dear students,
Since we have covered the mgf technique extensively already, here we only review the cdf
and the pdf techniques, first for univariate (one-to-one and more-to-one) and then for
bivariate (one-to-one and more-to-one) transform
Topic for review
Transformation methods
Cumulative distribution function approach
Probability density function formula approach
Exercise #6.23 (a),(c)
Exercise #6.34
Guideline/skill of doing a proof
Moment Generating Function
Exercise #6.46
Hint for homew
Kevin Gladstone
Ms. Duensing
British Literature I
2 February 2012
The Invisible Threat
The Invisible Man was written by H.G. Wells in 1897. The book is about how a man
named Griffin turns himself invisible. He finds a town where he sets up in to find a wa
Kevin Gladstone
CSIT 111
Orientation Exercise
For this exercise you will need to complete parts 1 through 4. Submit your answers for part 1
and your simple world from part 4 through blackboard by the due date.
Part 1. Introductions
For part 1, answer the
Gladstone 1
Kevin Gladstone
Mr. G
Theology 2
1/9/12
Every Young Mans Battle Overview
Young people all over the world go through a point in time where they are unsure on
what to do in certain situations. Teens do not think very clearly at this age and usua
Kevin Gladstone
MUSH 3
Mr. Coons
1/8/2012
Black Hawk Down
Black Hawk Down was written by Mark Bowden in 1999. Black Hawk Down is about
how one hundred U.S. elite soldiers get dropped into the middle of Mogadishu, Somalia on
October 3, 1993. The job involv
June 12, 2016
The Outdoor Connection, Inc.
3815 Schroeder Avenue
Perry Hall, MD 21128
410-256-7569
theoutdoorconnection@gmail.com
Salisbury Claim Center
Department of Labor, Licensing, and Regulation
P.O. Box 4278
Salisbury, MD 21803
Attn: Shannon
Dear St
Kevin Gladstone
Mr. G
Theology 12
5/20/12
Letter to Mike and Jan
Dear Mike and Jan,
I am Kevin Gladstone from Baltimore Lutheran High school and I am a student in Mr.
Gs theology class. I understand you guys are interested in pursuing a long-term relation
PH = 7.4 in arterial side and 7.35 in venous side due to excess CO2
Acidosis = PH <7.4
Alkalosis = PH> 7.4
PH of urine = 4.5 8
Kidneys regulate PH
PH = -log[H+]
Buffer system> Respiratory System> Kidneys
CO2 + H2O = H2CO3 (carbonic Anhydrase)
Carbonic an
Math 273 - Fall 2015
Homework 10
Due November 2nd, 2015
One of the biggest problems of mathematics is to explain to everyone else what it is all about. The
technical trappings of the subject, its symbolism and formality, its baffling terminology, its appa
Math 273 - Fall 2015
Homework 1
Due August 31, 2015
I must study politics and war that my sons may have liberty to study mathematics and philosophy.
John Adams
Turn in:
1.2.11 Find an expression for a cubic function f if f (1) = 6 and f (1) = f (0) = f (2
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
FIN 331 PRINCIPLES OF FINANCIAL MANAGEMENT
Fall 2015
FIN 331-008
Tu, Th .
9:30 - 10:45am
ST 309
FIN 331-009
Tu, Th.
11:00 - 12:15am
ST 309
FIN 331-010
Tu, Th.
3:30 - 4:45am
ST 309
INSTRUCTOR: Steven Jaworski, CFA, CPA
OFFICE HOUR:
Room ST 303
E-Mail:
Tues
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
MNGT 381 - Human Resource Management Online format
Department of Management
Towson University - First Five Week session - Summer 2016
Instructor:
Office and Phone:
Office Hours:
E-mail:
Nhung T. Hendy, Ph.D.
Room 118-D, Stephens Hall, 410-704-2900
by appo
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc
EBTM 306
Fundamentals of Project Management and Business Decisions
TOWSON UNIVERSITY
Department of e-Business & Technology Management
EBTM 306_020 - Summer 2016
Fundamentals of Project Management and Business Decisions
Credit Hours:
Prerequisites:
Instruc