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
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
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,