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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 inpu
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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 va
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 hav
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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 m
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 re
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
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
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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)
Examp
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 th
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
M