Outline
Transformation types
Exercises
Chapter 4 - Lecture 5
Transformations of a random variable
Andreas Artemiou
Novemer 4th, 2009
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
Transformation types
General
Theorem
Exercises
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Overview
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We will talk what happens when we know the pdf of a random
variable and then we want to learn the pdf of another variable.
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There is only one theorem in this section. If the conditions in
each case satisfy the theorem we will use it to find the answer.
If the conditions are not satisfied we will see those special
cases with tricks that will help us find the answer in similar
situations.
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Example 4.37 page 216
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Mean, Variance and mgf
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Let X have pdf fX (x) and let Y = g (X ), where g is
monotone so it has an inverse function X = h(Y ). Then:
fY (y ) = fX (h(y )) h0 (y )
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Example 4.37 page 216 (using theorem)
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Example 4.39 page 218
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Example 4.40 page 218
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Example 4.44 page 221
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
General
Theorem
Example 4.45 page 221
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable
Outline
Transformation types
Exercises
Exercises
I
Section 4.7 page 222
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Exercises 108, 109, 110, 111, 112, 113, 114, 121, 125, 126
Andreas Artemiou
Chapter 4 - Lecture 5 Transformations of a random variable