Analytic Aids
Probability Examples c7
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About the book
Description
In this book you find the basic mathematics that is needed by engineers and university students . The author will help you to understand the meaning and function of mathematical concepts. The best way to learn it, is by doing it, the exercises in this book will help you do just that.
Topics as Elementary probability calculus, density functions and stochastic processes are illustrated.
This book requires knowledge of Calculus 1 and Calculus 2.
Preface
This is the eight book of examples from the Theory of Probability. In general, this topic is not my favourite, but thanks to my former colleague, Ole Jørsboe, I somehow managed to get an idea of what it is all about. We shall, however, in this volume deal with some topics which are closer to my own mathematical fields.
The prerequisites for the topics can e.g. be found in the Ventus: Calculus 2 series and the Ventus: Complex Function Theory series, and all the previous Ventus: Probability c1c6.
Unfortunately errors cannot be avoided in a first edition of a work of this type. However, the author has tried to put them on a minimum, hoping that the reader will meet with sympathy the errors which do occur in the text.
Leif Mejlbro
27th October 2009
Content
 Generating functions; background
 Denition of the generating function of a discrete random variable
 Some generating functions of random variables
 Computation of moments
 Distribution of sums of mutually independent random variables
 Computation of probabilities
 Convergence in distribution
 The Laplace transformation; background
 Denition of the Laplace transformation
 Some Laplace transforms of random variables
 Computation of moments
 Distribution of sums of mutually independent random variables
 Convergence in distribution
 Characteristic functions; background
 Denition of characteristic functions
 Characteristic functions for some random variables
 Computation of moments
 Distribution of sums of mutually independent random variables
 Convergence in distribution
 Generating functions
 The Laplace transformation
 The characteristic function
 Index
About the Author
Leif Mejlbro was educated as a mathematician at the University of Copenhagen, where he wrote his thesis on Linear Partial Differential Operators and Distributions. Shortly after he obtained a position at the Technical University of Denmark, where he remained until his retirement in 2003. He has twice been on leave, first time one year at the Swedish Academy, Stockholm, and second time at the Copenhagen Telephone Company, now part of the Danish Telecommunication Company, in both places doing research.
At the Technical University of Denmark he has during more than three decades given lectures in such various mathematical subjects as Elementary Calculus, Complex Functions Theory, Functional Analysis, Laplace Transform, Special Functions, Probability Theory and Distribution Theory, as well as some courses where Calculus and various Engineering Sciences were merged into a bigger course, where the lecturers had to cooperate in spite of their different background. He has written textbooks to many of the above courses.
His research in Measure Theory and Complex Functions Theory is too advanced to be of interest for more than just a few specialist, so it is not mentioned here. It must, however, be admitted that the philosophy of Measure Theory has deeply in uenced his thinking also in all the other mathematical topics mentioned above.
After he retired he has been working as a consultant for engineering companies { at the latest for the Femern Belt Consortium, setting up some models for chloride penetration into concrete and giving some easy solution procedures for these models which can be applied straightforward without being an expert in Mathematics. Also, he has written a series of books on some of the topics mentioned above for the publisher Ventus/Bookboon.