Random variables I
Probability Examples c2
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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 second book of examples from the Theory of Probability. This topic is not my favourite, however, thanks to my former colleague, Ole Jørsboe, I somehow managed to get an idea of what it is all about. The way I have treated the topic will often diverge from the more professional treatment. On the other hand, it will probably also be closer to the way of thinking which is more common among many readers, because I also had to start from scratch.
The topic itself, Random Variables, is so big that I have felt it necessary to divide it into three books, of which this is the first one. We shall here deal with the basic stuff, i.e. frequencies and distribution functions in 1 and 2 dimensions, functions of random variables and inequalities between random variables, as well as means and variances.
The prerequisites for the topics can e.g. be found in the Ventus: Calculus 2 series, so I shall refer the reader to these books, concerning e.g. plane integrals. 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
25th October 2009
Content
 Introduction
 Some theoretical results
 Simple introducing examples
 Frequencies and distribution functions in 1 dimension
 Frequencies and distributions functions, 2 dimensions
 Functions of random variables, in general
 Inequalities between two random variables
 Functions Y = f(X) o f random variables
 Functions of two random variables, f(X; Y )
 Means and moments of higher order
 Mean and variance in special cases
 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.