Book/Libros

Leif Mejlbro

Probability Examples c-5 Discrete Distributions

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Probability Examples c-5 – Discrete Distributions © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 978-87-7681-521-9

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Discrete Distributions

Contents

Contents
Introduction 1 1.1 1.2 1.3 1.4 1.5 2 3 4 5 6 7 Some theoretical backgroundThe binomial distribution The Poisson distribution The geometric distribution The Pascal distribution The hypergeometric distribution The binomial distribution The Poisson distribution The geometric distribution The Pascal distribution The negative binomial distribution The hypergeometric distribution Index 5 6 6 8 8 9 11 12 26 33 51 54 56 72

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Discrete Distributions

Introduction

Introduction
This is the fifth 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 allabout. 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 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. planeintegrals. 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 26th October 2009

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Discrete Distributions

1. Some theoretical background

1
1.1

Some theoreticalbackground
The binomial distribution
α(k) α(α − 1) · · · (α − k + 1) = , k(k − 1) · · · 1 k!

It is well-known that the binomial coefficient, where α ∈ R and k ∈ N0 , is defined by α k where α(k) := α(α − 1) · · · (α − k + 1) denotes the k-th decreasing factorial. If in particular, α = n ∈ N, and n ≥ k, then n k = n! k!(n − k)! :=

is the number of combinations of k elements chosen from a set ofn elements without replacement. If n ∈ N0 and n < k, then n k = 0.

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Discrete Distributions

1. Some theoretical background

Also the following formulæ should be well-known to the reader n k k n k =...