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Continuous Distributions Probability Examples c-6

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Leif Mejlbro

Probability Examples c-6 Continuous Distributions

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Probability Examples c-6 – Continuous Distributions

© 2009 Leif Mejlbro & Ventus Publishing ApS

ISBN 978-87-7681-522-6

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Continuous Distributions

4

Contents

1 Some theoretical background 7

1.1 The exponential distribution 7

1.3 2-dimensional normal distributions 9

1.4 Conditional normal distribution 10

1.5 Sums of independent normal distributed random variables 11

2 The Exponential Distribution 20

3 The Normal Distribution 31

4 The Central Limit Theorem 46

5 The Maxwell distribution 80

6 The Gamma distribution 83

7 The normal distribution and the Gamma distribution 117

8 Convergence in distribution 122

Contents

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Continuous Distributions CHAPTER

5

9 The 2 distribution 126

10 The F distribution 127

11 The F distribution and the t distribution 130

12 Estimation of parameters 131

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Continuous Distributions

6

I ntroduction

Introduction

This is the sixth 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 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 27th October 2009

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Continuous Distributions

7

1 Some theoretical background

1 Some theoretical background

1.1 The exponential distribution

A random variable X follows an exponential distribution with parameter a > 0, if its distribution

function F (x) is given by

F(x) =

⎧

⎨

⎩

1 − e−ax

, for x ≥ 0,

0, for x < 0

The corresponding frequency f (x) is given by

f(x) =

⎧

⎨

⎩

a e−ax

, for x ≥ 0,

0, for x < 0

We have for an exponentially distributed random variable X with parameter a > 0,

E{X} = 1

a and V{X} = 1

a2

In general, if X is exponentially distributed, then

P{X > s + t | X > s} = P {X > t}, for s, t > 0,

which is equivalent with the formula

P{X > s + t} = P {X > s} · P {X > t}, for s, t > 0

We say that the exponential distribution is forgetful

In practice, the exponential distribution often occurs as a distribution of lifetimes, which is in particular

the case in queuing theory In this case the forgetfulness is of paramount importance

An exponentially distributed random variable X with parameter a > 0 is a special gamma distribution

(cf the following), so one also writes,

X ∈ Γ



1 , 1

a



for the exponential distribution

Another type of generalized exponential distributions is the Weibull distribution with parameters a,

b >0 This is given by the distribution function

F(x) =

⎧

⎨

⎩

1 − exp−a xb

 , for x ≥ 0,

0, for x < 0

We note that we get the exponential distribution for b = 1 The Weibull distribution is used in

connection with the theory of reliability

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