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ContinuousDistributions
ProbabilityExamplesc-6
LeifMejlbro

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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

Contents

Contents
Introduction

6

1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12

Some theoretical background
The exponential distribution
The normal distribution
2-dimensional normal distributions
Conditional normal distribution
Sums of independent normal distributed random variables
The Central Limit Theorem
The Maxwell distribution
The Gamma distribution
The 2 distribution
The t distribution
The F distribution
Estimation of parameters

7
7
8
9
10
11
11
13
13
14
15
17
17

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

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Continuous Distributions

CHAPTER

9

The 2 distribution

126

10

The F distribution

127

11

The F distribution and the t distribution

130

12

Estimation of parameters

131

Index

167

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Introduction

Continuous Distributions

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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1. Some theoretical background

Continuous Distributions

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
⎧
for x ≥ 0,
⎨ 1 − e−ax ,
F (x) =
⎩
0,
for x < 0.

The corresponding frequency f (x) is given by
⎧
for x ≥ 0,
⎨ a e−ax ,
f (x) =
⎩
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
⎧
for x ≥ 0,
⎨ 1 − exp −a xb ,
F (x) =
⎩
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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