TEMPUS PECUNIA EST

COLLANA DI MATEMATICA PER LE SCIENZE ECONOMICHEFINANZIARIE E AZIENDALI

Direttore

Beatrice VUniversità degli Studi di Cagliari

Comitato scientifico

Umberto NUniversity of Maryland

Russel Allan JUniversità degli Studi di Firenze

Gian Italo BUniversità degli Studi di Urbino

Giuseppe AUniversità degli Studi di Cagliari

TEMPUS PECUNIA EST

COLLANA DI MATEMATICA PER LE SCIENZE ECONOMICHEFINANZIARIE E AZIENDALI

Al suo livello più profondo la realtà è la matematica della natura.

P

Questa collana nasce dall’esigenza di offrire al lettore dei trattati cheaiutino la comprensione e l’approfondimento dei concetti matem-atici che caratterizzano le discipline dei corsi proposti nelle facoltà diScienze economiche, finanziarie e aziendali.

Fabio Gobbi

Probability theoryAn introduction

Copyright © MMXIVAracne editrice int.le S.r.l.

www.aracneeditrice.itinfo@aracneeditrice.it

via Quarto Negroni, Ariccia (RM)

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I diritti di traduzione, di memorizzazione elettronica,di riproduzione e di adattamento anche parziale,

con qualsiasi mezzo, sono riservati per tutti i Paesi.

Non sono assolutamente consentite le fotocopiesenza il permesso scritto dell’Editore.

I edizione: novembre

Contents

Introduction

Chapter IProbability

Chapter IIRandom variables

Chapter IIIMultivariate random variables

References

Introduction

This is a little book. A little book on probability. You will find every-thing that is necessary for a minimal understanding of the concept ofprobability from a mathematical point of view.Probability is reasoning. Or rather, if you prefer, reasoning has novalue if it does not have a probabilistic nature. All events that we faceeach day are subject to uncertainty and as such cannot be predictedwith certainty. All we can do is to make a decision based on the likeli-hood that we attribute to its possible consequences. And, pay attention,this is exactly what we do without, often, realizing.Think of financial markets. The price of an asset listed on the stockexchange is random and subject to many factors and as a consequenceit is not predictable. If the today’s closing price is, for example, $nobody (except God!) knows what will be the opening price tomor-row. Anyway, a rational investor considers the probability that such aprice is larger than $ definitely lower than the probability that theprice is within the range $-$ since the first event is a rare event(because it implies a daily increase of %). If you need to make aninvestment your reasoning will unavoidably be probabilistic.We can go on to say that probability is an essential part of rationality orbetter rationality has a practical dimension in probabilistic reasoning.My hope is that this little book will help you to begin to understandthe probability which is a topic as practical as it is fascinating.

Chapter I

Probability

Given a random experiment and a sample space of possible outcomesΩ, we define a σ -algebraF of subsets of Ω and a probability measureP on F . We present some properties of P and we introduce condi-tional probabilities and the independence among events which arethe hearth of the probabilistic reasoning.

.. Sets and operations

We define a set a collection of well-distinguished objects (also calledelements). We will use capital letters A, B, .. to indicate a set and lowercase letters a, b, ... for its elements. We say the a belongs to A if a is anelement of A and we write a ∈ A. Otherwise we write a /∈ A when adoes not belong to A. The empty set, that is a set without elements, isdenoted by ;. A class of sets is a set whose elements are themselvessets.

Example ... The set of all positive integers , , ... is usually denoted byN

N = , , ....

The set of all real numbers is R = ω :−∞ <ω < +∞; a closed intervalof R is [x, y] = ω ∈R : x ≤ω ≤ y.

A set A can be:

— finite, if there exists a one-to-one correspondence between Aand a finite set of positive integers , , ...., n;

— countable, if there exists a one-to-one correspondence betweenA and N;

Probability theory. An introduction

— uncountable, if it is not countable.

Let A and B be two sets. If every element of A is likewise an elementof B we say that A is a subset of B, or A is included in B. In symbolsA⊂ B. If both A⊂ B and B⊂ A, then A = B.

Remark For any set A, we have ; ⊂ A.

Two operations among sets are useful: the intersection and the union.If A and B are two sets, the intersection A∩ B is the set of all elementsbelonging to both A and B; the union A∪ B is the set of all elementsbelonging to either A or B.

Example ... Let A = a, b, c, d and B = a, b, d, e. We have

A∩ B = a, b, A∪ B = a, b, c, d, e.

It is very easy to check the following properties: for any sets A, Band C

a) A∩A = A and A∪A = A;b) A∩ ; = ; and A∪ ; = A;c) A∩ B⊂ A and A∩ B⊂ B;d) A⊂ A∪ B and B⊂ A∪ B;e) (A∩ B)∪C = (A∪ B)∩ (B∪C).

.. σ-algebras

Let Ω be a set. A classF of subsets of Ω is an algebra if the followingconditions hold

a. ; ∈ F ;b. if A ∈F then Ac ∈F , where Ac is the complement set of A, that is,

Ac contains all the elements that are not in A;c. F is closed under finite unions: if A, ..., An are inF then

⋃ni=

Ai ∈F .

. Probability

Remark It is not hard to prove that conditions a and b togetherimply that Ω ∈ F , whereas conditions b and c together imply that⋂n

i=Ai ∈F .

If condition c is replaced by

c’. F is closed under countable unions: if (An)n≥ is a sequence of setsinF then

⋃

n≥ An ∈F ,

thenF is a σ -algebra. If C ∈ Ω is a class of subsets of Ω the σ -algebragenerated byC and written σ(C ) is the smallest σ -algebra containingC .

Example ... F = ;,Ω is a σ -algebra. In fact, conditions a and b aretrivially satisfied. We verify condition c’: any sequence of sets ofF containsinfinite times ; or Ω so that

⋃

n≥ An = ; ∈ F if An = ; for all n ≥ ,whereas

⋃

n≥ An =Ω ∈F if An =Ω for some n.

Example ... F = A, Ac,;,Ω is a σ -algebra. The condition c’ is easilyverified taking into account that A∪Ac =Ω and A∩Ac = ;.

Example ... Let Ω = R. It can be proved that the class of all opensubsets of R is a σ -algebra. This class is known as the Borel σ -algebraB . InsymbolsB = E ⊂R : E is open.B contains all intervals of real numbers,open, closed, semi-closed and left-(right-)infinite

(x, y) ∈B , ∀x, y ∈R,

(−∞, x] ∈B , ∀x ∈R,

and so on. More rigorously,B is generated by intervals of the form (−∞, x]

B = σ(−∞, x], x ∈R.

Indeed, any semi-closed interval (x, y] can be obtained by left-infinite inter-vals

(x, y] = (−∞, y]∩ (−∞, x]c,

Probability theory. An introduction

and any open interval (x, y) is

(x, y) =⋃

n≥

(xn, yn],

where (xn)n≥ and (yn)n≥ are sequences of real numbers such that respectivelyxn ↓ x and yn ↑ y.

.. Probability

With the background introduced in the first two sections we can givethe definition of probability measure.

Definition Let Ω be any set and letF be a σ -algebra of subsets of Ω.A (set) function P which mapsF to [, ]

P :F → [, ]

is a probability measure (or simply a probability) if

a) axiom : P(Ω) = ;b) axiom : if (An)n≥ is a sequence of pairwise disjoint sets inF , that is An ∩ Am = ; for any n 6= m, then P

⋃

n≥ An

=∑

n≥P(An) (countable additivity).

The pair (Ω,F ) is called a measurable space and the triplet (Ω,F ,P) iscalled a probability space.

Proposition ... Let (Ω,F ,P) be a probability space. We have

— i. P(;) = ;— ii. P is additive.

Proof. First we prove that P(;) = . By the axiom , we set An = ; forn≥ ; then P

⋃

n≥ ;

=∑

n≥P(;). But⋃

n≥ ; = ;, therefore we get

P(;) =∑

n≥

P(;),

. Probability

which gives P(;) = since ≤ P(;)≤ .The second statement of the proposition may be proved by taking

A = E, A = E , ..., An = En, An+k = ;, f or k≥ .

Then⋃

n≥ An =⋃n

i=En, and by the axiom

P n⋃

i=

En

!

= P

⋃

n≥

An

=∑

n≥

P(An) =n∑

i=

P(En)+∞∑

i=n+

P(;) =n∑

i=

P(En),

as required.

The following proposition lists some properties of a probabilitymeasure.

Proposition ... Let (Ω,F ,P) be a probability space. We have

— p: ∀A ∈F , P(Ac) = −P(A);— p: ∀A, B ∈F such that A⊆ B, then P(A)≤ P(B);— p: ∀A, B ∈F ,

P (A∪ B) = P(A) +P(B)−P(A∩ B);

— p: (Boole’s inequalities) if (An)n≥ ⊂ F is a sequence of eventsthen

P n⋃

i=

Ai

!

≤n∑

i=

P(Ai), (finite sub-additivity),

P

⋃

n≥

An

≤∑

n≥

P(An), (countable sub-additivity);

— p: (Bonferroni’s inequality) if A, ..., An ∈F , thenP

⋂ni=

An

≥−∑n

i=P(Ac

i).

— p: (continuity property) if (An)n≥ ⊂ F is a sequence of events,then

limn→+∞

P(An) = P(A).