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Basic Probabilistic Tools for Stochastic Modeling

In this chapter, the readers will find a brief summary of the basic probability tools intensively used in this book. A more detailed version including proofs can be found in [JAN 06].

1.1. Probability space and random variables

Given a sample space O, the set of all possible events will be denoted by , which is assumed to have the structure of a s -field or a s -algebra. P will represent a probability measure.

DEFINITION 1.1.- A random variable (r.v.) with values in a topological space (E,?) is an application X from O to E such that:

where X-1(B) is called the inverse image of the set B defined by:

Particular cases:

a) If (E,?) = ( , ß), X is called a real random variable.

b) If (E, ?) = , where is the extended real line defined by and is the extended Borel s -field of , that is the minimal s -field containing all the elements of ß and the extended intervals:

X is called a real extended value random variable.

c) If E = (n>1) with the product s -field ß(n) of ß, X is called an n-dimensional real random variable.

d) If E = (n>1) with the product s -field ß(n) of ß, X is called a real extended n-dimensional real random variable.

A random variable X is called discrete or continuous accordingly as X takes at most a denumerable or a non-denumerable infinite set of values.

DEFINITION 1.2.- The distribution function of the r.v. X, represented by FX, is the function from [0,1] defined by:

Briefly, we write:

This last definition can be extended to the multi-dimensional case with a r.v. X being an n-dimensional real vector: X = (X1,., Xn), a measurable application from (O, , P) to .

DEFINITION 1.3.- The distribution function of the r.v. X = (X1,., Xn) , represented by FX, is the function from to [0,1] defined by:

Briefly, we write:

Each component Xi (i = 1,.,n) is itself a one-dimensional real r.v. whose d.f., called the marginal d.f., is given by:

The concept of random variable is stable under a lot of mathematical operations; so any Borel function of a r.v. X is also a r.v.

Moreover, if X and Y are two r.v., so are:

provided, in the last case, that Y does not vanish.

Concerning the convergence properties, we must mention the property that, if (Xn, n = 1) is a convergent sequence of r.v. - that is, for all ??O, the sequence (Xn (?)) converges to X (?) - then the limit X is also a r.v. on O. This convergence, which may be called the sure convergence, can be weakened to give the concept of almost sure (a.s.) convergence of the given sequence.

DEFINITION 1.4.- The sequence (Xn (?)) converges a.s. to X (?) if:

This last notion means that the possible set where the given sequence does not converge is a null set, that is, a set N belonging to such that:

In general, let us remark that, given a null set, it is not true that every subset of it belongs to but of course if it belongs to , it is clearly a null set. To avoid unnecessary complications, we will assume from here onward that any considered probability space is complete, i.e. all the subsets of a null set also belong to and thus their probability is zero.

1.2. Expectation and independence

Using the concept of integral, it is possible to define the expectation of a random variable X represented by:

provided that this integral exists. The computation of the integral:

can be done using the induced measure µ on ( , ß), defined by [1.4] and then using the distribution function F of X.

Indeed, we can write:

and if FX is the d.f. of X, it can be shown that:

The last integral is a Lebesgue-Stieltjes integral.

Moreover, if FX is absolutely continuous with fX as density, we obtain:

If g is a Borel function, then we also have (see, e.g. [CHU 00] and [LOÈ 63]):

and with a density for X:

It is clear that the expectation is a linear operator on integrable functions.

DEFINITION 1.5.- Let a be a real number and r be a positive real number, then the expectation:

is called the absolute moment of X, of order r, centered on a.

The moments are said to be centered moments of order r if a=E(X). In particular, for r = 2, we get the variance of X represented by s2 (var(X)) :

REMARK 1.1.- From the linearity of the expectation, it is easy to prove that:

and so:

and, more generally, it can be proved that the variance is the smallest moment of order 2, whatever the number a is.

The set of all real r.v. such that the moment of order r exists is represented by Lr.

The last fundamental concept that we will now introduce in this section is stochastic independence, or more simply independence.

DEFINITION 1.6.- The events A1,., An, (n > 1) are stochastically independent or independent iff:

For n = 2, relation [1.23] reduces to:

Let us remark that piecewise independence of the events A1,., An, (n > 1) does not necessarily imply the independence of these sets and, thus, not the stochastic independence of these n events.

From relation [1.23], we find that:

If the functions FX, FX1,., FX n are the distribution functions of the r.v. X = (X1,., Xn), X1,., Xn, we can write the preceding relation as follows:

It can be shown that this last condition is also sufficient for the independence of X = (X1,., Xn), X1,., Xn. If these d.f. have densities fX, fX1,., fXn, relation [1.24]is equivalent to:

In case of the integrability of the n real r.v X1,X2,.,Xn,, a direct consequence of relation [1.26] is that we have a very important property for the expectation of the product of n independent r.v.:

The notion of independence gives the possibility of proving the result called the strong law of large numbers, which states that if (Xn, n = 1) is a sequence of integrable independent and identically distributed r.v., then:

The next section will present the most useful distribution functions for stochastic modeling.

DEFINITION 1.7 (SKEWNESS AND KURTOSIS COEFFICIENTS).-

a) The skewness coefficient of Fisher is defined as follows:

From the odd value of this exponent, it follows that:

-?1>0 gives a left dissymmetry giving a maximum of the density function situated to the left and a distribution with a right heavy queue, ?1 = 0 gives symmetric distribution with respect to the mean;

-?1<0 gives a right dissymmetry giving a maximum of the density function situated to the right and a distribution with a left heavy queue.

b) The kurtosis coefficient also due to Fisher is defined as follows:

Its interpretation refers to the normal distribution for which its value is 3. Also some authors refer to the excess of kurtosis given by ?1-3 of course null in the normal case.

For ?2<3, distributions are called leptokurtic, being more plated around the mean than in the normal case and with heavy queues.

For ?2>3, distributions are less plated around the mean than in the normal case and with heavy...