Theory of probability - SpeedyLook encyclopedia

The Theory of probability is the mathematical study phenomena characterized by the chance and uncertainty. The central objects of the theory of probability are the random variable , the stochastic Processus S, and the events: they translate in an abstract way of the nondeterministic events or the measured quantities which can sometimes evolve/move in the time in an apparently random way. As a mathematical base of the Statistical , the theory of probability is essential with the majority of the human activities which require a quantitative analysis of a great number of measurements. The methods of the theory of probability also apply to the description of complex systems which one knows only partly the state, as in Mécanique statistics. A great discovery of the Physique of the twentieth century was the probabilistic nature of physical phenomena on a microscopic scale, described by the quantum Mécanique.

History

The mathematical theory of the Probabilité S finds its origins in the analysis of games of chance by Gerolamo Cardano at the sixteenth century, and by Pierre de Fermat and Blaise Pascal with the seventeenth century. Although a simple pile or face or a throw of as of is a random event, by repeating them of many times one obtains a series of results which will have certain statistical properties, that one can study and envisage. Two fundamental mathematical results on this subject are the Loi of the great numbers and the Théorème of the central limit.

Initially, the theory of probability considered especially the discrete events , and its methods were mainly Combinatoire S. But analytical considerations forced the introduction of random variables continuous into the theory. This idea takes all its rise in the modern theory of the probabilities, whose foundations were posed by Andreï Nikolaevich Kolmogorov. Kolmogorov combined the concept of universe, introduced by Richard von Mises and the Théorie of measurement to present its system of axioms for the theory of probability in 1933. Very quickly, its approach became the uncontested base of the modern probabilities.

Discrete theory of probability

The discrete theory of the probabilities deals with events within the framework of countable universe.

Examples: To launch Die S, experiments with packages of charts, and random Walk.

traditional Definition : Initially, the probability of an event was defined like the number of outcomes successful for the event, divided by the full number of possible exits to the random experiment.

For example, if the event is to obtain an even number while launching it as of , its probability is given by $\ tfrac {3} {6} = \ tfrac {1} {2}$ , since three faces out of six have an even number.

modern Definition : The modern definition starts with a Ensemble called universe, which corresponds to all the possible exits to the experiment in the traditional definition. It is noted $\ Omega= \ left \ {x_1, x_2, \ dowries \ right \}$ . Then, one thus needs a function $f$ definite on $\ Omega$ , which will associate with each element $\ Omega$ its probability, satisfactory the following properties:

$f (X) \ in \ mbox {for all} X \ in \ Omega$
$\ sum_ {X \ in \ Omega} F (X) = 1$

One defines then an event as a whole of exits, i.e. a subset of $\ Omega$ . The probability of an event $E$ is then defined in a natural way by:

P (E) = \ sum_ {X \ in E} F (X) \,

Thus, the probability of the universe is 1, and the probability of the null event (the Empty set) is 0.

To return the throw following the example of as of, one can model this experiment by giving oneself a universe $\ Omega= \ {1; 2; 3; 4; 5; 6 \}$ corresponding to the possible values of the die, and a function $f$ who with each $i \ in \ Omega$ associates $f (I) = \ tfrac {1} {6}$ .

Continuous theory of probability

The theory of probability continues deals with the events which occur in a continuous universe (for example the real right ).

traditional Definition : The traditional definition is put in failure when she is confronted with the continuous case (cf Paradoxe of Bertrand).

modern Definition If the universe is the real line $\ mathbb {R}$ , then one assumes the existence of a function called Fonction of distribution $F \,$ , which gives $P (X \ X) = F (X) \,$ for a Random variable $X$ . In other words, $F (X)$ turns over the probability that $X$ is lower or equal to $x$ .

The function of distribution must satisfy the following properties:

$F \,$ is an increasing function and continuous on the right.
$\ lim_ {X \ rightarrow - \ infty} F (X) =0$
$\ lim_ {X \ rightarrow \ infty} F (X) =1$

If $F \,$ is Dérivable, then one says that the random variable $X$ has a Densité of probability $f (X) = \ frac {dF (X)}{dx} \,$ .

For a unit $E \ subseteq \ mathbb {R}$ , the probability that the random variable $X$ is in $E \,$ is defined like:

P (X \ in E) = \ int_ {X \ in E} dF (X) \,

If the density of probability exists, one can then rewrite it:

P (X \ in E) = \ int_ {X \ in E} F (X) \, dx

While the density of probability exists only for the continuous random variables, the function of distribution exists for any random variable (including the discrete variables) with values in $\ mathbb {R}$ .

These concepts can be generalized in the multidimensional cases on $\ mathbb {R} ^n$ and other continuous universes.

The theory of probability today

Certain distributions can be a mixture of discrete distributions and continuous , and thus not to have neither density of probability nor function of mass. The distribution of Cantor constitutes such an example. The modern approach of the probabilities solves these problems by the use of the Théorie of measurement to define a Espace probabilized:

Being given a unit $\ Omega \,$ (also called universe) provided with a σ-algebra $\ mathcal {F} \,$ , a measurement $\ driven \,$ is called measurement of probability if:

$\ driven \,$ is a positive measurement
$\ driven (\ Omega) =1 \,$

For each function of distribution there exists a single measurement of probability on the Borélien S, and vice versa. Measurement corresponding to a function of distribution is known as induced by the function. In the continuous case, this measurement coincides with measurement $fd \ lambda$ with $\ lambda$ the Mesure of Lebesgue if F is the density of probability associated with the function of distribution (which does not exist inevitably) while in the discrete case, it coincides with the definite function F previously.

In addition to allowing a better comprehension and a unification of the discrete and continuous theories of the probabilities, the approach of the theory of measurement also enables us to speak about probabilities apart from $\ mathbb {R} ^n$ , in particular in the theory of the stochastic Processus S. For example for the study of the Brownian Movement, the probability is defined on a space of functions.

Laws of probability

Certain random variables are frequently met in theory of probability because one finds them in many natural processes. Their law thus have an particular importance. The discrete laws most frequent are the discrete uniform Loi, the Loi of Bernoulli, as well as the laws binomial, of geometrical Poisson and . The laws uniform continues, normal, exponential and gamma is among the most important continuous laws.

Convergence of random variables

In theory of probability, there are several concepts of convergence for the Random variable. Here is the list:

Convergence in law : a succession of random variables

(X_n) _ {N \ in \ mathbb {NR}}

converges in law towards the random variable

X \,

if and only if the continuation of measurements images

(\ mu_ {X_n}) _n \ in \ mathbb {NR}

converges narrowly towards measurement image

\ mu_X

. In particular in the real case, it is enough that the functions of distributions converge simply towards the function of distribution of X.

Convergence in probabilities :

(X_n) _ {N \ in \ mathbb {NR}}

converges in probabilities towards

X \,

\ forall \ epsilon >0

\ lim_ {N \ rightarrow \ infty} P \ left (\ left|X_n-X \ right|\ geq \ varepsilon \ right) =0

. This convergence implies convergence in law.

almost sure Convergence :

(X_n) _ {N \ in \ mathbb {NR}}

almost surely converges towards

X \,

P (\ {\ Omega \ lim_ {N \ rightarrow \ infty} X_n (\ Omega) =X (\ Omega) \}) =1.

. It implies convergence in probabilities, therefore convergence in law.

Convergence in $\ mathcal {L} ^1$ :

(X_n) _ {N \ in \ mathbb {NR}}

converges in

\ mathcal {L} ^1

towards

X \,

\ lim_ {N \ rightarrow \ infty} E (|X_n-X|) =0

. It implies also convergence in probabilities.

See too

External bonds

Jean-François random Gall, '' Intégration, probabilities and processes '', course of ENS.

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