PROBABILITY THEORY

(Redirected from Theory of probability)
'Probability theory' is the branch of mathematics concerned with analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

Contents

History

Treatment

Discrete probability distributions

Continuous probability distributions

Measure theoretic probability theory

Probability distributions

Convergence of random variables

Law of large numbers

Central limit theorem

See also

Bibliography

References

History

The mathematical theory of probability has its roots in attempts to analyse games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").
Initially, probability theory mainly considered 'discrete' events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of 'continuous' variables into the theory. This culminated in modern probability theory, the foundations of which were laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and 'measure theory' and presented his axiom system for probability theory in 1933. Fairly quickly this became the undisputed axiomatic basis for modern probability theory.^[2]

Treatment

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continous, any mix of these two and more.

Discrete probability distributions

'Discrete probability theory' deals with events that occur in countable sample spaces.
Examples: Throwing dice, experiments with decks of cards, and random walk.
'Classical definition:'
Initially the probability of an event to occur was defined as number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space.
For example, if the event is "occurrence of an even number when a die is rolled", the probability is given by $frac\{3\}\{6\}=\; frac\{1\}\{2\}$, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.
'Modern definition:'
The modern definition starts with a set called the 'sample space', which relates to the set of all ''possible outcomes'' in classical sense, denoted by $Omega=left\; \{\; x\_1,x\_2,dots$
ight }. It is then assumed that for each element $x\; in\; Omega,$, an intrinsic "probability" value $f(x),$ is attached, which satisfies the following properties:
#$f(x)in[0,1]mbox\{\; for\; all\; \}xin\; Omega$
#$sum\_\{xin\; Omega\}\; f(x)\; =\; 1$
That is, the probability function ''f''(''x'') lies between zero and one for every value of ''x'' in the sample space ''Ω'', and the sum of ''f''(''x'') over all values ''x'' in the sample space ''Ω'' is exactly equal to 1. An 'event' is defined as any subset $E,$ of the sample space $Omega,$. The 'probability' of the event $E,$ defined as
:$P(E)=sum\_\{xin\; E\}\; f(x),$
So, the probability of the entire sample space is 1, and the probability of the null event is 0.
The function $f(x),$ mapping a point in the sample space to the "probability" value is called a 'probability mass function' abbreviated as 'pmf'. The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence.

Continuous probability distributions

'Continuous probability theory' deals with events that occur in a continuous sample space.
'Classical definition:'
The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox.
'Modern definition:'
If the sample space is the real numbers ($mathbb\{R\}$), then a function called the 'cumulative distribution function' (or 'cdf') $F,$ is assumed to exist, which gives $P(Xle\; x)\; =\; F(x),$ for a random variable ''X''. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.
The cdf must satisfy the following properties.
#$F,$ is a monotonically non-decreasing, right-continuous function
#$lim\_\{x$
ightarrow -infty} F(x)=0
#$lim\_\{x$
ightarrow infty} F(x)=1
If $F,$ is differentiable, then the random variable ''X'' is said to have a 'probability density function' or 'pdf' or simply 'density' .
For a set $E\; subseteq\; mathbb\{R\}$, the probability of the random variable ''X'' being in $E,$ is defined as
:$P(Xin\; E)\; =\; int\_\{xin\; E\}\; dF(x),$
In case the probability density function exists, then it can be written as
:$P(Xin\; E)\; =\; int\_\{xin\; E\}\; f(x),dx$
Whereas the ''pdf'' exists only for continuous random variables, the ''cdf'' exists for all random variables (including discrete random variables) that take values on $mathbb\{R\}$.
These concepts can be generalized for multidimensional cases on $mathbb\{R\}^n$ and other continuous sample spaces.

Measure theoretic probability theory

The raison d'être of the measure theoretic treatment of probability is that it unifies the discrete and the continous, and makes the difference a question of which meassure is used. Furthermore it covers distributions that are neither discrete or continous. An example of such distributions could be a mix of ''discrete'' and ''continuous'' distributions e.g. a sum of a discrete and continuous random variable will neither have a pmf nor a pdf. Other distributions may not even be a mix: For example, the Cantor distribution has no point mass and no density. The modern approach to probability theory solves these problems using measure theory to define the probability space:
Given any set $Omega,$ (also called 'sample space') and a σ-algebra $mathcal\{F\},$ on it, a measure $mu,$ is called a 'probability measure' if
#$mu,$ is non-negative
#$mu(Omega)=1,$
For any cdf there is a unique probability measure on the Borel sigma field, and vice versa. The measure corresponding to a cdf is said to be 'induced' by the cdf. This measure coincides with the pmf for discrete variables, and pdf for continuous variables, making the measure theoretic approach free of fallacies.
The ''probability'' of a set $E,$ in the sigma field $mathcal\{F\},$ is defined as
:$P(Xin\; E)\; =\; int\_\{xin\; E\}\; dF(x),$
where the integration is with respect to the measure induced by $F,$.
Along with providing better understanding and unification of discrete and continuous probabilities, measure theoretic treatment also allows us to work on probabilities outside $mathbb\{R\}^n$, as in the theory of stochastic processes. For example to study Brownian motion, probability is defined on a space of functions.

Probability distributions

Main articles: Probability distributions

Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions therefore have gained ''special importance'' in probability theory. Some fundamental ''discrete distributions'' are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important ''continuous distributions'' include the continuous uniform, normal, exponential, gamma and beta distributions.

Convergence of random variables

Main articles: Convergence of random variables

In probability theory, there are several notions of convergence for random variables. They are listed below in the order of strength, i.e., any subsequent notion convergence in the list implies convergence according to all of the preceding notions.
:'Convergence in distribution:' As the name implies, a sequence of random variables $X\_1,X\_2,dots,,$ converges to the random variable $X,$ 'in distribution' if their respective cumulative ''distribution functions'' $F\_1,F\_2,dots,$ converge to the cumulative distribution function $F,$ of $X,$, wherever $F,$ is continuous.
:'Weak convergence:' The sequence of random variables $X\_1,X\_2,dots,$ is said to converge towards the random variable $X,$ 'weakly' if $lim\_\{n$
ightarrowinfty}Pleft(left|X_n-X
ight|geqarepsilon
ight)=0 for every ε > 0. Weak convergence is also called 'convergence in probability'.
:'Strong convergence:' The sequence of random variables $X\_1,X\_2,dots,$ is said to converge towards the random variable $X,$ 'strongly' if $P(lim\_\{n$
ightarrowinfty} X_n=X)=1. Strong convergence is also known as 'almost sure convergence'.
Intuitively, ''strong'' convergence is a stronger version of the ''weak'' convergence, and in both cases the random variables $X\_1,X\_2,dots,$ show an increasing correlation with $X,$. However, in case of ''convergence in distribution'', the realized values of the random variables do not need to converge, and any possible correlation among them is immaterial.

Law of large numbers

Main articles: Law of large numbers

Common intuition suggests that if a fair coin is tossed many times, then ''roughly'' half of the time it will turn up ''heads'', and the other half it will turn up ''tails''. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of ''heads'' to the number of ''tails'' will approach unity. Modern probability provides a formal version of this intuitive idea, known as the 'law of large numbers'. This law is remarkable because it is nowhere assumed in the foundations of probability theory, but instead emerges out of these foundations as a theorem. Since it links theoretically-derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory.[1]
The 'strong law of large numbers' (SLLN) states that if an event of probability ''p'' is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards ''p'' strongly in probability.
In other words, if $X\_1,X\_2,...,$ are independent Bernoulli random variables taking values 1 with probability ''p'' and 0 with probability 1-''p'', then the sequence of random numbers converges to ''p'' almost surely, i.e.
:$Pleft(\; lim\_\{n$
ightarrow infty} rac{sum_{i=1}^n X_i}{n}=p
ight)=1.,

Central limit theorem

Main articles: Central limit theorem

The 'central limit theorem' is the reason for the ubiquitous occurrence of the normal distribution in nature; it is one of the most celebrated theorems in probability and statistics.
The theorem states that the average of many independent and identically distributed random variables tends towards a normal distribution ''irrespective'' of the distribution followed by the original random variables. Formally,
let $X\_1,X\_2,dots,$ be independent random variables with means $mu\_1,mu\_2,dots,$, and variances $sigma\_1^2,sigma\_2^2,dots.,$ Then the sequence of random variables
:
converges in distribution to a standard normal random variable.

See also

★ Expected value and Variance

★ Fuzzy logic and Fuzzy measure theory

★ Glossary of probability and statistics

★ Likelihood function

★ List of probability topics

★ List of publications in statistics

★ List of statistical topics

★ Notation in probability

★ Predictive modelling

★ Probabilistic logic - A combination of probability theory and logic

★ Probability interpretations

★ Statistical independence

Bibliography

★ Analytical Theory of Probability, Pierre Simon de Laplace, , , , 1812,
:: The first major treatise blending calculus with probability theory, originally in French: ''Théorie Analytique des Probabilités''.

★ Foundations of the Theory of Probability, Andrei Nikolajevich Kolmogorov, , , , 1950,
:: The modern measure-theoretic foundation of probability theory; the original German version (''Grundbegriffe der Wahrscheinlichkeitrechnung'') appeared in 1933.

★ Probability and Measure, Patrick Billingsley, , , John Wiley and Sons, 1979,

★ Understanding Probability, Henk Tijms, , , Cambridge Univ. Press, 2004,
:: A lively introduction to probability theory for the beginner.

★ Probability: A Graduate Course, , Allan, Gut, Springer-Verlag, 2005, ISBN 0387228330

References

1. Probability theory, Encyclopaedia Britannica
2. "The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk