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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. 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 continuous, any mix of these two and more. Discrete probability distributions Main article: Discrete probability distribution 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 3/6=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 1. 2. 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
So, the probability of the entire sample space is 1, and the probability of the null event is 0. The function Continuous probability distributions Main article: Continuous probability distribution 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 The cdf must satisfy the following properties. 1. 2. 3. If For a set
In case the probability density function exists, then it can be written as
Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables (including discrete random variables) that take values on These concepts can be generalized for multidimensional cases on Measure theoretic probability theory The raison d'être of the measure theoretic treatment of probability is that it unifies the discrete and the continuous, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous. An example of such distributions could be a mix of discrete and continuous distributions, for example, a random variable which is 0 with probability 1/2, and takes a value from random normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a pdf of (δ[x] + φ(x)) / 2, where δ[x] is the Kronecker delta function. Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space: Given any set Ω, (also called sample space) and a σ-algebra 1. 2. If The probability of a set E\, in the σ-algebra
where the integration is with respect to the measure induced by Along with providing better understanding and unification of discrete and continuous probabilities, measure theoretic treatment also allows us to work on probabilities outside Probability distributions Main article: 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 article: 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 of 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 Most common short hand notation: Weak convergence: The sequence of random variables Most common short hand notation: Strong convergence: The sequence of random variables Most common short hand notation: Intuitively, strong convergence is a stronger version of the weak convergence, and in both cases the random variables Law of large numbers Main article: 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 law of large numbers (LLN) states that the sample average It is in the different forms of convergence of random variables that separates the weak and the strong law of large numbers Weak law: Strong law: It follows from LLN 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. Putting this in terms of random variables and LLN we have Central limit theorem Main article: 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.[citation needed] The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let
converges in distribution to a standard normal random variable. See also * Expected value and Variance Bibliography * Pierre Simon de Laplace (1812). Analytical Theory of Probability. The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités. * Andrei Nikolajevich Kolmogorov (1950). Foundations of the Theory of Probability. The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933. * Patrick Billingsley (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons. * Henk Tijms (2004). Understanding Probability. Cambridge Univ. Press. A lively introduction to probability theory for the beginner. * Gut, Allan (2005). Probability: A Graduate Course. Springer-Verlag. ISBN 0387228330. References 1. ^ Probability theory, Encyclopaedia Britannica 2. ^ "The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk Retrieved from "http://en.wikipedia.org/" |
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. It is then assumed that for each element 
,, an intrinsic "probability" value 
, is attached, which satisfies the following properties:
, of the sample space Ω,. The probability of the event 
,
, then a function called the cumulative distribution function (or cdf) 
, is assumed to exist, which gives 
, for a random variable X. That is, F(x) returns the probability that X will be less than or equal to x.
.
, the probability of the random variable X being in 
and other continuous sample spaces.
, on it, a measure P is called a probability measure if
, is non-negative
,
, converges to the random variable 
, in distribution if their respective cumulative distribution functions 
, converge to the cumulative distribution function 
. Weak convergence is also called convergence in probability.
. Strong convergence is also known as almost sure convergence.
, (independent and identically distributed random variables with finite expectation μ) converges towads the theoretical expectation μ.
.
.
, are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p. E(Yi) = p for all i and it follows from LLN that 
, converges to p almost surely.
, and variance 
, Then the sequence of random variables
,