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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. "The origins and legacy of Kolmogorov's Grundbegriffe", by Glenn Shafer and Vladimir Vovk 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 $\backslash Omega=\backslash left\; \backslash \{\; x\_1,x\_2,\backslash dots\; ight\; \backslash \}$. It is then assumed that for each Element $x\; \backslash in\; \backslash Omega\backslash ,$, an intrinsic "probability" value $f(x)\backslash ,$ is attached, which satisfies the following properties: #$f(x)\backslash in${Link without Title} \mbox{ for all }x\in \Omega #$\backslash sum\_\{x\backslash in\; \backslash Omega\}\; f(x)\; =\; 1$ 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 $E\backslash ,$ of the sample space $\backslash Omega,$. The '''probability''' of the event $E\backslash ,$ defined as :$P(E)=\backslash sum\_\{x\backslash in\; E\}\; f(x)\backslash ,$ So, the probability of the entire sample space is 1, and the probability of the null event is 0. The function $f(x)\backslash ,$ 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 ($\backslash mathbb\{R\}$), then a function called the Cumulative Distribution Function (or '''cdf''') $F\backslash ,$ is assumed to exist, which gives $P(X\backslash le\; x)\; =\; F(x)\backslash ,$ 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. #$F\backslash ,$ is a Monotonically Non-decreasing , Right-continuous function #$\backslash lim\_\{x\; ightarrow\; -\backslash infty\}\; F(x)=0$ #$\backslash lim\_\{x\; ightarrow\; \backslash infty\}\; F(x)=1$ If $F\backslash ,$ is Differentiable , then the random variable ''X'' is said to have a Probability Density Function or '''pdf''' or simply '''density''' . For a set $E\; \backslash subseteq\; \backslash mathbb\{R\}$, the probability of the random variable ''X'' being in $E\backslash ,$ is defined as :$P(X\backslash in\; E)\; =\; \backslash int\_\{x\backslash in\; E\}\; dF(x)\backslash ,$ In case the probability density function exists, then it can be written as :$P(X\backslash in\; E)\; =\; \backslash int\_\{x\backslash in\; E\}\; f(x)\backslash ,dx$ Whereas the ''pdf'' exists only for continuous random variables, the ''cdf'' exists for all random variables (including discrete random variables) that take values on $\backslash mathbb\{R\}$. These concepts can be generalized for Multidimensional cases on $\backslash mathbb\{R\}^n$ 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 $\backslash Omega,$ (also called sample space) and a σ-algebra $\backslash mathcal\{F\}\backslash ,$ on it, a Measure $\backslash mu\backslash ,$ is called a '''probability measure''' if #$\backslash mu\backslash ,$ is non-negative #$\backslash mu(\backslash Omega)=1\backslash ,$ 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 $E\backslash ,$ in the sigma field $\backslash mathcal\{F\}\backslash ,$ is defined as :$P(X\backslash in\; E)\; =\; \backslash int\_\{x\backslash in\; E\}\; dF(x)\backslash ,$ where the integration is with respect to the measure induced by $F\backslash ,$. Along with providing better understanding and unification of discrete and continuous probabilities, measure theoretic treatment also allows us to work on probabilities outside $\backslash mathbb\{R\}^n$, as in the theory of Stochastic Process es. For example to study Brownian Motion , probability is defined on a space of functions. PROBABILITY DISTRIBUTIONS See Also: 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 See Also: Convergence of random variables In probability theory, there are several notions of convergence for Random Variable s. 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 $X\_1,X\_2,\backslash dots,\backslash ,$ converges to the random variable $X\backslash ,$ '''in distribution''' if their respective cumulative ''distribution functions'' $F\_1,F\_2,\backslash dots\backslash ,$ converge to the cumulative distribution function $F\backslash ,$ of $X\backslash ,$, wherever $F\backslash ,$ is Continuous .

converges in distribution to a Standard Normal random variable.


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