Probability Theory  

In General > s.a. integration; measure theory; statistics.
* Idea: The branch of mathematics that deals with computing the odds of an event's occurrence.
* History: The idea that rigorous math can be used to calculate odds was introduced by Pascal, who conferred with Fermat; De Moivre found the Bell curve for distributions of random variables; Theory founded by Laplace.
* Applications: Telecommunications (e.g., Dublin IAS applied probability group).
$ Definition (measure theory): A probability is a normalized measure on a set of possible events .
$ Definition (logic): A probability is a map P: X × X → [0,1], where X is a set of propositions, closed under , , and , such that (1) P(b c|a) = P(b|a) P(c|a b); (2) P(b|a) + P(b|a) = 1; (3) P(a|a) = 0.

Interpretations and Applications > s.a. probability in physics.
* Bayesian approach: An approach to the problem of inferring something about a parameter or state of nature s after observing a random variable x whose distribution p depends on s; Probabilities are "degrees of belief," and refer to our confidence in certain statements based on previous experience; Useful for measurements and updating our predictions, allows us to assign probabilities that numbers be "true values" and to use induction; & Bayes, Bernoulli, Gauss, Laplace; > s.a. foundations of quantum mechanics, statistics.
* Frequentist approach: Probabilities are frequencies of occurrence of values in the ensembles of all observations; It is considered the conventional one in physics, but if strictly followed will lead nowhere.
* Combinatorial approach: Probabilities are ratios of favorable cases to some statement over all cases, in a series of performed tests.
@ Frequentist: Brody in(81); Wall SHPSA(06) [Venn's opposition to degree of belief].
@ Bayesian: Hartigan 83; D'Agostini hp/95 [primer]; Howson BJPS(97) [and logic]; Weintraub BJPS(01) [paradox]; Appleby O&S(05)qp/04-in [vs frequentist]; Howson & Urbach 05; D'Agostini ISBA-phy/05 ["Fermi's Bayes theorem"]; Mielczarek et al a0901 [introduction, conceptual]; Fuchs & Schack AIP-a0906 [priors in quantum Bayesian inference].

Distributions > s.a. Binomial and Gamma Distribution.
* Moments of a distribution: The n-th moment of a probability distribution for x is the mean value of xn.
* Poisson distribution: The distribution on N given by P(n) = ea an/n!; Has mean a, and standard deviation a1/2.
@ Poisson: de Groot 75, ch5; Elizalde & Gaztañaga PLA(88) [of galaxies].
@ Combining probability distributions: Eliazar & Klafter PhyA(08) [limit laws and non-linear scaling schemes]; > s.a. Central Limit Theorem.

References > s.a. probabilistic combinatorics.
@ General: Bernoulli 1713; Howson BJPS(95) [survey]; Williams SHPSA(05) [history, Cardano].
@ Books, I: Ekeland 93; Nahin 00.
@ Books: Marle 74; de Groot 75; Bitsakis & Nicolaides ed-89; Fristedt & Grey 97; Stirzaker 99.
@ Books, special emphasis: Billingsley 68 [convergence]; Mackey 78 [and group representations]; Stroock 93 [analytic].
@ Results: Tribelsky PRL(02)m.PR/01 [sums of variables].
@ Fuzzy: in Gudder IJTP(00), Gudder FP(00) [rev]; Habil & Nasr IJTP(02).
@ Complex: Salcedo JMP(97); Weingarten PRL(02) [and path integrals].
@ Related topics: ENelson 87 [non-standard]; Friedman AAM(99) [frequency interpretation]; Streater mp/00 [historical survey].


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