Probability
Theory| 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) = e–a 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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send feedback and suggestions to bombelli at olemiss.edu – modified 11
jun 2009