Wikipedia:WikiProject Probability/Learn

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A Guide to Learning Probability from the Wikipedia

The Wikipedia is not a textbook; it is a reference, but does not have an overtly pedagogical intent (despite the etymological links between encyclopedia and pedagogy!). But, in order to be truly useful, the Wikipedia does need to be accessible to those without much knowledge in the field they are reading about, while retaining the detail to be useful, and trustworthy, to those who already have such knowledge. Wiki's satisfy this apparent double-bind admirably, but again, this is an encyclopedia, not a textbook.

I hope anyone and everyone interested in probability will join me in organising this guide to learning probability from the Wikipedia. It is NOT intended to be a list in the style of the list of mathematics topics, but an annotated table of contents, ordered in some more-or-less linear fashion, which could (at least hypothetically) serve as an entry point for the interested student to learn probability theory from the Wikipedia. I hope, too, that it will identify holes in the subject matter covered by the Wikipedia, and help us improve it. Finally, I hope that one day it might form the basis for a Wikibook.

Please feel free to add your contributions to the list below, and your comments to the talk page.

Currently this guide has over 120 links to articles on probability theory, stochastic processes and related topics.

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* Just because a page linked from here doesn't exist yet, doesn't mean that it should exist! Some of these topics should probably (!) be incorporated into an existing page, and some should be combined with others to form new pages. Perhaps some deserve their own, new page.

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Contents 1 Introduction 2 Basic probability 2.1 Events 2.2 Elementary probability 2.3 Conditional probability 2.4 Independence 3 Probability theory 3.1 Measure-theoretic probability 3.2 Independence 4 Random variables 4.1 Discrete and continuous random variables 4.2 Expectation 4.3 Independence 4.4 Some common distributions 4.5 Some other distributions 4.6 Functions of random variables 5 Generating functions 5.1 Common generating functions 5.2 Applications 6 Convergence of random variables 6.1 Modes of convergence 6.2 Applications 7 Stochastic processes 7.1 Simple stochastic processes 7.2 Markov processes 7.3 Stochastic differential equations 7.4 Time series 7.5 |Martingales

Introduction

• Probability and randomness.

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Basic probability

(Related topics: set theory, simple theorems in the algebra of sets)

Events

• Events in probability theory
• Elementary events, sample spaces, Venn diagrams
• Mutual exclusivity

Elementary probability

• The axioms of probability
• Boole's inequality

Conditional probability

• The law of total probability
• Likelihood
• Bayes' theorem
• Bayesian probability

Independence

• Independent events

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Probability theory

(Related topics: measure theory)

Measure-theoretic probability

• Sample spaces, σ-algebras and probability measures
• Probability spaces
• "Almost surely"

Independence

• The Borel-Cantelli lemmas and Kolmogorov's zero-one law

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Random variables

Discrete and continuous random variables

• Discrete random variables: Probability mass functions
• Continuous random variables: Probability density functions
• Normalizing constants
• Cumulative distribution functions
• Joint*, marginal and conditional distributions

Expectation

• Expectation (or mean), variance and covariance
  • Jensen's inequality
• General moments about the mean
• Correlated and uncorrelated random variables
• Conditional expectation:
  • law of total expectation, law of total variance
• Fatou's lemma and the monotone and dominated convergence theorems
• Markov's inequality and Chebyshev's inequality

Independence

• Independent random variables

Some common distributions

• Discrete:
  • constant (see also degenerate distribution),
  • Bernoulli and binomial,
  • negative binomial,
  • (discrete) uniform,
  • geometric,
  • Poisson, and
  • hypergeometric.
• Continuous:
  • (continuous) uniform,
  • exponential,
  • gamma,
  • beta,
  • normal (or Gaussian) and multivariate normal,
  • χ-squared (or chi-squared),
  • F-distribution,
  • Student's t-distribution, and
  • Cauchy.

Some other distributions

• Cantor
• Fisher-Tippett (or Gumbel)
• Pareto
• Benford's law

Functions of random variables

• Sums of random variables*
• General functions of random variables*
• Borel's paradox

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Generating functions

(Related topics: integral transforms)

Common generating functions

• Probability-generating functions
• Moment-generating functions
• Laplace transforms and Laplace-Stieltjes transforms
• Characteristic functions

Applications

• A proof of the central limit theorem*
• Random sums of random variables*

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Convergence of random variables

(Related topics: convergence)

Modes of convergence

• Convergence in distribution and convergence in probability,
• Convergence in mean, mean square and rth mean
• Almost sure convergence
• Skorokhod's representation theorem*

Applications

• Central limit theorem and Laws of large numbers
  • Illustration of the central limit theorem and a 'concrete' illustration
  • Berry-Esséen theorem
• Law of the iterated logarithm

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Stochastic processes

Simple stochastic processes

• Poisson process
• Random walk

Markov processes

• Markov property
• Branching process
  • Galton-Watson process
• Markov chain
  • Examples of Markov chains
• Population processes
• Applications to queueing theory
  • Erlang distribution

Stochastic differential equations

• Stochastic calculus
• Diffusions
  • Brownian motion
  • Wiener equation
  • Wiener process

Time series

• Moving-average* and autoregressive* processes
• Correlation function and autocorrelation

|Martingales

• Martingale central limit theorem*
• Azuma's inequality