Outline of probability

Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition whose truth is not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain is it that the event will occur?" The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

Probability

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Introduction

Probability and randomness.

Basic probability

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

Events

Elementary probability

Meaning of probability

Calculating with probabilities

Independence

Independence (probability theory)

Probability theory

(Related topics: measure theory)

Measure-theoretic probability

Independence

Conditional probability

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

Functions of random variables

Generating functions

(Related topics: integral transforms)

Common generating functions

Applications

A proof of the central limit theorem

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

Stochastic processes

Some common stochastic processes

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

Time series

Moving-average and autoregressive processes
Correlation function and autocorrelation

Martingales

See also

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