# Outline of probability

__: Overview of and topical guide to probability__

**Short description**Probability |
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**Probability** is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition of whose truth we are 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 are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call 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.

## Introduction

- Probability and randomness.

## 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

### Meaning of probability

- Probability interpretations
- Bayesian probability
- Frequency probability

### Calculating with probabilities

### Independence

## Probability theory

(Related topics: measure theory)

### Measure-theoretic probability

- Sample spaces, σ-algebras and probability measures
- Probability space
- Probability axioms
- Event (probability theory)
- Elementary event
- "Almost surely"

### Independence

### Conditional probability

- Conditional probability
- Conditioning (probability)
- Conditional expectation
- Conditional probability distribution
- Regular conditional probability
- Disintegration theorem
- Bayes' theorem
- Rule of succession
- Conditional independence
- Conditional event algebra

## 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
- General moments about the mean
- Correlated and uncorrelated random variables
- Conditional expectation:
- 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

- Sum of normally distributed random variables
- Borel's paradox

## Generating functions

(Related topics: integral transforms)

### Common generating functions

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

### Applications

## Convergence of random variables

(Related topics: convergence)

### Modes of convergence

- Convergence in distribution and convergence in probability,
- Convergence in mean, mean square and
*r*th 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

## Stochastic processes

### Some common stochastic processes

- Random walk
- Poisson process
- Compound Poisson process
- Wiener process
- Geometric Brownian motion
- Fractional Brownian motion
- Brownian bridge
- Ornstein–Uhlenbeck process
- Gamma process

### Markov processes

- Markov property
- Branching process
- Markov chain
- Population processes
- Applications to queueing theory

### Stochastic differential equations

### Time series

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

### Martingales

## See also

- Catalog of articles in probability theory
- Glossary of probability and statistics
- Notation in probability and statistics
- List of mathematical probabilists
- List of probability distributions
- List of probability topics
- List of scientific journals in probability
- Timeline of probability and statistics
- Topic outline of statistics

Original source: https://en.wikipedia.org/wiki/Outline of probability.
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