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.
Introduction edit
- Probability and randomness.
Basic probability edit
(Related topics: set theory, simple theorems in the algebra of sets)
Events edit
Elementary probability edit
Meaning of probability edit
Calculating with probabilities edit
Independence edit
Probability theory edit
(Related topics: measure theory)
Measure-theoretic probability edit
- Sample spaces, σ-algebras and probability measures
- Probability space
- Probability axioms
- Event (probability theory)
- Elementary event
- "Almost surely"
Independence edit
Conditional probability edit
Random variables edit
Discrete and continuous random variables edit
- Discrete random variables: Probability mass functions
- Continuous random variables: Probability density functions
- Normalizing constants
- Cumulative distribution functions
- Joint, marginal and conditional distributions
Expectation edit
- 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 edit
Some common distributions edit
- 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 edit
- Cantor
- Fisher–Tippett (or Gumbel)
- Pareto
- Benford's law
Functions of random variables edit
Generating functions edit
(Related topics: integral transforms)
Common generating functions edit
- Probability-generating functions
- Moment-generating functions
- Laplace transforms and Laplace–Stieltjes transforms
- Characteristic functions
Applications edit
Convergence of random variables edit
(Related topics: convergence)
Modes of convergence edit
- Convergence in distribution and convergence in probability,
- Convergence in mean, mean square and rth mean
- Almost sure convergence
- Skorokhod's representation theorem
Applications edit
Stochastic processes edit
Some common stochastic processes edit
- Random walk
- Poisson process
- Compound Poisson process
- Wiener process
- Geometric Brownian motion
- Fractional Brownian motion
- Brownian bridge
- Ornstein–Uhlenbeck process
- Gamma process
Markov processes edit
- Markov property
- Branching process
- Markov chain
- Population processes
- Applications to queueing theory
Stochastic differential equations edit
Time series edit
- Moving-average and autoregressive processes
- Correlation function and autocorrelation
Martingales edit
See also edit
- 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