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 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
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
- 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
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
Stochastic processes
Some common stochastic processes
Markov processes
Stochastic differential equations
Time series
- Moving-average and autoregressive processes
- Correlation function and autocorrelation
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