Outline of probability
확률(Probability)은 사건이 발생할 가능성의 측정입니다. 확률은 우리가 확신 할 수 없는 진리의 어떤 명제를 향한 마음 자세를 정량화하기 위해서 사용됩니다. 관심의 명제는 보통 "특정 사건이 발생할 것이다"라는 형태입니다. 마음의 자세는 "그 사건이 발생할 것은 얼마나 확실한 것인가?"라는 형태입니다. 우리가 채택하는 확실성은 수치적인 측정 그리고, 우리가 확률이라고 이름짓는, 0에서 1 사이의 이들 숫자 (0은 불가능, 1은 확실성을 나타냄)의 관점에서 표현될 수 있습니다. 확률 이론은 잠재적 사건의 가능성과 복잡한 시스템의 기본 역학에 대한 결론을 도출하기 위해서 통계학(statistics), 수학(mathematics), 과학(science) 및 철학(philosophy)에서 광범위하게 사용됩니다.
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
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
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
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 rth mean
- Almost sure convergence
- Skorokhod's representation theorem
Applications
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