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, 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
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
- Conditional probability
- The law of total probability
- Bayes' theorem
Independence
- Independence
[Probability theory]
Measure-theoretic probability
- Sample spaces, σ-algebras and probability measures
- Probability space
- * Sample space
- * Standard probability space
- * Random element
- ** Random compact set
- * Dynkin system
- Probability axioms
- Event
- * Complementary event
- Elementary event
- "Almost surely"
Independence
- Independence
- The Borel–Cantelli lemmas and Kolmogorov's zero–one law
Conditional probability
- Conditional probability
- Conditioning
- Conditional expectation
- Conditional probability distribution
- Regular conditional probability
- Disintegration theorem
- Bayes' theorem
- Rule of succession
- Conditional independence
- Conditional event algebra
- * Goodman–Nguyen–van Fraassen algebra
[Random variable]s
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, 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,
- * Bernoulli and binomial,
- * negative binomial,
- * uniform,
- * geometric,
- * Poisson, and
- * hypergeometric.
- Continuous:
- * uniform,
- * exponential,
- * gamma,
- * beta,
- * normal and multivariate normal,
- * χ-squared,
- * F-distribution,
- * Student's t-distribution, and
- * Cauchy.
Some other distributions
- Cantor
- Fisher–Tippett
- Pareto
- Benford's law
Functions of random variables
- Sum of normally distributed random variables
- Borel's paradox
Generating functions
Common generating functions
- Probability-generating functions
- Moment-generating functions
- Laplace transforms and Laplace–Stieltjes transforms
- Characteristic functions
Applications
- A proof of the central limit theorem
Convergence of random variables
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 process]es
- 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
- * Galton–Watson process
- Markov chain
- * Examples of Markov chains
- Population processes
- Applications to queueing theory
- * Erlang distribution
Stochastic differential equations
- Stochastic calculus
- Diffusions
- * Brownian motion
- * Wiener equation
- * Wiener process
[Time series]
- Moving-average and autoregressive processes
- Correlation function and autocorrelation
Martingales">Martingale (probability theory)">Martingales
- Martingale central limit theorem
- Azuma's inequality