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 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 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
- Probability and randomness.
Basic probability
Events
Elementary probability
Meaning of probability
Calculating with probabilities
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 (probability theory)
- * Complementary event
- 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
- * Goodman–Nguyen–van Fraassen algebra
[Random variable]s
Discrete and continuous random variables
- Discrete [random variable]s: 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 convergence theorem|monotone] and dominated convergence theorems
- Markov's inequality and Chebyshev's inequality
Independence
Some common distributions
- Discrete:
- * constant,
- * Bernoulli and binomial,
- * negative binomial,
- * (discrete) uniform,
- * geometric,
- * Poisson, and
- * hypergeometric.
- Continuous:
- * (continuous) uniform,
- * exponential,
- * gamma,
- * beta,
- * normal and multivariate normal,
- * χ-squared,
- * F-distribution,
- * Student's t-distribution, and
- * Cauchy.
Some other distributions
Functions of random variables
Generating functions
Common generating functions
- Probability-generating functions
- Moment-generating functions
- Laplace transforms and Laplace–Stieltjes transforms
- Characteristic functions
Applications
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 process]es
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
[Time series]
- Moving-average and autoregressive processes
- Correlation function and autocorrelation