outline of probability

{{Short description|1=Overview of and topical guide to 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 (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

Probability and randomness.

Basic probability

=Events=

=Elementary probability=

=Meaning of probability=

=Calculating with probabilities=

=Independence=

Independence (probability theory)

[[Probability theory]]

(Related topics: measure theory)

=Measure-theoretic probability=

=Independence=

=Conditional probability=

[[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 (or mean), 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 (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=

=Functions of random variables=

Generating functions

(Related topics: integral transforms)

=Common generating functions=

=Applications=

A proof of the central limit theorem

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 process]]es

=Some common [[stochastic process]]es=

=Markov processes=

=Stochastic differential equations=

=[[Time series]]=

Moving-average and autoregressive processes
Correlation function and autocorrelation

outline of probability

Introduction

Basic probability

=Events=

=Elementary probability=

=Meaning of probability=

=Calculating with probabilities=

=Independence=

[[Probability theory]]

=Measure-theoretic probability=

=Independence=

=Conditional probability=

[[Random variable]]s

=Discrete and continuous random variables=

=Expectation=

=Independence=

=Some common distributions=

=Some other distributions=

=Functions of random variables=

Generating functions

=Common generating functions=

=Applications=

Convergence of random variables

=Modes of convergence=

=Applications=

[[Stochastic process]]es

=Some common [[stochastic process]]es=

=Markov processes=

=Stochastic differential equations=

=[[Time series]]=

=[[Martingale (probability theory)|Martingales]]=

See also