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Probability Distribution
*** Shopping-Tip: Probability Distribution
In
mathematics and
statistics, a '''probability distribution''', more properly called a '''probability density''', assigns to every
interval (mathematics) interval of the
real numbers a
probability, so that the
probability axioms are satisfied. In technical terms, a probability distribution is a
probability measure whose domain is the
Borel algebra on the reals.
A probability distribution is a special case of the more general notion of a
probability measure, which is a function that assigns probabilities satisfying the
Kolmogorov axioms to the measurable sets of a
measurable space. Additionally, some authors define a distribution generally as the probability measure induced by a
random variable ''X'' on its
range (mathematics) range - the probability of a set ''B'' is
. However, this article discusses only probability measures over the real numbers.
Every random variable gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If ''X'' is a random variable, the corresponding probability distribution assigns to the interval [''a'', ''b''] the probability Pr[''a'' ≤ ''X'' ≤ ''b''], i.e. the probability that the variable ''X'' will take a value in the interval [''a'', ''b''].
The probability distribution of the variable ''X'' can be uniquely described by its
cumulative distribution function ''F''(''x''), which is defined by
:
for any ''x'' in '''R'''.
A distribution is called ''discrete'' if its cumulative distribution function consists of a sequence of finite jumps, which means that it belongs to a
discrete random variable ''X'': a variable which can only attain values from a certain finite or
countable set. By one convention, a distribution is called ''continuous'' if its cumulative distribution function is
continuous function continuous, which means that it belongs to a random variable ''X'' for which Pr[ ''X'' = ''x'' ] = 0 for all ''x'' in '''R'''. Another convention reserves the term ''continuous probability distribution'' for
absolute continuity absolutely continuous distributions. These can be expressed by a
probability density function: a non-negative
Lebesgue integration Lebesgue integrable function ''f'' defined on the real numbers such that
:
for all ''a'' and ''b''. Of course, discrete distributions do not admit such a density. Also some continuous distributions like the
devil's staircase that do not admit a density.
* The ''support'' of a distribution is the smallest closed set whose complement has probability zero.
* The probability distribution of the sum of two independent random variables is the
convolution of each of their distributions.
* vThe probability distribution of the difference of two random variables is the
cross-correlation of each of their distributions.
List of important probability distributions
Several probability distributions are so important in theory or applications that they have been given specific names:
Discrete distributions
=With finite support
=
* The
Bernoulli distribution, which takes value 1 with probability ''p'' and value 0 with probability ''q'' = 1 − ''p''.
** The
Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
* The
binomial distribution describes the number of successes in a series of independent Yes/No experiments.
* The
degenerate distribution at ''x''
0, where ''X'' is certain to take the value ''x
0''. This does not look random, but it satisfies the definition of
random variable. This is useful because it puts deterministic variables and random variables in the same formalism.
* The
Uniform distribution (discrete) discrete uniform distribution, where all elements of a finite
set theory set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette or a well-shuffled deck. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers,
Pseudorandom number sequence pseudo-random number generators are used to produced a
randomness statistically random discrete uniform distribution.
* The
hypergeometric distribution, which describes the number of successes in the first ''m'' of a series of ''n'' independent Yes/No experiments, if the total number of successes is known.
*
Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
* The
Zipf-Mandelbrot law is a discrete power law distribution which is a generalization of the
Zipf distribution.
=With infinite support
=
* The
Boltzmann distribution, a discrete distribution important in
statistical physics which describes the probabilities of the various discrete energy levels of a system in
thermal equilibrium. It has a continuous analogue. Special cases include:
** The
Gibbs distribution
** The
Maxwell-Boltzmann distribution
** The
Bose-Einstein distribution
** The
Fermi-Dirac distribution
* The
geometric distribution, a discrete distribution which describes the number of attempts needed to get the first success in a series of independent Yes/No experiments.
Image:Poisson distribution PMF.png Poisson_distribution.html" title="Meaning of 150px 150px|thumb|[[Poisson distribution.html" title="Meaning of thumb|[[Poisson distribution">150px|thumb|[[Poisson distribution">thumb|[[Poisson distribution">150px|thumb|[[Poisson distribution
* The
logarithmic distribution logarithmic (series) distribution
* The
negative binomial distribution, a generalization of the geometric distribution to the ''n''th success.
* The
parabolic fractal distribution
* The
Poisson distribution, which describes a very large number of individually unlikely events that happen in a certain time interval.
Image:SkellamDistribution.png Skellam_distribution.html" title="Meaning of 150px 150px|thumb|[[Skellam distribution.html" title="Meaning of thumb|[[Skellam distribution">150px|thumb|[[Skellam distribution">thumb|[[Skellam distribution">150px|thumb|[[Skellam distribution
* The
Skellam distribution, the distribution of the difference between two independent Poisson-distributed random variables.
* The
Yule-Simon distribution
* The
zeta distribution has uses in applied statistics and statistical mechanics, and perhaps may be of interest to number theorists. It is the
Zipf distribution for an infinite number of elements.
Continuous distributions
=Supported on a bounded interval
=
Image:Beta distribution pdf.png Beta_distribution.html" title="Meaning of thumb thumb|150px|[[Beta distribution.html" title="Meaning of 150px|[[Beta distribution">thumb|150px|[[Beta distribution">150px|[[Beta distribution">thumb|150px|[[Beta distribution
* The
Beta distribution on [0,1], of which the uniform distribution is a special case, and which is useful in estimating success probabilities.
Image:Uniform_distribution_PDF.png Uniform_distribution (continuous) thumb|150px|[[Uniform distribution (continuous)|continuous uniform distribution.html" title="Meaning of continuous uniform distribution.html" title="Meaning of thumb|150px|[[Uniform distribution (continuous)|continuous uniform distribution">thumb|150px|[[Uniform distribution (continuous)|continuous uniform distribution">continuous uniform distribution.html" title="Meaning of thumb|150px|[[Uniform distribution (continuous)|continuous uniform distribution">thumb|150px|[[Uniform distribution (continuous)|continuous uniform distribution
* The
Uniform distribution (continuous) continuous uniform distribution on [''a'',''b''], where all points in a finite interval are equally likely.
** The
rectangular distribution is a uniform distribution on [-1/2,1/2].
* The
Dirac delta function although not strictly a function, is a limiting form of many continuous probability functions. It represents a ''discrete'' probability distribution concentrated at 0 — a
degenerate distribution — but the notation treats it as if it were a continuous distribution.
* The
Kumaraswamy distribution is as versatile as the Beta distribution but has simple closed forms for both the cdf and the pdf.
* The
logarithmic distribution (continuous)
* The
triangular distribution on [''a'', ''b''], a special case of which is the distribution of the sum of two uniformly distributed random variables (the ''convolution'' of two uniform distributions).
* The
von Mises distribution
*The
Wigner semicircle distribution is important in the theory of
random matrices.
=Supported on semi-infinite intervals, usually [0,∞)
=
Image:Chi-square distributionPDF.png chi-square_distribution.html" title="Meaning of thumb thumb|150px|[[chi-square distribution.html" title="Meaning of 150px|[[chi-square distribution">thumb|150px|[[chi-square distribution">150px|[[chi-square distribution">thumb|150px|[[chi-square distribution
* The
chi distribution
* The
noncentral chi distribution
* The
chi-square distribution, which is the sum of the squares of ''n'' independent Gaussian random variables. It is a special case of the Gamma distribution, and it is used in
goodness-of-fit tests in
statistics.
** The
inverse-chi-square distribution
** The
noncentral chi-square distribution
** The
scale-inverse-chi-square distribution
Image:Exponential distribution pdf.png Exponential_distribution.html" title="Meaning of thumb thumb|150px|[[Exponential distribution.html" title="Meaning of 150px|[[Exponential distribution">thumb|150px|[[Exponential distribution">150px|[[Exponential distribution">thumb|150px|[[Exponential distribution
* The
exponential distribution, which describes the time between consecutive rare random events in a process with no memory.
* The
F-distribution, which is the distribution of the ratio of two (normalized) chi-square distributed random variables, used in the
analysis of variance.
** The
noncentral F-distribution
Image:Gamma distribution pdf.png Gamma_distribution.html" title="Meaning of thumb thumb|150px|[[Gamma distribution.html" title="Meaning of 150px|[[Gamma distribution">thumb|150px|[[Gamma distribution">150px|[[Gamma distribution">thumb|150px|[[Gamma distribution
* The
Gamma distribution, which describes the time until ''n'' consecutive rare random events occur in a process with no memory.
** The
Erlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in
queuing systems.
** The
inverse-gamma distribution
*
Fisher's z-distribution
* The
half-normal distribution
* The
Lévy distribution
* The
log-logistic distribution
* The
log-normal distribution, describing variables which can be modelled as the product of many small independent positive variables.
Image:Pareto distributionPDF.png Pareto_distribution.html" title="Meaning of thumb thumb|150px|[[Pareto distribution.html" title="Meaning of 150px|[[Pareto distribution">thumb|150px|[[Pareto distribution">150px|[[Pareto distribution">thumb|150px|[[Pareto distribution
* The
Pareto distribution, or "power law" distribution, used in the analysis of financial data and critical behavior.
* The
Rayleigh distribution
* The
Rayleigh mixture distribution
* The
Rice distribution
* The
type-2 Gumbel distribution
* The
Wald distribution
* The
Weibull distribution, of which the exponential distribution is a special case, is used to model the lifetime of technical devices.
=Supported on the whole real line
=
Image:Cauchy distribution pdf.png Cauchy_distribution.html" title="Meaning of 150px 150px|thumb|[[Cauchy distribution.html" title="Meaning of thumb|[[Cauchy distribution">150px|thumb|[[Cauchy distribution">thumb|[[Cauchy distribution">150px|thumb|[[Cauchy distribution
Image:Laplace distribution pdf.png Laplace_distribution.html" title="Meaning of 150px 150px|thumb|[[Laplace distribution.html" title="Meaning of thumb|[[Laplace distribution">150px|thumb|[[Laplace distribution">thumb|[[Laplace distribution">150px|thumb|[[Laplace distribution
Image:LevyDistribution.png Levy_distribution.html" title="Meaning of 150px 150px|thumb|[[Levy distribution.html" title="Meaning of thumb|[[Levy distribution">150px|thumb|[[Levy distribution">thumb|[[Levy distribution">150px|thumb|[[Levy distribution
Image:Normal distribution pdf.png Normal_distribution.html" title="Meaning of thumb thumb|150px|[[Normal distribution.html" title="Meaning of 150px|[[Normal distribution">thumb|150px|[[Normal distribution">150px|[[Normal distribution">thumb|150px|[[Normal distribution
* The
Beta prime distribution
* The
Cauchy distribution, an example of a distribution which does not have an
expected value or a
variance. In physics it is usually called a
Lorentzian function Lorentzian profile, and is associated with many processes, including
resonance energy distribution, impact and natural
spectral line broadening and quadratic
stark effect stark line broadening.
* The
Fisher-Tippett distribution Fisher-Tippett, extreme value, or log-Weibull distribution
** The
Gumbel distribution, a special case of the Fisher-Tippett distribution
* The
generalized extreme value distribution
* The
hyperbolic secant distribution
* The
Landau distribution
* The
Laplace distribution
* The
Lévy skew alpha-stable distribution is often used to characterize financial data and critical behavior.
* The
map-Airy distribution
* The
normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the
central limit theorem: every variable that can be modelled as a sum of many small independent variables is approximately normal.
*
Student's t-distribution, useful for estimating unknown means of Gaussian populations.
** The
noncentral t-distribution
* The
type-1 Gumbel distribution
* The
Voigt profile Voigt distribution, or Voigt profile, is the convolution of a
normal distribution and a
Cauchy distribution. It is found in spectroscopy when
spectral line profiles are broadened by a mixture of
Lorentz function Lorentzian and
Doppler profile Doppler broadening mechanisms.
Joint distributions
For any set of
Statistical independence independent random variables the
probability density function of the joint distribution is the product of the individual ones.
=Two or more random variables on the same sample space
=
*
Dirichlet distribution, a generalization of the
beta distribution.
*The
Ewens's sampling formula is a probability distribution on the set of all
integer partition partitions of an integer ''n'', arising in
population genetics.
*
multinomial distribution, a generalization of the
binomial distribution.
*
multivariate normal distribution, a generalization of the
normal distribution.
=Matrix-valued distributions
=
*
Wishart distribution
*
matrix normal distribution
*
matrix t-distribution
*
Hotelling's T-square distribution
Miscellaneous distributions
* The
Cantor distribution
See also
*
copula (statistics)
*
cumulative distribution function
-
Interactive Discrete and Continuous Probability Distributions
*
likelihood function
*
list of statistical topics
*
probability density function
*
random variable
*
histogram
Category:Probability and statistics
Category:Probability distributions
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es:Distribución de probabilidad
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|+ style="font-size: larger; margin-left: inherit;" | '''{{{name}}}'''
|-
| style="background: white; text-align: center; font-size: smaller;" colspan="2" | Probability {{{type}}} function
{{{pdf_image}}}
|-
| style="background: white; text-align: center; font-size: smaller;" colspan="2" | Cumulative distribution function
{{{cdf_image}}}
|- valign="top"
| style="text-align: right;" | '''Parameters'''
| bgcolor="#FFFFFF" | {{{parameters}}}
|- valign="top"
| style="text-align: right;" | '''
Support (mathematics) Support'''
| bgcolor="#FFFFFF" | {{{support}}}
|- valign="top"
| style="text-align: right;" | '''{{Probability distribution/link {{{type}}}}}'''
| bgcolor="#FFFFFF" | {{{pdf}}}
|- valign="top"
| style="text-align: right;" | '''
cumulative distribution function cdf'''
| bgcolor="#FFFFFF" | {{{cdf}}}
|- valign="top"
| style="text-align: right;" | '''
expected value Mean'''
| bgcolor="#FFFFFF" | {{{mean}}}
|- valign="top"
| style="text-align: right;" | '''
Median'''
| bgcolor="#FFFFFF" | {{{median}}}
|- valign="top"
| style="text-align: right;" | '''
Mode (statistics) Mode'''
| bgcolor="#FFFFFF" | {{{mode}}}
|- valign="top"
| style="text-align: right;" | '''
Variance'''
| bgcolor="#FFFFFF" | {{{variance}}}
|- valign="top"
| style="text-align: right;" | '''
Skewness'''
| bgcolor="#FFFFFF" | {{{skewness}}}
|- valign="top"
| style="text-align: right;" | '''
Kurtosis'''
| bgcolor="#FFFFFF" | {{{kurtosis}}}
|- valign="top"
| style="text-align: right;" | '''
Information entropy Entropy'''
| bgcolor="#FFFFFF" | {{{entropy}}}
|- valign="top"
| style="text-align: right;" | '''
moment-generating function mgf'''
| bgcolor="#FFFFFF" | {{{mgf}}}
|- valign="top"
| style="text-align: right;" | '''
Characteristic function (probability theory) Char. func.'''
| bgcolor="#FFFFFF" | {{{char}}}
|}
see
Probability distribution
*** Shopping-Tip: Probability Distribution
