Probability distribution

This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics). For other uses, see Distribution.

In probability and statistics, a probability distribution is a mathematical function that, stated in simple terms, can be thought of as providing the probability of occurrence of different possible outcomes in an experiment. For instance, if the random variable X is used to denote the outcome of a coin toss ('the experiment'), then the probability distribution of X would take the value 0.5 for ${\text{X=Heads}}$ , and 0.5 for ${\text{X=Tails}}$ .

In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. Examples of random phenomena can include the results of an experiment or survey. A probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a higher-dimensional vector space, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be $\{{\text{Heads}},{\text{Tails}}\}$ .

Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenario where the set of possible outcomes is discrete, such as in a coin toss or a flip of a dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g., real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution represents a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution whose sample space is the set of real numbers is called univariate, while a distribution whose sample space is a vector space is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a list of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

Introduction

The probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.

To define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function assigning a probability to each possible outcome: for example, when throwing a fair dice, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is $\operatorname {Prob} (2)+\operatorname {Prob} (4)+\operatorname {Prob} (6)=1/6+1/6+1/6=1/2.$

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of our instruments.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. On the other hand, the cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists.

The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.

Terminology

As probability theory is used in quite diverse applications, terminology is not uniform and sometimes confusing. The following terms are used for non-cumulative probability distribution functions:

Probability mass, Probability mass function, p.m.f.: for discrete random variables.
Categorical distribution: for discrete random variables with a finite set of values.
Probability density, Probability density function, p.d.f.: most often reserved for continuous random variables.

The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:

Probability distribution function: continuous or discrete, non-cumulative or cumulative.
Probability function: even more ambiguous, can mean any of the above or other things.

Finally,

Probability distribution: sometimes the same as probability distribution function, but usually refers to the more complete assignment of probabilities to all measurable subsets of outcomes (i.e. the corresponding probability measure), not just to specific outcomes or ranges of outcomes.

Basic terms

Mode: for a discrete random variable, the value with highest probability (the location at which the probability mass function has its peak); for a continuous random variable, the location at which the probability density function has its peak.
Support: the smallest closed set whose complement has probability zero.
Head: the range of values where the pmf or pdf is relatively high.
Tail: the complement of the head within the support; the large set of values where the pmf or pdf is relatively low.
Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
Median: the value such that the set of values less than the median has a probability of one-half.
Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
Standard deviation: the square root of the variance, and hence another measure of dispersion.
Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value is a mirror image of the portion to its right.
Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.
Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

Cumulative distribution function

Because a probability distribution Pr on the real line is determined by the probability of a scalar random variable X being in a half-open interval (-∞, x], the probability distribution is completely characterized by its cumulative distribution function:

F(x)=\Pr[X\leq x]\qquad {\text{ for all }}x\in \mathbb {R} .

Discrete probability distribution

The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.

The cdf of a discrete probability distribution, ...

... of a continuous probability distribution, ...

... of a distribution which has both a continuous part and a discrete part.

A discrete probability distribution is a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, and X is called a discrete random variable, if

\sum _{u}\Pr(X=u)=1

as u runs through the set of all possible values of X. A discrete random variable can assume only a finite or countably infinite number of values. For the number of potential values to be countably infinite, even though their probabilities sum to 1, the probabilities have to decline to zero fast enough. For example, if $\Pr(X=n)={\tfrac {1}{2^{n}}}$ for n = 1, 2, ..., we have the sum of probabilities 1/2 + 1/4 + 1/8 + ... = 1.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

Measure theoretic formulation

A measurable function $X\colon A\to B$ between a probability space $(A,{\mathcal {A}},P)$ and a measurable space $(B,{\mathcal {B}})$ is called a discrete random variable provided its image is a countable set and the pre-image of singleton sets are measurable, i.e., $X^{-1}(b)\in {\mathcal {A}}$ for all $b\in B$ . The latter requirement induces a probability mass function $f_{X}\colon X(A)\to \mathbb {R}$ via $f_{X}(b):=P(X^{-1}(b))$ . Since the pre-images of disjoint sets are disjoint

\sum _{b\in X(A)}f_{X}(b)=\sum _{b\in X(A)}P(X^{-1}(b))=P\left(\bigcup _{b\in X(A)}X^{-1}(b)\right)=P(A)=1.

This recovers the definition given above.

Cumulative density

Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. The points where jumps occur are precisely the values which the random variable may take.

Delta-function representation

Consequently, a discrete probability distribution is often represented as a generalized probability density function involving Dirac delta functions, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.

Indicator-function representation

For a discrete random variable X, let u₀, u₁, ... be the values it can take with non-zero probability. Denote

\Omega _{i}=X^{-1}(u_{i})=\{\omega :X(\omega )=u_{i}\},\,i=0,1,2,\dots

These are disjoint sets, and by formula (1)

\Pr \left(\bigcup _{i}\Omega _{i}\right)=\sum _{i}\Pr(\Omega _{i})=\sum _{i}\Pr(X=u_{i})=1.

It follows that the probability that X takes any value except for u₀, u₁, ... is zero, and thus one can write X as

X(\omega )=\sum _{i}u_{i}1_{\Omega _{i}}(\omega )

except on a set of probability zero, where $1_{A}$ is the indicator function of A. This may serve as an alternative definition of discrete random variables.

Continuous probability distribution

Some properties

The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
Probability distributions are not a vector space—they are not closed under linear combinations, as these do not preserve non-negativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures).

Kolmogorov definition

Main articles: Probability space and Probability measure

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function X from a probability space $\scriptstyle (\Omega ,{\mathcal {F}},\operatorname {P} )$ to measurable space $\scriptstyle ({\mathcal {X}},{\mathcal {A}})$ . A probability distribution of X is the pushforward measure X_*P of X , which is a probability measure on $\scriptstyle ({\mathcal {X}},{\mathcal {A}})$ satisfying X_*P = PX⁻¹.

Random number generation

Main article: Pseudo-random number sampling

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of pseudo-random numbers that are distributed in a given way. Most algorithms are based on a pseudorandom number generator that produces numbers X that are uniformly distributed in the interval [0,1]. These random variates X are then transformed via some algorithm to create a new random variate having the required probability distribution.

Applications

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics many processes are described probabilistically,from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

As a more specific example of an application, the cache language models and other statistical language models used in natural language processing to assign probabilities to the occurrence of particular words and word sequences do so by means of probability distributions.

Common probability distributions

Main article: List of probability distributions

The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)

Note also that all of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.

Related to real-valued quantities that grow linearly (e.g. errors, offsets)

Normal distribution (Gaussian distribution), for a single such quantity; the most common continuous distribution

Related to positive real-valued quantities that grow exponentially (e.g. prices, incomes, populations)

Log-normal distribution, for a single such quantity whose log is normally distributed
Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical power law distribution

Related to real-valued quantities that are assumed to be uniformly distributed over a (possibly unknown) region

Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die)
Continuous uniform distribution, for continuously distributed values

Related to Bernoulli trials (yes/no events, with a given probability)

Basic distributions:
- Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
- Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences
- Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
- Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special case of the negative binomial distribution
Related to sampling schemes over a finite population:
- Hypergeometric distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, using sampling without replacement
- Beta-binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed number of total occurrences, sampling using a Polya urn scheme (in some sense, the "opposite" of sampling without replacement)

Related to categorical outcomes (events with K possible outcomes, with a given probability for each outcome)

Categorical distribution, for a single categorical outcome (e.g. yes/no/maybe in a survey); a generalization of the Bernoulli distribution
Multinomial distribution, for the number of each type of categorical outcome, given a fixed number of total outcomes; a generalization of the binomial distribution
Multivariate hypergeometric distribution, similar to the multinomial distribution, but using sampling without replacement; a generalization of the hypergeometric distribution

Related to events in a Poisson process (events that occur independently with a given rate)

Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time
Exponential distribution, for the time before the next Poisson-type event occurs
Gamma distribution, for the time before the next k Poisson-type events occur

Related to the absolute values of vectors with normally distributed components

Rayleigh distribution, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
Rice distribution, a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in Rician fading of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.

Related to normally distributed quantities operated with sum of squares (for hypothesis testing)

Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test)
Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (see Student's t-test)
F-distribution, the distribution of the ratio of two scaled chi squared variables; useful e.g. for inferences that involve comparing variances or involving R-squared (the squared correlation coefficient)

Useful as conjugate prior distributions in Bayesian inference

Main article: Conjugate prior

Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution and binomial distribution
Gamma distribution, for a non-negative scaling parameter; conjugate to the rate parameter of a Poisson distribution or exponential distribution, the precision (inverse variance) of a normal distribution, etc.
Dirichlet distribution, for a vector of probabilities that must sum to 1; conjugate to the categorical distribution and multinomial distribution; generalization of the beta distribution
Wishart distribution, for a symmetric non-negative definite matrix; conjugate to the inverse of the covariance matrix of a multivariate normal distribution; generalization of the gamma distribution

References

B. S. Everitt: The Cambridge Dictionary of Statistics, Cambridge University Press, Cambridge (3rd edition, 2006). ISBN 0-521-69027-7
Bishop: Pattern Recognition and Machine Learning, Springer, ISBN 0-387-31073-8
den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", Physica Medica,

External links

Wikimedia Commons has media related to Probability distribution.

Hazewinkel, Michiel, ed. (2001), "Probability distribution", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Theory of probability distributions

probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function

raw moment central moment mean variance standard deviation skewness kurtosis L-moment

moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic Median Mode

Dispersion	Variance Standard deviation Coefficient of variation Percentile Range Interquartile range

Shape	Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Sample size determination Missing data

Survey methodology	Sampling Standard error stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in

Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife

Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F

Goodness of fit	Chi-squared Kolmogorov–Smirnov Anderson–Darling Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC

Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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Probability distribution

Introduction

Terminology

Basic terms

Cumulative distribution function

Discrete probability distribution

Measure theoretic formulation

Cumulative density

Delta-function representation

Indicator-function representation

Continuous probability distribution

Some properties

Kolmogorov definition

Random number generation

Applications

Common probability distributions

Related to real-valued quantities that grow linearly (e.g. errors, offsets)

Related to positive real-valued quantities that grow exponentially (e.g. prices, incomes, populations)

Related to real-valued quantities that are assumed to be uniformly distributed over a (possibly unknown) region

Related to Bernoulli trials (yes/no events, with a given probability)

Related to categorical outcomes (events with K possible outcomes, with a given probability for each outcome)

Related to events in a Poisson process (events that occur independently with a given rate)

Related to the absolute values of vectors with normally distributed components

Related to normally distributed quantities operated with sum of squares (for hypothesis testing)

Useful as conjugate prior distributions in Bayesian inference

See also

References

External links