Dictionary - Definition of Bernoulli distribution

Definition: Bernoulli distribution


Part of Speech	Definition
Noun	1. A theoretical distribution of the number of successes in a finite set of independent trials with a constant probability of success.[Wordnet].
Source: WordNet 3.0 Copyright © 2006 by Princeton University. All rights reserved.		Top

Extended Definition: Bernoulli distribution

Bernoulli distribution

Bernoulli
Probability mass function
Cumulative distribution function
Parameters	$1>p>0, p\in\R$
Support	$k=\{0,1\}\,$
Probability mass function (pmf)	$\begin{matrix} q=(1-p) & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$
Cumulative distribution function (cdf)	$\begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix}$
Mean	$p\,$
Median	N/A
Mode	$\begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}$
Variance	$pq\,$
Skewness	$\frac{q-p}{\sqrt{pq}}$
Excess kurtosis	$\frac{6p2-6p+1}{p(1-p)}$
Entropy	$-q\ln(q)-p\ln(p)\,$
Moment-generating function (mgf)	$q+pet\,$
Characteristic function	$q+pe{it}\,$

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability $p$ and value 0 with failure probability $q = 1 - p$ . So if X is a random variable with this distribution, we have:

$\Pr(X=1) = 1 - \Pr(X=0) = 1 - q = p.\!$

The probability mass function f of this distribution is

$f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\ 1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.$

The expected value of a Bernoulli random variable X is $E\left(X\right)=p$ , and its variance is

$\textrm{var}\left(X\right)=p\left(1-p\right).\,$

The kurtosis goes to infinity for high and low values of p, but for $p = 1 / 2$ the Bernoulli distribution has a lower kurtosis than any other probability distribution, namely -2.

The Bernoulli distribution is a member of the exponential family.

Related distributions

If $X_1,\dots,X_n$ are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then $Y = \sum_{k=1}n X_k \sim \mathrm{Binomial}(n,p)$ (binomial distribution).
The Categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
The Beta distribution is the conjugate prior of the Bernoulli distribution.

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Statistics

Descriptive statistics	Mean (Arithmetic, Geometric, Harmonic) - Median - Mode - Range - Variance - Standard deviation

Inferential statistics	Hypothesis testing - Significance - Power - Null hypothesis/Alternative hypothesis - Error - Z-test - Student's t-test - Maximum likelihood - Standard score/Z score - P-value - Analysis of variance - Meta-analysis

Survival analysis	Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models

Correlation	Confounding variable - Pearson product-moment correlation coefficient - Rank correlation (Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient)

Regression analysis	Linear regression - Nonlinear regression - Logistic regression

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Source: adapted by the editor from Wikipedia, the free encyclopedia; from the article "Bernoulli distribution". Image Credit.
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Translations: Bernoulli distribution

Language	Translations (or nearest inflections or synonyms, in parentheses)
Íslenska	tvíkostadreifing (Bernouilli distribution, Bernoulli distribution). Additional references: Íslenska, Iceland, Bernoulli distribution. (volunteer & more translations)
Central Danish	binominalfordeling (Bernoulli distribution, binomial distribution), Bernoullifordeling (Bernoulli distribution). Additional references: Central Danish, Denmark, Germany, Bernoulli distribution. (volunteer & more translations)
Chinese Simplified	伯怒里多项式 (Bernoulli distribution). Additional references: Chinese Simplified, China, Brunei, Bernoulli distribution. (volunteer & more translations)
Danish	binominalfordeling (Bernoulli distribution, binomial distribution), Bernoullifordeling (Bernoulli distribution). Additional references: Danish, Denmark, Germany, Bernoulli distribution. (volunteer & more translations)
Dansk	binominalfordeling (Bernoulli distribution, binomial distribution), Bernoullifordeling (Bernoulli distribution). Additional references: Dansk, Denmark, Germany, Bernoulli distribution. (volunteer & more translations)
Deutsch	Bernoullische Verteilung (Bernoulli distribution). Additional references: Deutsch, Germany, Austria, Bernoulli distribution. (volunteer & more translations)
Dutch	Bernoulli-verdeling (Bernoulli distribution), binomiaalverdeling (Bernoulli distribution, binomial distribution), Bernoulliverdeling (Bernoulli distribution, binomial distribution), Bernoulli verdeling (Bernoulli distribution). Additional references: Dutch, Netherlands, Aruba, Bernoulli distribution. (volunteer & more translations)
Euskara	Bernoulli-ren banaketa (Bernoulli distribution). Additional references: Euskara, Spain, Bernoulli distribution. (volunteer & more translations)
Finnish	Bernoullin jakauma (Bernoulli distribution), binomijakauma (binomial distribution, Bernoulli distribution). Additional references: Finnish, Finland, Russia (Europe), Bernoulli distribution. (volunteer & more translations)
Français	distribution binomiale (binomial distribution, Bernoulli distribution, Bernouilli distribution, point binomial), Distribution de Bernoulli (Bernoulli distribution), série binomiale (binomial series, Bernoulli distribution, binomial distribution). Additional references: Français, France, Algeria, Bernoulli distribution. (volunteer & more translations)
French	distribution binomiale (binomial distribution, Bernoulli distribution, Bernouilli distribution, point binomial), Distribution de Bernoulli (Bernoulli distribution), série binomiale (binomial series, Bernoulli distribution, binomial distribution). Additional references: French, France, Algeria, Bernoulli distribution. (volunteer & more translations)
German	Bernoullische Verteilung (Bernoulli distribution). Additional references: German, Germany, Austria, Bernoulli distribution. (volunteer & more translations)
Greek	διωνυμική κατανομή (Bernoulli distribution, binomial distribution, point binomial), διωνυμική σειρά (Bernoulli distribution, binomial distribution). Additional references: Greek, Greece, Albania, Bernoulli distribution. (volunteer & more translations)
Greek (transliteration)	dhionimiki katanomi (Bernoulli distribution, binomial distribution, point binomial), dhionimiki seira (Bernoulli distribution, binomial distribution). Additional references: Greek, Greece, Albania, Bernoulli distribution. (volunteer & more translations)
High German	Bernoullische Verteilung (Bernoulli distribution). Additional references: High German, Germany, Austria, Bernoulli distribution. (volunteer & more translations)
Hochdeutsch	Bernoullische Verteilung (Bernoulli distribution). Additional references: Hochdeutsch, Germany, Austria, Bernoulli distribution. (volunteer & more translations)
Icelandic	tvíkostadreifing (Bernouilli distribution, Bernoulli distribution). Additional references: Icelandic, Iceland, Bernoulli distribution. (volunteer & more translations)
Italian	distribuzione di Bernoulli (Bernoulli distribution), Variabile casuale Bernoulliana (Bernoulli distribution). Additional references: Italian, Italy, Croatia, Bernoulli distribution. (volunteer & more translations)
Japanese	ベルヌーイ分布 (Bernoulli distribution). Additional references: Japanese, Japan, Taiwan, Bernoulli distribution. (volunteer & more translations)
Polish	rozkład Bernoulliego (Bernoulli distribution). Additional references: Polish, Poland, Czech Republic, Bernoulli distribution. (volunteer & more translations)
Polnisch	rozkład Bernoulliego (Bernoulli distribution). Additional references: Polnisch, Poland, Czech Republic, Bernoulli distribution. (volunteer & more translations)
Polski	rozkład Bernoulliego (Bernoulli distribution). Additional references: Polski, Poland, Czech Republic, Bernoulli distribution. (volunteer & more translations)
Ruotsi	Bernoullifördelning (Bernoulli distribution). Additional references: Ruotsi, Sweden, Finland, Bernoulli distribution. (volunteer & more translations)
Sjaelland	binominalfordeling (Bernoulli distribution, binomial distribution), Bernoullifordeling (Bernoulli distribution). Additional references: Sjaelland, Denmark, Germany, Bernoulli distribution. (volunteer & more translations)
Spanish	distribución de Bernoulli (Bernoulli distribution). Additional references: Spanish, Spain, Mexico, Bernoulli distribution. (volunteer & more translations)
Suomea	Bernoullin jakauma (Bernoulli distribution), binomijakauma (binomial distribution, Bernoulli distribution). Additional references: Suomea, Finland, Russia (Europe), Bernoulli distribution. (volunteer & more translations)
Suomi	Bernoullin jakauma (Bernoulli distribution), binomijakauma (binomial distribution, Bernoulli distribution). Additional references: Suomi, Finland, Russia (Europe), Bernoulli distribution. (volunteer & more translations)
Svenska	Bernoullifördelning (Bernoulli distribution). Additional references: Svenska, Sweden, Finland, Bernoulli distribution. (volunteer & more translations)
Swedish	Bernoullifördelning (Bernoulli distribution). Additional references: Swedish, Sweden, Finland, Bernoulli distribution. (volunteer & more translations)
Source: Eve, based on a combination of meta analysis and graph theory (for near and back translations).		Top

Definition: Bernoulli distribution

Extended Definition: Bernoulli distribution

Bernoulli distribution

Related distributions

See also

Translations: Bernoulli distribution