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WordNet:

Bernoulli distribution

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The noun has one meaning:

Meaning #1: a theoretical distribution of the number of successes in a finite set of independent trials with a constant probability of success
  Synonym: binomial distribution


 
 
Wikipedia: Bernoulli distribution
Bernoulli
Probability mass function
Cumulative distribution function
Parameters p>0\, (real)
Support k=\{0,1\}\,
Probability mass function (pmf) \begin{matrix}     q & \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{6p^2-6p+1}{p(1-p)}
Entropy -q\ln(q)-p\ln(p)\,
Moment-generating function (mgf) q+pe^t\,
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

See also


Image:Bvn-small.png Probability distributions []
Univariate Multivariate
Discrete: BenfordBernoullibinomialBoltzmanncategoricalcompound Poissondiscrete phase-typedegenerateGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomialmultivariate Polya
Continuous: BetaBeta primeCauchychi-squareDirac delta functionCoxianErlangexponentialexponential powerFfadingFermi-DiracFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHalf-logisticHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-square (scaled inverse chi-square) • inverse Gaussianinverse gamma (scaled inverse gamma) • KumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speedNakagaminormal (Gaussian)normal-gammanormal inverse GaussianParetoPearsonphase-typepolarraised cosineRayleighrelativistic Breit-WignerRiceshifted GompertzStudent's ttriangulartruncated normaltype-1 Gumbeltype-2 GumbeluniformVariance-GammaVoigtvon MisesWeibullWigner semicircleWilks' lambda DirichletGeneralized Dirichlet distribution . inverse-WishartKentmatrix normalmultivariate normalmultivariate Studentvon Mises-FisherWigner quasiWishart
Miscellaneous: bimodalCantorconditionalequilibriumexponential familyInfinite divisibility (probability)location-scale familymarginalmaximum entropyposteriorpriorquasisamplingsingular



nov:Distributione de Bernoulli


 
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