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Bernoulli
Probability mass function
Cumulative distribution function
Parameters $ p>0\, $ (real)
$ q\equiv 1-p\, $
Support $ k=\{0,1\}\, $
Template:Probability distribution/link mass $ \begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix} $
cdf $ \begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0<k<1\\1 & \mbox{for }k>1 \end{matrix} $
Mean $ p\, $
Median N/A
Mode $ \textrm{max}(p,q)\, $
Variance $ pq\, $
Skewness $ \frac{q-p}{\sqrt{pq}} $
Kurtosis $ \frac{6p^2-6p+1}{p(1-p)} $
Entropy $ -q\ln(q)-p\ln(p)\, $
mgf $ q+pe^t\, $
Char. func. $ 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) = 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 Bernoulli distribution is a member of the exponential family.

Related distributionsEdit

  • 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).

See alsoEdit

fr:Distribution de Bernoullihe:התפלגות ברנולי nl:Bernoulli-verdelingfi:Bernoullin jakauma zh:伯努利分布

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