Bernoulli distribution
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Template:Probability distribution</math>|
kurtosis =<math>\frac{6p^2-6p+1}{p(1-p)}</math>|
entropy =<math>-q\ln(q)-p\ln(p)\,</math>|
mgf =<math>q+pe^t\,</math>|
char =<math>q+pe^{it}\,</math>|
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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 <math>p</math> and value 0 with failure probability <math>q=1-p</math>. So if X is a random variable with this distribution, we have:
- <math> \Pr(X=1) = 1- \Pr(X=0) = p.\!</math>
The probability mass function f of this distribution is
- <math> f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\
1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.</math>
The expected value of a Bernoulli random variable X is <math>E\left(X\right)=p</math>, and its variance is
- <math>\textrm{var}\left(X\right)=p\left(1-p\right).\,</math>
The Bernoulli distribution is a member of the exponential family.
Related distributions
- If <math>X_1,\dots,X_n</math> are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then <math>Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)</math> (binomial distribution).
See also
fr:Distribution de Bernoulli it:Variabile casuale Bernoulliana he:התפלגות ברנולי nl:Bernoulli-verdeling ja:ベルヌーイ分布 fi:Bernoullin jakauma zh:伯努利分布