Bernoulli distribution
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Template:Probability distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James 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>. The probability mass function of this distribution is
- <math> f_k(p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\
q & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.<math>
The expected value of a Bernoulli random variable is <math>p<math>, and its variance is
- <math>\textrm{var}(k)=pq=p(1-p).\,<math>
The Bernoulli distribution is a member of the exponential family.
Related distributions
- <math>Y \sim \mathrm{Binomial}(n,p)<math> is a binomial distribution if <math>Y = \sum_{k=1}^n X_k<math> where <math>X_k \sim \mathrm{Bernoulli}_k(p)<math> is a Bernoulli distribution.
See also
fi:Bernoullin jakauma it:Variabile casuale Bernoulliana ja:ベルヌーイ分布 zh:伯努利分布