Bernoulli distribution

Bernoulli

Probability mass function
Cumulative distribution function
Parameters	<math>p>0\,</math> (real)
Support	<math>k=\{0,1\}\,</math>
Probability mass function (pmf)	<math> \begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix} </math>
Cumulative distribution function (cdf)	<math> \begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix} </math>
Mean	<math>p\,</math>
Median	N/A
Mode	<math>\begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}</math>
Variance	<math>pq\,</math>
Skewness	<math>\frac{q-p}{\sqrt{pq
Excess kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
Moment-generating function (mgf)	{{{mgf}}}
Characteristic function	{{{char}}}

</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>|

}}

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) = 1 - q = 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 kurtosis goes to infinity for high and low values of p, but for <math>p=1/2</math> 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 <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).
• 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.

See also

• Bernoulli trial
• Bernoulli process
• Sample size

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