# Bernoulli distribution

 Parameters Probability mass function Cumulative distribution function $p>0\,$ (real) $k=\{0,1\}\,$ $ \begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$  $ \begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix}$  $p\,$ N/A $\begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}$ $pq\,$ $\frac{q-p}{\sqrt{pq {{{kurtosis}}} {{{entropy}}} {{{mgf}}} {{{char}}}$|
 kurtosis   =$\frac{6p^2-6p+1}{p(1-p)}$|
entropy    =$-q\ln(q)-p\ln(p)\,$|
mgf        =$q+pe^t\,$|
char       =$q+pe^{it}\,$|


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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 $p$ 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

• 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).
• 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.