# 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 $pq\,$ ${\frac {q-p}{\sqrt {pq}}}$ ${\frac {6p^{2}-6p+1}{p(1-p)}}$ $-q\ln(q)-p\ln(p)\,$ $q+pe^{t}\,$ $q+pe^{it}\,$ 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:

$\Pr(X=1)=1-\Pr(X=0)=1-q=p.\!$ 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. 