# Bernoulli distribution

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 Probability mass function Cumulative distribution function Parameters $p>0\,$ (real)$q\equiv 1-p\,$ Support $k=\{0,1\}\,$ Template:Probability distribution/link mass $\begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$ cdf $\begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }01 \end{matrix}$ Mean $p\,$ Median N/A Mode $\textrm{max}(p,q)\,$ Variance $pq\,$ Skewness $\frac{q-p}{\sqrt{pq}}$ Kurtosis $\frac{6p^2-6p+1}{p(1-p)}$ Entropy $-q\ln(q)-p\ln(p)\,$ mgf $q+pe^t\,$ Char. func. $q+pe^{it}\,$

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:

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

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 Bernoulli distribution is a member of the exponential family.

## Related distributionsEdit

• 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).