Bernoulli distribution

Probability distribution
name =Bernoulli
type =mass
pdf_

cdf_

parameters =$1>p>0,\; pinR$
support =$k=\{0,1\},$
pdf =
cdf =
mean =$p,$
median =N/A
mode =
variance =$pq,$
skewness =$frac\{q-p\}\{sqrt\{pq$
kurtosis =$frac\{6p^2-6p+1\}\{p(1-p)\}$
entropy =$-qln(q)-pln(p),$
mgf =$q+pe^t,$
char =$q+pe^\{it\},$

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

:

The expected value of a Bernoulli random variable "X" is $Eleft(X\; ight)=p$, and its variance is

:$extrm\{var\}left(X\; ight)=pleft(1-p\; ight).,$

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 (i.i.d.) 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.

ee also

*Bernoulli trial
*Bernoulli process
*Bernoulli sampling
*Sample size