# Bernoulli distribution

Probability distribution
name =Bernoulli
type =mass
pdf_

cdf_

parameters =$1>p>0, pinR$
support =$k=\left\{0,1\right\},$
pdf =
cdf =
mean =$p,$
median =N/A
mode =
variance =$pq,$
skewness =$frac\left\{q-p\right\}\left\{sqrt\left\{pq$

kurtosis =$frac\left\{6p^2-6p+1\right\}\left\{p\left(1-p\right)\right\}$
entropy =$-qln\left(q\right)-pln\left(p\right),$
mgf =$q+pe^t,$
char =$q+pe^\left\{it\right\},$

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:

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

The probability mass function "f" of this distribution is

:

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

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

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 (i.i.d.) random variables, all Bernoulli distributed with success probability p, then $Y = sum_\left\{k=1\right\}^n X_k sim mathrm\left\{Binomial\right\}\left(n,p\right)$ (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

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### Look at other dictionaries:

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