# Bernoulli distribution

Probability distribution

name =Bernoulli

type =mass

pdf_

cdf_

parameters =$1>p>0,\; pinR$

support =$k=\{0,1\},$

pdf =$egin\{matrix\}\; q=(1-p)\; mbox\{for\; \}k=0\; \backslash p~~\; mbox\{for\; \}k=1\; end\{matrix\}$

cdf =$egin\{matrix\}\; 0\; mbox\{for\; \}k0\; \backslash q\; mbox\{for\; \}0leq\; k1\backslash 1\; mbox\{for\; \}kgeq\; 1\; end\{matrix\}$

mean =$p,$

median =N/A

mode =$egin\{matrix\}0\; mbox\{if\; \}\; q\; p\backslash 0,\; 1\; mbox\{if\; \}\; q=p\backslash 1\; mbox\{if\; \}\; q\; pend\{matrix\}$

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 andstatistics , the**Bernoulli distribution**, named after Swiss scientistJakob Bernoulli , is a discreteprobability 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:$f(k;p)\; =\; left\{egin\{matrix\}\; p\; mbox\; \{if\; \}k=1,\; \backslash 1-p\; mbox\; \{if\; \}k=0,\; \backslash 0\; mbox\; \{otherwise.\}end\{matrix\}\; ight.$

The

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

*TheCategorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.

*TheBeta distribution is theconjugate prior of the Bernoulli distribution.**ee also***

Bernoulli trial

*Bernoulli process

*Bernoulli sampling

*Sample size

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Bernoulli distribution**— noun a theoretical distribution of the number of successes in a finite set of independent trials with a constant probability of success • Syn: ↑binomial distribution • Topics: ↑statistics • Hypernyms: ↑distribution, ↑statistical distribution * *… … Useful english dictionary**Bernoulli distribution**— Statistics. See binomial distribution. [named after Jakob BERNOULLI] * * * … Universalium**Bernoulli distribution**— noun A discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability … Wiktionary**Bernoulli distribution**— binomial d … Medical dictionary**Bernoulli distribution theorem trial**— Ber·noul·li distribution, theorem, trial (bər nooґle) (bār noo eґ) [Jakob Bernoulli, Swiss mathematician, 1654â€“1705] see under distribution, theorem, and trial … Medical dictionary**Bernoulli**— can refer to: *any one or more of the Bernoulli family of Swiss mathematicians in the eighteenth century, including: ** Daniel Bernoulli (1700–1782), developer of Bernoulli s principle ** Jakob Bernoulli (1654–1705), also known as Jean or Jacques … Wikipedia**Bernoulli trial**— In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, success and failure .In practice it refers to a single experiment which can have one of two… … Wikipedia**Bernoulli sampling**— In the theory of finite population sampling, Bernoulli sampling is a sampling process where each element of the population that is sampled is subjected to an independent Bernoulli trial which determines whether the element becomes part of the… … Wikipedia**Bernoulli**— Famille Bernoulli Les Bernoulli qui se sont illustrés dans les mathématiques et la physique, sont issus de Nicolas Bernoulli (1623 1708), descendant d une famille ayant émigré d Anvers à Bâle à la fin du XVIe siècle. Les représentants les… … Wikipédia en Français**Bernoulli family**— The Bernoullis were a family of traders and scholars from Basel, Switzerland. The founder of the family, Leon Bernoulli, immigrated to Basel from Antwerp in the Flanders in the 16th century.The Bernoulli family has produced many notable artists… … Wikipedia