Bernoulli distribution

Probability distribution
name =Bernoulli
type =mass
pdf_

cdf_

parameters =$$1>p>0, pinR
support =$$k={0,1},
pdf =$$egin{matrix} q=(1-p) & mbox{for }k=0 \p~~ & mbox{for }k=1 end{matrix}
cdf =$$egin{matrix} 0 & mbox{for }k<0 \q & mbox{for }0leq k<1\1 & mbox{for }kgeq 1 end{matrix}
mean =$$p,
median =N/A
mode =$$egin{matrix}0 & mbox{if } q > p\0, 1 & mbox{if } q=p\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 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{egin{matrix} p & mbox {if }k=1, \1-p & mbox {if }k=0, \0 & mbox {otherwise.}end{matrix} ight.

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