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Bernoulli distribution

**Bernoulli**
Probability mass function
Cumulative distribution function
Parameters	$1>p>0\,$ (real)
Support	$k=\{0,1\}\,$
Probability mass function (pmf)	$\begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$
Cumulative distribution function (cdf)	$\begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix}$
Mean	$p\,$
Median	N/A
Mode	$\begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}$
Variance	$pq\,$
Skewness	$\frac{q-p}{\sqrt{pq}}$
Excess kurtosis	$\frac{6p^2-6p+1}{p(1-p)}$
Entropy	$-q\ln(q)-p\ln(p)\,$
Moment-generating function (mgf)	$q+pe^t\,$
Characteristic function	$q+pe^{it}\,$

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:

The probability mass function f of this distribution is

The expected value of a Bernoulli random variable X is , and its variance is

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 are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then (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.