Bernoulli distribution

Bernoulli
parameters:	$0<p<1, p\in\R$
support:	$k=\{0,1\}\,$
pmf:	$\begin{matrix} q=(1-p) & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$
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:	$\frac{q-p}{\sqrt{pq}}$
ex.kurtosis:	$\frac{6p^2-6p+1}{p(1-p)}$
entropy:	$-q\ln(q)-p\ln(p)\,$
mgf:	$q+pe^t\,$
cf:	$q+pe^{it}\,$

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob 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

$f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\ 1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.$

This can also be expressed as

$f(k;p) = p^k (1-p)^{1-k}\!\quad \text{for }k\in\{0,1\}$ .

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

$\textrm{var}\left(X\right)=p\left(1-p\right).\,$

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 $Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)$ (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.
The Geometric distribution is the number of Bernoulli trials needed to get one success.

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell–Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · noncentral t · normal (Gaussian) · normal-inverse Gaussian · skew normal · slash · stable · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Polya · negative multinomial Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional:Circular Uniform · bivariate von Mises · Kent · univariate von Mises · von Mises–Fisher · Wrapped normal · Wrapped Cauchy · Wrapped Lévy Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

Circular · compound Poisson · elliptical · exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · Tweedie

Some common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • Laplace • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

Bernoulli distribution

Related distributions

See also