Bernoulli distribution

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Bernoulli
Parameters	$0<p<1, p\in\R$
Support	$k=\{0,1\}\,$
pmf	$\begin{cases} q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1 \end{cases}$
CDF	$\begin{cases} 0 & \text{for }k<0 \\ q & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1 \end{cases}$
Mean	$p\,$
Median	$\begin{cases} 0 & \text{if } q > p\\ 0.5 & \text{if } q=p\\ 1 & \text{if } q<p \end{cases}$
Mode	$\begin{cases} 0 & \text{if } q > p\\ 0, 1 & \text{if } q=p\\ 1 & \text{if } q < p \end{cases}$
Variance	$p(1-p)\,$
Skewness	$\frac{q-p}{\sqrt{pq}}$
Ex. kurtosis	$\frac{1-6pq}{pq}$
Entropy	$-q\ln(q)-p\ln(p)\,$
MGF	$q+pe^t\,$
CF	$q+pe^{it}\,$
PGF	$q+pz\,$
Fisher information	$\frac{1}{p(1-p)}$

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:

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

A classical example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability $p$ and tails with probability $1-p$ . The experiment is called fair if $p=0.5$ , indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability).

The probability mass function $f$ of this distribution is

$f(k;p) = \begin{cases} p & \text{if }k=1, \\[6pt] 1-p & \text {if }k=0.\end{cases}$

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

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

Bernoulli distribution is a special case of the Binomial distribution with $n = 1$ .^[1]

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 distributions for $0 \le p \le 1$ form an exponential family.

The maximum likelihood estimator of $p$ based on a random sample is the sample mean.

1 Related distributions
2 See also
3 Notes
4 References
5 External links

Related distributions[edit]

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 Bernoulli distribution is simply $\mathrm{Binomial}(1,p)$ .
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.

Notes[edit]

^ McCullagh and Nelder (1989), Section 4.2.2.

References[edit]

McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.

Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9

External links[edit]

Hazewinkel, Michiel, ed. (2001), "Binomial distribution", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Bernoulli Distribution", MathWorld.
Interactive graphic: Univariate Distribution Relationships

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

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

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

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

v t e Some common univariate probability distributions

Continuous	beta Cauchy chi-squared 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

Contents

Related distributions[edit]

See also[edit]

Notes[edit]

References[edit]

External links[edit]