Bernoulli distribution

Bernoulli
Parameters	;
Support
pmf
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF
PGF
Fisher information

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,^[1] is the discrete probability distribution of a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q=1-p,$ that is, the probability distribution of any single experiment that asks a yes–no question; the question results in a boolean-valued outcome, a single bit of information whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails" (or vice versa), respectively, and p would be the probability of the coin landing on heads or tails, respectively. In particular, unfair coins would have $p\neq 1/2.$

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1.

Properties of the Bernoulli distribution

If $X$ is a random variable with this distribution, then:

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

The probability mass function $f$ of this distribution, over possible outcomes k, is

f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}

^[2]

This can also be expressed as

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

or as

f(k;p)=pk+(1-p)(1-k)\quad {\text{for }}k\in \{0,1\}.

The Bernoulli distribution is a special case of the binomial distribution with $n=1.$ ^[3]

The kurtosis goes to infinity for high and low values of $p,$ but for $p=1/2$ the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2.

The Bernoulli distributions for $0\leq p\leq 1$ form an exponential family.

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

Mean

The expected value of a Bernoulli random variable $X$ is

\operatorname {E} \left(X\right)=p

This is due to the fact that for a Bernoulli distributed random variable $X$ with $\Pr(X=1)=p$ and $\Pr(X=0)=q$ we find

\operatorname {E} [X]=\Pr(X=1)\cdot 1+\Pr(X=0)\cdot 0=p\cdot 1+q\cdot 0=p.

^[2]

Variance

The variance of a Bernoulli distributed $X$ is

\operatorname {Var} [X]=pq=p(1-p)

We first find

\operatorname {E} [X^{2}]=\Pr(X=1)\cdot 1^{2}+\Pr(X=0)\cdot 0^{2}=p\cdot 1^{2}+q\cdot 0^{2}=p

From this follows

\operatorname {Var} [X]=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=p-p^{2}=p(1-p)=pq

^[2]

Skewness

The skewness is ${\frac {q-p}{\sqrt {pq}}}={\frac {1-2p}{\sqrt {pq}}}$ . When we take the standardized Bernoulli distributed random variable ${\frac {X-\operatorname {E} [X]}{\sqrt {\operatorname {Var} [X]}}}$ we find that this random variable attains ${\frac {q}{\sqrt {pq}}}$ with probability $p$ and attains $-{\frac {p}{\sqrt {pq}}}$ with probability $q$ . Thus we get

{\begin{aligned}\gamma _{1}&=\operatorname {E} \left[\left({\frac {X-\operatorname {E} [X]}{\sqrt {\operatorname {Var} [X]}}}\right)^{3}\right]\\&=p\cdot \left({\frac {q}{\sqrt {pq}}}\right)^{3}+q\cdot \left(-{\frac {p}{\sqrt {pq}}}\right)^{3}\\&={\frac {1}{{\sqrt {pq}}^{3}}}\left(pq^{3}-qp^{3}\right)\\&={\frac {pq}{{\sqrt {pq}}^{3}}}(q-p)\\&={\frac {q-p}{\sqrt {pq}}}\end{aligned}}

Related distributions

If $X_{1},\dots ,X_{n}$ are independent, identically distributed (i.i.d.) random variables, all Bernoulli trials with success probability p, then their sum is distributed according to a binomial distribution with parameters n and p:
$\sum _{k=1}^{n}X_{k}\sim \operatorname {B} (n,p)$ (binomial distribution).^[2]

The Bernoulli distribution is simply

\operatorname {B} (1,p)

, also written as

{\textstyle \mathrm {Bernoulli} (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 models the number of independent and identical Bernoulli trials needed to get one success.
If ${\textstyle Y\sim \mathrm {Bernoulli} \left({\frac {1}{2}}\right)}$ , then ${\textstyle 2Y-1}$ has a Rademacher distribution.

Notes

↑ James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
1 2 3 4 Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
↑ McCullagh and Nelder (1989), Section 4.2.2.

References

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

Hazewinkel, Michiel, ed. (2001) [1994], "Binomial distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Bernoulli Distribution". MathWorld.
Interactive graphic: Univariate Distribution Relationships

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[1] James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45

[:0-2] 1 2 3 4 Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.

[McCullagh1989Ch422-3] McCullagh and Nelder (1989), Section 4.2.2.

Probability distributions
List
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 Borel 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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-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 asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Parameters	$0 \leq p \leq 1$ $q=1-p$
Support	$k\in \{0,1\}$
pmf	${\begin{cases}q=1-p&{\text{if }}k=0\\p&{\text{if }}k=1\end{cases}}$
CDF	${\begin{cases}0&{\text{if }}k<0\\1-p&{\text{if }}0\leq k<1\\1&{\text{if }}k\geq 1\end{cases}}$
Mean	$p$
Median	${\begin{cases}0&{\text{if }}p<1/2\\1/2&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}$
Mode	${\begin{cases}0&{\text{if }}p<1/2\\0,1&{\text{if }}p=1/2\\1&{\text{if }}p>1/2\end{cases}}$
Variance	$p(1-p)=pq$
Skewness	${\frac {1-2p}{\sqrt {pq}}}$
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	${\frac {1}{pq}}$

Bernoulli distribution

Properties of the Bernoulli distribution

Mean

Variance

Skewness

Related distributions

See also

Notes

References

External links