Bernoulli distribution

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Short description: Probability distribution modeling a coin toss which need not be fair
Bernoulli distribution
Probability mass function
Funzione di densità di una variabile casuale normale

Three examples of Bernoulli distribution:

  [math]\displaystyle{ P(x=0) = 0{.}2 }[/math] and [math]\displaystyle{ P(x=1) = 0{.}8 }[/math]
  [math]\displaystyle{ P(x=0) = 0{.}8 }[/math] and [math]\displaystyle{ P(x=1) = 0{.}2 }[/math]
  [math]\displaystyle{ P(x=0) = 0{.}5 }[/math] and [math]\displaystyle{ P(x=1) = 0{.}5 }[/math]

[math]\displaystyle{ 0 \leq p \leq 1 }[/math]

[math]\displaystyle{ q = 1 - p }[/math]
Support [math]\displaystyle{ k \in \{0,1\} }[/math]
pmf [math]\displaystyle{ \begin{cases} q=1-p & \text{if }k=0 \\ p & \text{if }k=1 \end{cases} }[/math]
CDF [math]\displaystyle{ \begin{cases} 0 & \text{if } k \lt 0 \\ 1 - p & \text{if } 0 \leq k \lt 1 \\ 1 & \text{if } k \geq 1 \end{cases} }[/math]
Mean [math]\displaystyle{ p }[/math]
Median [math]\displaystyle{ \begin{cases} 0 & \text{if } p \lt 1/2\\ \left[0, 1\right] & \text{if } p = 1/2\\ 1 & \text{if } p \gt 1/2 \end{cases} }[/math]
Mode [math]\displaystyle{ \begin{cases} 0 & \text{if } p \lt 1/2\\ 0, 1 & \text{if } p = 1/2\\ 1 & \text{if } p \gt 1/2 \end{cases} }[/math]
Variance [math]\displaystyle{ p(1-p) = pq }[/math]
Skewness [math]\displaystyle{ \frac{q - p}{\sqrt{pq}} }[/math]
Kurtosis [math]\displaystyle{ \frac{1 - 6pq}{pq} }[/math]
Entropy [math]\displaystyle{ -q\ln q - p\ln p }[/math]
MGF [math]\displaystyle{ q+pe^t }[/math]
CF [math]\displaystyle{ q+pe^{it} }[/math]
PGF [math]\displaystyle{ q+pz }[/math]
Fisher information [math]\displaystyle{ \frac{1}{pq} }[/math]

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 [math]\displaystyle{ p }[/math] and the value 0 with probability [math]\displaystyle{ q = 1-p }[/math]. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit 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", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have [math]\displaystyle{ p \neq 1/2. }[/math]

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. [2]


If [math]\displaystyle{ X }[/math] is a random variable with a Bernoulli distribution, then:

[math]\displaystyle{ \Pr(X=1) = p = 1 - \Pr(X=0) = 1 - q. }[/math]

The probability mass function [math]\displaystyle{ f }[/math] of this distribution, over possible outcomes k, is

[math]\displaystyle{ f(k;p) = \begin{cases} p & \text{if }k=1, \\ q = 1-p & \text {if } k = 0. \end{cases} }[/math][3]

This can also be expressed as

[math]\displaystyle{ f(k;p) = p^k (1-p)^{1-k} \quad \text{for } k\in\{0,1\} }[/math]

or as

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

The Bernoulli distribution is a special case of the binomial distribution with [math]\displaystyle{ n = 1. }[/math][4]

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

The Bernoulli distributions for [math]\displaystyle{ 0 \le p \le 1 }[/math] form an exponential family.

The maximum likelihood estimator of [math]\displaystyle{ p }[/math] based on a random sample is the sample mean.

The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.


The expected value of a Bernoulli random variable [math]\displaystyle{ X }[/math] is

[math]\displaystyle{ \operatorname{E}[X]=p }[/math]

This is due to the fact that for a Bernoulli distributed random variable [math]\displaystyle{ X }[/math] with [math]\displaystyle{ \Pr(X=1)=p }[/math] and [math]\displaystyle{ \Pr(X=0)=q }[/math] we find

[math]\displaystyle{ \operatorname{E}[X] = \Pr(X=1)\cdot 1 + \Pr(X=0)\cdot 0 = p \cdot 1 + q\cdot 0 = p. }[/math][3]


The variance of a Bernoulli distributed [math]\displaystyle{ X }[/math] is

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

We first find

[math]\displaystyle{ \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 = \operatorname{E}[X] }[/math]

From this follows

[math]\displaystyle{ \operatorname{Var}[X] = \operatorname{E}[X^2]-\operatorname{E}[X]^2 = \operatorname{E}[X]-\operatorname{E}[X]^2 = p-p^2 = p(1-p) = pq }[/math][3]

With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside [math]\displaystyle{ [0,1/4] }[/math].


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

[math]\displaystyle{ \begin{align} \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{align} }[/math]

Higher moments and cumulants

The raw moments are all equal due to the fact that [math]\displaystyle{ 1^k=1 }[/math] and [math]\displaystyle{ 0^k=0 }[/math].

[math]\displaystyle{ \operatorname{E}[X^k] = \Pr(X=1)\cdot 1^k + \Pr(X=0)\cdot 0^k = p \cdot 1 + q\cdot 0 = p = \operatorname{E}[X]. }[/math]

The central moment of order [math]\displaystyle{ k }[/math] is given by

[math]\displaystyle{ \mu_k =(1-p)(-p)^k +p(1-p)^k. }[/math]

The first six central moments are

[math]\displaystyle{ \begin{align} \mu_1 &= 0, \\ \mu_2 &= p(1-p), \\ \mu_3 &= p(1-p)(1-2p), \\ \mu_4 &= p(1-p)(1-3p(1-p)), \\ \mu_5 &= p(1-p)(1-2p)(1-2p(1-p)), \\ \mu_6 &= p(1-p)(1-5p(1-p)(1-p(1-p))). \end{align} }[/math]

The higher central moments can be expressed more compactly in terms of [math]\displaystyle{ \mu_2 }[/math] and [math]\displaystyle{ \mu_3 }[/math]

[math]\displaystyle{ \begin{align} \mu_4 &= \mu_2 (1-3\mu_2 ), \\ \mu_5 &= \mu_3 (1-2\mu_2 ), \\ \mu_6 &= \mu_2 (1-5\mu_2 (1-\mu_2 )). \end{align} }[/math]

The first six cumulants are

[math]\displaystyle{ \begin{align} \kappa_1 &= p, \\ \kappa_2 &= \mu_2 , \\ \kappa_3 &= \mu_3 , \\ \kappa_4 &= \mu_2 (1-6\mu_2 ), \\ \kappa_5 &= \mu_3 (1-12\mu_2 ), \\ \kappa_6 &= \mu_2 (1-30\mu_2 (1-4\mu_2 )). \end{align} }[/math]

Related distributions

  • If [math]\displaystyle{ X_1,\dots,X_n }[/math] 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:
    [math]\displaystyle{ \sum_{k=1}^n X_k \sim \operatorname{B}(n,p) }[/math] (binomial distribution).[3]
The Bernoulli distribution is simply [math]\displaystyle{ \operatorname{B}(1, p) }[/math], also written as [math]\displaystyle{ \mathrm{Bernoulli} (p). }[/math]

See also


  1. Uspensky, James Victor (1937). Introduction to Mathematical Probability. New York: McGraw-Hill. p. 45. OCLC 996937. 
  2. Dekking, Frederik; Kraaikamp, Cornelis; Lopuhaä, Hendrik; Meester, Ludolf (9 October 2010). A Modern Introduction to Probability and Statistics (1 ed.). Springer London. pp. 43–48. ISBN 9781849969529. 
  3. 3.0 3.1 3.2 3.3 Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829. 
  4. McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. Section 4.2.2. ISBN 0-412-31760-5. 
  5. Orloff, Jeremy; Bloom, Jonathan. "Conjugate priors: Beta and normal". 

Further reading

  • Johnson, N. L.; Kotz, S.; Kemp, A. (1993). Univariate Discrete Distributions (2nd ed.). Wiley. ISBN 0-471-54897-9. 
  • Peatman, John G. (1963). Introduction to Applied Statistics. New York: Harper & Row. pp. 162–171. 

External links