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Bernoulli distribution

Probability distribution modeling a coin toss which need not be fair

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated February 16, 2026
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In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, 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. 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 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.

01Properties

If X is a random variable with a Bernoulli distribution, then:

{\begin{aligned}\Pr(X{=}1)&=p,\\\Pr(X{=}0)&=q=1-p.\end{aligned}}

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}}

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.

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, namely −2, than any other probability distribution.

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.

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

02Mean

The expected value of a Bernoulli random variable X is

\operatorname {E} [X]=p

This is because for a Bernoulli distributed random variable X with \Pr(X{=}1)=p and {\textstyle \Pr(X{=}0)=q we find

{\begin{aligned}\operatorname {E} [X]&=\Pr(X{=}1)\cdot 1+\Pr(X{=}0)\cdot 0\\[1ex]&=p\cdot 1+q\cdot 0\\[1ex]&=p.\end{aligned}}

03Variance

The variance of a Bernoulli distributed X is

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

We first find

{\begin{aligned}\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]\end{aligned}}

From this follows

{\begin{aligned}\operatorname {Var} [X]&=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=\operatorname {E} [X]-\operatorname {E} [X]^{2}\\[1ex]&=p-p^{2}=p(1-p)=pq\end{aligned}}

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

04Skewness

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^{2}-p^{2})\\&={\frac {(1-p)^{2}-p^{2}}{\sqrt {pq}}}\\&={\frac {1-2p}{\sqrt {pq}}}={\frac {q-p}{\sqrt {pq}}}.\end{aligned}}

05Higher moments and cumulants

The raw moments are all equal because 1^{k}=1 and 0^{k}=0.

\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].

The central moment of order k is given by \mu _{k}=(1-p)(-p)^{k}+p(1-p)^{k}. The first six central moments are {\begin{aligned}\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{aligned}} The higher central moments can be expressed more compactly in terms of \mu _{2} and \mu _{3} {\begin{aligned}\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{aligned}} The first six cumulants are {\begin{aligned}\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{aligned}}

06Entropy and Fisher's Information

Entropy

Entropy is a measure of uncertainty or randomness in a probability distribution. For a Bernoulli random variable X with success probability p and failure probability q=1-p, the entropy H(X) is defined as:

{\begin{aligned}H(X)&=\mathbb {E} _{p}\ln {\frac {1}{\Pr(X)}}\\[1ex]&=-\Pr(X{=}0)\ln \Pr(X{=}0)-\Pr(X{=}1)\ln \Pr(X{=}1)\\[1ex]&=-(q\ln q+p\ln p).\end{aligned}}

The entropy is maximized when p=0.5, indicating the highest level of uncertainty when both outcomes are equally likely. The entropy is zero when p=0 or p=1, where one outcome is certain.

Fisher's Information

Fisher information measures the amount of information that an observable random variable X carries about an unknown parameter p upon which the probability of X depends. For the Bernoulli distribution, the Fisher information with respect to the parameter p is given by:

I(p)={\frac {1}{pq}}

Proof:

  • The Likelihood Function for a Bernoulli random variableX is: L(p;X)=p^{X}(1-p)^{1-X} This represents the probability of observing X given the parameter p.
  • The Log-Likelihood Function is: \ln L(p;X)=X\ln p+(1-X)\ln(1-p)
  • The Score Function (the first derivative of the log-likelihood with respect to p is: {\frac {\partial }{\partial p}}\ln L(p;X)={\frac {X}{p}}-{\frac {1-X}{1-p}}
  • The second derivative of the log-likelihood function is: {\frac {\partial ^{2}}{\partial p^{2}}}\ln L(p;X)=-{\frac {X}{p^{2}}}-{\frac {1-X}{(1-p)^{2}}}
  • Fisher information is calculated as the negative expected value of the second derivative of the log-likelihood:{\begin{aligned}I(p)=-E\left[{\frac {\partial ^{2}}{\partial p^{2}}}\ln L(p;X)\right]=-\left(-{\frac {p}{p^{2}}}-{\frac {1-p}{(1-p)^{2}}}\right)={\frac {1}{p(1-p)}}={\frac {1}{pq}}\end{aligned}}

It is maximized when p=0.5, reflecting maximum uncertainty and thus maximum information about the parameter p.

08Author's mention

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Sources and credits

This article is adapted from the Wikipedia article “Bernoulli distribution”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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