Bernoulli distribution

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 ${\displaystyle p}${\displaystyle p} and the value 0 with probability ${\displaystyle q=1-p}${\displaystyle 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 ${\displaystyle p\neq 1/2.}${\displaystyle 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.^[2]

If ${\displaystyle X}${\displaystyle X} is a random variable with a Bernoulli distribution, then:

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

The probability mass function ${\displaystyle f}${\displaystyle f} of this distribution, over possible outcomes k, is^[3]

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

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

or as

$${\displaystyle f(k;p)=pk+(1-p)(1-k)\quad {\text{for }}k\in \{0,1\}.}$${\displaystyle 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 ${\displaystyle n=1.}${\displaystyle n=1.}^[4]

The kurtosis goes to infinity for high and low values of ${\displaystyle p,}${\displaystyle p,} but for ${\displaystyle p=1/2}${\displaystyle 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 ${\displaystyle 0\leq p\leq 1}${\displaystyle 0\leq p\leq 1} form an exponential family.

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

The expected value of a Bernoulli random variable ${\displaystyle X}${\displaystyle X} is

$${\displaystyle \operatorname {E} [X]=p}$${\displaystyle \operatorname {E} [X]=p}

This is because for a Bernoulli distributed random variable ${\displaystyle X}${\displaystyle X} with ${\displaystyle \Pr(X{=}1)=p}${\displaystyle \Pr(X{=}1)=p} and ${\textstyle \Pr(X{=}0)=q}${\textstyle \Pr(X{=}0)=q} we find^[3]

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

The variance of a Bernoulli distributed ${\displaystyle X}${\displaystyle X} is

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

We first find

$${\displaystyle {\begin{aligned}\operatorname {E} [X^{2}]&=\Pr(X{=}1)\cdot 1^{2}+\Pr(X{=}0)\cdot 0^{2}\\&=p\cdot 1^{2}+q\cdot 0^{2}\\&=p=\operatorname {E} [X]\end{aligned}}}$${\displaystyle {\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^[3]

$${\displaystyle {\begin{aligned}\operatorname {Var} [X]&=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=\operatorname {E} [X]-\operatorname {E} [X]^{2}\\[1ex]&=p-p^{2}=p(1-p)=pq\end{aligned}}}$${\displaystyle {\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 ${\displaystyle [0,1/4]}${\displaystyle [0,1/4]}.

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

$${\displaystyle {\begin{aligned}\gamma _{1}&=\operatorname {E} \left[\left({\frac {X-\operatorname {E} [X]}{\sqrt {\operatorname {Var} [X]}}}\right)^{3}\right]\\&=p\cdot \left({\frac {q}{\sqrt {pq}}}\right)^{3}+q\cdot \left(-{\frac {p}{\sqrt {pq}}}\right)^{3}\\&={\frac {1}{{\sqrt {pq}}^{3}}}\left(pq^{3}-qp^{3}\right)\\&={\frac {pq}{{\sqrt {pq}}^{3}}}(q^{2}-p^{2})\\&={\frac {(1-p)^{2}-p^{2}}{\sqrt {pq}}}\\&={\frac {1-2p}{\sqrt {pq}}}={\frac {q-p}{\sqrt {pq}}}.\end{aligned}}}$${\displaystyle {\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}}}

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

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

The central moment of order ${\displaystyle k}${\displaystyle k} is given by $${\displaystyle \mu _{k}=(1-p)(-p)^{k}+p(1-p)^{k}.}$${\displaystyle \mu _{k}=(1-p)(-p)^{k}+p(1-p)^{k}.} The first six central moments are $${\displaystyle {\begin{aligned}\mu _{1}&=0,\\\mu _{2}&=p(1-p),\\\mu _{3}&=p(1-p)(1-2p),\\\mu _{4}&=p(1-p)(1-3p(1-p)),\\\mu _{5}&=p(1-p)(1-2p)(1-2p(1-p)),\\\mu _{6}&=p(1-p)(1-5p(1-p)(1-p(1-p))).\end{aligned}}}$${\displaystyle {\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 ${\displaystyle \mu _{2}}${\displaystyle \mu _{2}} and ${\displaystyle \mu _{3}}${\displaystyle \mu _{3}} $${\displaystyle {\begin{aligned}\mu _{4}&=\mu _{2}(1-3\mu _{2}),\\\mu _{5}&=\mu _{3}(1-2\mu _{2}),\\\mu _{6}&=\mu _{2}(1-5\mu _{2}(1-\mu _{2})).\end{aligned}}}$${\displaystyle {\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 $${\displaystyle {\begin{aligned}\kappa _{1}&=p,\\\kappa _{2}&=\mu _{2},\\\kappa _{3}&=\mu _{3},\\\kappa _{4}&=\mu _{2}(1-6\mu _{2}),\\\kappa _{5}&=\mu _{3}(1-12\mu _{2}),\\\kappa _{6}&=\mu _{2}(1-30\mu _{2}(1-4\mu _{2})).\end{aligned}}}$${\displaystyle {\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}}}

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

$${\displaystyle {\begin{aligned}H(X)&=\mathbb {E} _{p}\ln {\frac {1}{\Pr(X)}}\\[1ex]&=-\Pr(X{=}0)\ln \Pr(X{=}0)-\Pr(X{=}1)\ln \Pr(X{=}1)\\[1ex]&=-(q\ln q+p\ln p).\end{aligned}}}$${\displaystyle {\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 ${\displaystyle p=0.5}${\displaystyle p=0.5}, indicating the highest level of uncertainty when both outcomes are equally likely. The entropy is zero when ${\displaystyle p=0}${\displaystyle p=0} or ${\displaystyle p=1}${\displaystyle p=1}, where one outcome is certain.

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

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

Proof:

• The Likelihood Function for a Bernoulli random variable${\displaystyle X}${\displaystyle X} is: $${\displaystyle L(p;X)=p^{X}(1-p)^{1-X}}$${\displaystyle L(p;X)=p^{X}(1-p)^{1-X}} This represents the probability of observing ${\displaystyle X}${\displaystyle X} given the parameter ${\displaystyle p}${\displaystyle p}.
• The Log-Likelihood Function is: $${\displaystyle \ln L(p;X)=X\ln p+(1-X)\ln(1-p)}$${\displaystyle \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 ${\displaystyle p}${\displaystyle p} is: $${\displaystyle {\frac {\partial }{\partial p}}\ln L(p;X)={\frac {X}{p}}-{\frac {1-X}{1-p}}}$${\displaystyle {\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: $${\displaystyle {\frac {\partial ^{2}}{\partial p^{2}}}\ln L(p;X)=-{\frac {X}{p^{2}}}-{\frac {1-X}{(1-p)^{2}}}}$${\displaystyle {\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:$${\displaystyle {\begin{aligned}I(p)=-E\left[{\frac {\partial ^{2}}{\partial p^{2}}}\ln L(p;X)\right]=-\left(-{\frac {p}{p^{2}}}-{\frac {1-p}{(1-p)^{2}}}\right)={\frac {1}{p(1-p)}}={\frac {1}{pq}}\end{aligned}}}$${\displaystyle {\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 ${\displaystyle p=0.5}${\displaystyle p=0.5}, reflecting maximum uncertainty and thus maximum information about the parameter ${\displaystyle p}${\displaystyle p}.

• If ${\displaystyle X_{1},\dots ,X_{n}}${\displaystyle 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:

  ${\displaystyle \sum _{k=1}^{n}X_{k}\sim \operatorname {B} (n,p)}${\displaystyle \sum _{k=1}^{n}X_{k}\sim \operatorname {B} (n,p)} (binomial distribution).^[3]

The Bernoulli distribution is simply ${\displaystyle \operatorname {B} (1,p)}${\displaystyle \operatorname {B} (1,p)}, also written as ${\textstyle \mathrm {Bernoulli} (p).}${\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.^[5]
• The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.
• If ${\textstyle Y\sim \mathrm {Bernoulli} \left({\frac {1}{2}}\right)}${\textstyle Y\sim \mathrm {Bernoulli} \left({\frac {1}{2}}\right)}, then ${\textstyle 2Y-1}${\textstyle 2Y-1} has a Rademacher distribution.

• Bernoulli process, a random process consisting of a sequence of independent Bernoulli trials
• Bernoulli sampling
• Binary entropy function
• Binary decision diagram

• Abhirath, dwivedi; Kotz, Samuel; Kemp, Adrienne W. (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.

• "Binomial distribution", Encyclopedia of Mathematics , EMS Press, 2001 [1994].
• Weisstein, Eric W. "Bernoulli Distribution". MathWorld .
• Interactive graphic: Univariate Distribution Relationships.

• Discrete distributions
• Conjugate prior distributions
• Exponential family distributions

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