Parameters Probability mass function Three examples of Bernoulli distribution: .mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}  $P(x=0)=0{.}2$ and $P(x=1)=0{.}8$ $P(x=0)=0{.}8$ and $P(x=1)=0{.}2$ $P(x=0)=0{.}5$ and $P(x=1)=0{.}5$ $0\leq p\leq 1$ $q=1-p$ $k\in \{0,1\)$ ${\begin{cases}q=1-p&{\text{if ))k=0\\p&{\text{if ))k=1\end{cases))$ ${\begin{cases}0&{\text{if ))k<0\\1-p&{\text{if ))0\leq k<1\\1&{\text{if ))k\geq 1\end{cases))$ $p$ ${\begin{cases}0&{\text{if ))p<1/2\\\left[0,1\right]&{\text{if ))p=1/2\\1&{\text{if ))p>1/2\end{cases))$ ${\begin{cases}0&{\text{if ))p<1/2\\0,1&{\text{if ))p=1/2\\1&{\text{if ))p>1/2\end{cases))$ $p(1-p)=pq$ ${\frac {1}{2))$ ${\frac {q-p}{\sqrt {pq)))$ ${\frac {1-6pq}{pq))$ $-q\ln q-p\ln p$ $q+pe^{t)$ $q+pe^{it)$ $q+pz$ ${\frac {1}{pq))$ 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.

## Properties

If $X$ is a random variable with a Bernoulli 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

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

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

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

The variance of a Bernoulli distributed $X$ is

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

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

$\operatorname {Var} [X]=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=\operatorname {E} [X]-\operatorname {E} [X]^{2}=p-p^{2}=p(1-p)=pq$ With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside $[0,1/4]$ .

## 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)) ## Higher moments and cumulants

The raw moments are all equal due to the fact that $1^{k}=1$ 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)) ## 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).
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.