# Bernoulli distribution

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In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is the probability distribution of a random variable which takes value 1 with success probability $p$ and value 0 with failure probability $q=1-p$ . It can be used, for example, to represent the toss of a coin, where "1" is defined to mean "heads" and "0" is defined to mean "tails" (or vice versa).

## Properties

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

$Pr(X=1)=1-Pr(X=0)=1-q=p.\!$ A classical example of a Bernoulli experiment is a single toss of a coin. The coin might come up heads with probability $p$ and tails with probability $1-p$ . The experiment is called fair if $p=0.5$ , indicating the origin of the terminology in betting (the bet is fair if both possible outcomes have the same probability).

$f(k;p)={\begin{cases}p&{\text{if }}k=1,\\[6pt]1-p&{\text{if }}k=0.\end{cases}}$ This can also be expressed as

$f(k;p)=p^{k}(1-p)^{1-k}\!\quad {\text{for }}k\in \{0,1\}.$ ${\textrm {Var}}\left(X\right)=p\left(1-p\right).$ 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 Bernoulli distribution has a lower excess kurtosis than any other probability distribution, namely −2.

## Related distributions

$Y=\sum _{k=1}^{n}X_{k}\sim \mathrm {B} (n,p)$ (binomial distribution).