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).

The probability mass function $f$ of this distribution is

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

This can also be expressed as

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

The expected value of a Bernoulli random variable $X$ is $E\left(X\right)=p$, and its variance is

${\textrm {Var}}\left(X\right)=p\left(1-p\right).$

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.

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.

Related distributions

• If $X_{1},\dots ,X_{n}$ are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with success probability p, then

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

The Bernoulli distribution is simply $\mathrm {B} (1,p)$.

• 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.
• The bernoulli distribution is a special case of binomial distribution.
• If Y ~ Bernoulli(0.5), then (2Y-1) has a Rademacher distribution.

See also

• Bernoulli process
• Bernoulli sampling
• Bernoulli trial
• Binary entropy function

References

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• Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9

External links

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• Weisstein, Eric W., "Bernoulli Distribution", MathWorld.
• Interactive graphic: Univariate Distribution Relationships

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