In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45 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" (or vice versa), respectively, and p would be the probability of the coin landing on heads or tails, respectively.
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 this 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 \le p \le 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}\left(X\right)=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{align} \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{align}
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{align} \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{align}
The higher central moments can be expressed more compactly in terms of \mu_2 and \mu_3
\begin{align} \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{align}
The first six cumulants are
\begin{align} \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{align}
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 \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 Y \sim \mathrm{Bernoulli}\left(\frac{1}{2}\right), then 2Y - 1 has a Rademacher distribution.
See also
Bernoulli process, a random process consisting of a sequence of independent Bernoulli trials
Bernoulli sampling
Binary entropy function
Binary decision diagram
References
Further reading
External links
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Interactive graphic: Univariate Distribution Relationships.
