Editing Bernoulli distribution (section)

==Properties==
If <math>X</math> is a random variable with a Bernoulli distribution, then:

:<math>\Pr(X=1) = p, \Pr(X=0) = q =1 - p.</math>

The [[probability mass function]] <math>f</math> of this distribution, over possible outcomes ''k'', is

:<math> f(k;p) = \begin{cases}
   p & \text{if }k=1, \\
   q = 1-p & \text {if } k = 0.
 \end{cases}</math><ref name=":0">{{Cite book|title=Introduction to Probability|last=Bertsekas|author-link=Dimitri Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=[[John Tsitsiklis|Tsitsiklis, John N.]], Τσιτσικλής, Γιάννης Ν.|isbn=188652940X|location=Belmont, Mass.|oclc=51441829}}</ref>

This can also be expressed as

:<math>f(k;p) = p^k (1-p)^{1-k} \quad \text{for } k\in\{0,1\}</math>

or as

:<math>f(k;p)=pk+(1-p)(1-k) \quad \text{for } k\in\{0,1\}.</math>

The Bernoulli distribution is a special case of the [[binomial distribution]] with <math>n = 1.</math><ref name="McCullagh1989Ch422">{{cite book | last = McCullagh | first = Peter | author-link= Peter McCullagh |author2=Nelder, John |author-link2=John Nelder | title = Generalized Linear Models, Second Edition | publisher = Boca Raton: Chapman and Hall/CRC | year = 1989 | isbn = 0-412-31760-5 |ref=McCullagh1989 |at=Section 4.2.2 }}</ref>

The [[kurtosis]] goes to infinity for high and low values of <math>p,</math> but for <math>p=1/2</math> the two-point distributions including the Bernoulli distribution have a lower [[excess kurtosis]], namely −2, than any other probability distribution.

The Bernoulli distributions for <math>0 \le p \le 1</math> form an [[exponential family]].

The [[maximum likelihood estimator]] of <math>p</math> based on a random sample is the [[sample mean]].

[[File:PMF and CDF of a bernouli distribution.png|thumb|The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.]]