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Bernoulli distribution

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability that is, the probability distribution of any single experiment that asks a yes–no question; the question results in a boolean-valued outcome, a single bit of information 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

Bernoulli
Parameters
Support
pmf
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF
PGF
Fisher information

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.

Contents

Properties of the Bernoulli distributionEdit

If   is a random variable with this distribution, then:

 

The probability mass function   of this distribution, over possible outcomes k, is

 [2]

This can also be expressed as

 

or as

 

The Bernoulli distribution is a special case of the binomial distribution with  [3]

The kurtosis goes to infinity for high and low values of   but for   the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2.

The Bernoulli distributions for   form an exponential family.

The maximum likelihood estimator of   based on a random sample is the sample mean.

MeanEdit

The expected value of a Bernoulli random variable   is

 

This is due to the fact that for a Bernoulli distributed random variable   with   and   we find

 [2]

VarianceEdit

The variance of a Bernoulli distributed   is

 

We first find

 

From this follows

 [2]

SkewnessEdit

The skewness is  . When we take the standardized Bernoulli distributed random variable   we find that this random variable attains   with probability   and attains   with probability  . Thus we get

 

Related distributionsEdit

The Bernoulli distribution is simply  , also written as  
  • 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  , then   has a Rademacher distribution.

See alsoEdit

NotesEdit

  1. ^ James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
  2. ^ a b c d Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
  3. ^ McCullagh and Nelder (1989), Section 4.2.2.

ReferencesEdit

External linksEdit