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In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,[1] is the probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability — i.e., 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 coin toss where 1 and 0 would represent "head" and "tail" (or vice versa), 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 experiment/trial is conducted (n=1). It is also a special case of the two-point distribution, for which the outcome need not be a bit, i.e., the two possible outcomes need not be 0 and 1.

Contents

Properties of the Bernoulli DistributionEdit

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

 

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

 

This can also be expressed as

 

or as

 

The Bernoulli distribution is a special case of the binomial distribution with  .[2]

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

 

VarianceEdit

The variance of a Bernoulli distributed   is

 

We first find

 

From this follows

 

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

  • If   are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with success probability p, then
  (binomial distribution).

The Bernoulli distribution is simply  .

See alsoEdit

NotesEdit

  1. ^ James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
  2. ^ McCullagh and Nelder (1989), Section 4.2.2.

ReferencesEdit

External linksEdit