Bernoulli Distribution
Bernoulli
Parameters


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

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

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 of the Bernoulli distribution

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.

Mean

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]

Variance

The variance of a Bernoulli distributed is

We first find

From this follows

[2]

Skewness

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 distributions

The Bernoulli distribution is simply , also written as

See also

Notes

  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., ?, ? ?. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
  3. ^ McCullagh and Nelder (1989), Section 4.2.2.

References

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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