Bernoulli distribution

In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli,[1] is the probability distribution of a random variable which takes the value 1 with success probability of and the value 0 with failure probability of . 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 .

The Bernoulli distribution is a special case of the two-point distribution, for which the two possible outcomes need not be 0 and 1. It is also a special case of the binomial distribution; the Bernoulli distribution is a binomial distribution where n=1.

Properties of the Bernoulli Distribution

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

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.

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

Variance

The variance of a Bernoulli distributed is

We first find

From this follows

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

• 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 .

Notes

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

References

• McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.
• Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9