In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, 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 .
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 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
The variance of a Bernoulli distributed is
We first find
From this follows
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
- If are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with success probability p, then
The Bernoulli distribution is simply .
- 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 Y ~ Bernoulli(0.5), then (2Y-1) has a Rademacher distribution.
- James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
- McCullagh and Nelder (1989), Section 4.2.2.
- 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
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- Hazewinkel, Michiel, ed. (2001), "Binomial distribution", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Bernoulli Distribution". MathWorld.
- Interactive graphic: Univariate Distribution Relationships