In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, 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 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 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.
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 maximum likelihood estimator of based on a random sample is the sample 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
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
- (binomial distribution).
The Bernoulli distribution is simply .
- ^ James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
- ^ McCullagh and Nelder (1989), Section 4.2.2.