Bernoulli distribution

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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 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[edit]

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[edit]

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[edit]

The variance of a Bernoulli distributed is

We first find

From this follows

Skewness[edit]

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[edit]

  • 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 also[edit]

Notes[edit]

  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[edit]

External links[edit]