Bernoulli Distribution

The Bernoulli distribution is a discrete distribution having two possible outcomes labelled by  and  in which  ("success") occurs with probability  and  ("failure") occurs with probability , where . It therefore has probability density function

(1)

which can also be written

(2)

The corresponding distribution function is

(3)

The Bernoulli distribution is implemented in the Wolfram Language as BernoulliDistribution[p].

The performance of a fixed number of trials with fixed probability of success on each trial is known as a Bernoulli trial.

The distribution of heads and tails in coin tossing is an example of a Bernoulli distribution with . The Bernoulli distribution is the simplest discrete distribution, and it the building block for other more complicated discrete distributions. The distributions of a number of variate types defined based on sequences of independent Bernoulli trials that are curtailed in some way are summarized in the following table (Evans et al. 2000, p. 32).

distribution	definition
binomial distribution	number of successes in  trials
geometric distribution	number of failures before the first success
negative binomial distribution	number of failures before the th success

The characteristic function is

(4)

and the moment-generating function is

			(5)
			(6)
			(7)

so

			(8)
			(9)
			(10)
			(11)

These give raw moments

			(12)
			(13)
			(14)

and central moments

			(15)
			(16)
			(17)

The mean, variance, skewness, and kurtosis excess are then

			(18)
			(19)
			(20)
			(21)

To find an estimator  for the mean of a Bernoulli population with population mean , let  be the sample size and suppose  successes are obtained from the  trials. Assume an estimator given by

(22)

so that the probability of obtaining the observed  successes in  trials is then

(23)

The expectation value of the estimator  is therefore given by

			(24)
			(25)
			(26)

so  is indeed an unbiased estimator for the population mean .

The mean deviation is given by

(27)

REFERENCES:

Evans, M.; Hastings, N.; and Peacock, B. "Bernoulli Distribution." Ch. 4 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 31-33, 2000.