Bernoulli distribution

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 . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit 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 (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1. If is a random variable with a Bernoulli distribution, then: The probability mass function of this distribution, over possible outcomes k, is This can also be expressed as or 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 With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside .