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 mathematician Jacob Bernoulli,^{[1]} 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 yesno question; the question results in a booleanvalued 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 (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails" (or vice versa), respectively, and p would be the probability of the coin landing on heads or tails, respectively. 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 twopoint distribution, for which the possible outcomes need not be 0 and 1.
If is a random variable with this 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 ^{[3]}
The kurtosis goes to infinity for high and low values of but for the twopoint 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
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