Bernoulli distribution
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Probability mass function |
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Cumulative distribution function |
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Cumulative distribution function (cdf) | |
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Median | N/A |
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Moment-generating function (mgf) | |
Characteristic function |
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability p and value 0 with failure probability q = 1 − p. So if X is a random variable with this distribution, we have:
The probability mass function f of this distribution is
The expected value of a Bernoulli random variable X is , and its variance is
The kurtosis goes to infinity for high and low values of p, but for p = 1 / 2 the Bernoulli distribution has a lower kurtosis than any other probability distribution, namely -2.
The Bernoulli distribution is a member of the exponential family.
[edit] Related distributions
- If are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then (binomial distribution).