Parameters 0 < p < 1 , p ∈ R {\displaystyle 0 Support k ∈ { 0 , 1 } {\displaystyle k\in \{0,1\}\,} pmf { q = ( 1 − p ) for k = 0 p for k = 1 {\displaystyle {\begin{cases}q=(1-p)&{\text{for }}k=0\\p&{\text{for }}k=1\end{cases}}} CDF { 0 for k < 0 1 − p for 0 ≤ k < 1 1 for k ≥ 1 {\displaystyle {\begin{cases}0&{\text{for }}k<0\\1-p&{\text{for }}0\leq k<1\\1&{\text{for }}k\geq 1\end{cases}}} Mean p {\displaystyle p\,} Median { 0 if q > p 0.5 if q = p 1 if q < p {\displaystyle {\begin{cases}0&{\text{if }}q>p\\0.5&{\text{if }}q=p\\1&{\text{if }}q |
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is the probability distribution of a random variable which takes the value 1 with probability
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The Bernoulli distribution is a special case of the binomial distribution where a single experiment/trial is conducted (n=1). It is also a special case of the two-point distribution, for which the two possible outcomes need not be 0 and 1.
Properties of the Bernoulli Distribution
If
The probability mass function
This can also be expressed as
The Bernoulli distribution is a special case of the binomial distribution with
The kurtosis goes to infinity for high and low values of
The Bernoulli distributions for
The maximum likelihood estimator of
Mean
The expected value of a Bernoulli random variable
This is due to the fact that for a Bernoulli distributed random variable
Variance
The variance of a Bernoulli distributed
We first find
From this follows
Skewness
The skewness is
Related distributions
The Bernoulli distribution is simply