Bernoulli distribution

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Bernoulli
Probability mass function
Cumulative distribution function
Parameters p>0\, (real)
Support k=\{0,1\}\,
Probability mass function (pmf) \begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}
Cumulative distribution function (cdf) \begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix}
Mean p\,
Median N/A
Mode \begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}
Variance pq\,
Skewness \frac{q-p}{\sqrt{pq}}
Excess kurtosis \frac{6p^2-6p+1}{p(1-p)}
Entropy -q\ln(q)-p\ln(p)\,
Moment-generating function (mgf) q+pe^t\,
Characteristic function q+pe^{it}\,

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:

\Pr(X=1) = 1- \Pr(X=0) = p.\!

The probability mass function f of this distribution is

f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\ 1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.

The expected value of a Bernoulli random variable X is E\left(X\right)=p, and its variance is

\textrm{var}\left(X\right)=p\left(1-p\right).\,

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.

  • If X_1,\dots,X_n are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p) (binomial distribution).

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