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Bernoulli distribution

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Probability mass function
Cumulative distribution function
Parameters p>0\, (real)
q\equiv 1-p\,
Support k=\{0,1\}\,
Template:Probability distribution/link mass 
    q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1
    0 & \mbox{for }k<0 \\q & \mbox{for }0<k<1\\1 & \mbox{for }k>1
Mean p\,
Median N/A
Mode \textrm{max}(p,q)\,
Variance pq\,
Skewness \frac{q-p}{\sqrt{pq}}
Kurtosis \frac{6p^2-6p+1}{p(1-p)}
Entropy -q\ln(q)-p\ln(p)\,
mgf q+pe^t\,
Char. func. 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


The Bernoulli distribution is a member of the exponential family.

Related distributionsEdit

  • 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).

See alsoEdit

fr:Distribution de Bernoullihe:התפלגות ברנולי nl:Bernoulli-verdelingfi:Bernoullin jakauma zh:伯努利分布

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