Psychology Wiki

Bernoulli distribution

34,203pages on
this wiki
Add New Page
Talk0 Share

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory

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

This page uses Creative Commons Licensed content from Wikipedia (view authors).

Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random Wiki