World Library  
Flag as Inappropriate
Email this Article
 

Bernoulli distribution

Bernoulli
Parameters 0
Support k \in \{0,1\}\,
pmf \begin{cases} q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1 \end{cases}
CDF \begin{cases} 0 & \text{for }k<0 \\ q & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1 \end{cases}
Mean p\,
Median \begin{cases} 0 & \text{if } q > p\\ 0.5 & \text{if } q=p\\ 1 & \text{if } q

Mode \begin{cases} 0 & \text{if } q > p\\ 0, 1 & \text{if } q=p\\ 1 & \text{if } q < p \end{cases}
Variance p(1-p) (=pq)\,
Skewness \frac{1-2p}{\sqrt{pq}}
Ex. kurtosis \frac{1-6pq}{pq}
Entropy -q\ln(q)-p\ln(p)\,
MGF q+pe^t\,
CF q+pe^{it}\,
PGF q+pz\,
Fisher information \frac{1}{p(1-p)}

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 success probability of p and the value 0 with failure probability of q=1-p. It can be used to represent a coin toss where 1 and 0 would represent "head" and "tail" (or vice versa), respectively. In particular, unfair coins would have p \neq 0.5.

The Bernoulli distribution is a special case of the two-point distribution, for which the two possible outcomes need not be 0 and 1.

Contents

  • Properties 1
  • Related distributions 2
  • See also 3
  • Notes 4
  • References 5
  • External links 6

Properties

Plot of Bernoulli distribution probability mass function

If X is a random variable with this distribution, we have:

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

The probability mass function f of this distribution, over possible outcomes k, is

f(k;p) = \begin{cases} p & \text{if }k=1, \\[6pt] 1-p & \text {if }k=0.\end{cases}

This can also be expressed as

f(k;p) = p^k (1-p)^{1-k}\!\quad \text{for }k\in\{0,1\}.

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 Bernoulli distribution is a special case of the binomial distribution with n = 1.[1]

The kurtosis goes to infinity for high and low values of p, but for p=1/2 the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2.

The Bernoulli distributions for 0 \le p \le 1 form an exponential family.

The maximum likelihood estimator of p based on a random sample is the sample mean.

Related distributions

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

The Bernoulli distribution is simply \mathrm{B}(1,p).

See also

Notes

  1. ^ McCullagh and Nelder (1989), Section 4.2.2.

References

  •  
  • Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9
  • Doctor Professor Patrick McDikkButte McGeep, Auschvitz 1943 get gud productions llc.

External links

  • Hazewinkel, Michiel, ed. (2001), "Binomial distribution",  
  • Weisstein, Eric W., "Bernoulli Distribution", MathWorld.
  • Interactive graphic: Univariate Distribution Relationships
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.