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Bernoulli distribution

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.


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


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


and its variance is


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


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


  • 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
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