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# Bernoulli distribution

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### Bernoulli distribution

 Parameters 0 k \in \{0,1\}\, \begin{cases} q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1 \end{cases} \begin{cases} 0 & \text{for }k<0 \\ q & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1 \end{cases} p\, \begin{cases} 0 & \text{if } q > p\\ 0.5 & \text{if } q=p\\ 1 & \text{if } q \begin{cases} 0 & \text{if } q > p\\ 0, 1 & \text{if } q=p\\ 1 & \text{if } q < p \end{cases} p(1-p) (=pq)\, \frac{1-2p}{\sqrt{pq}} \frac{1-6pq}{pq} -q\ln(q)-p\ln(p)\, q+pe^t\, q+pe^{it}\, q+pz\, \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
• Notes 4
• References 5

## 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).

## 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.