Bernoulli distribution

Bernoulli

Parameters	$0<p<1, p\in\R$
Support	$k \in \{0,1\}\,$
pmf	$\begin{cases} q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1 \end{cases}$
CDF	$\begin{cases} 0 & \text{for }k<0 \\ 1 - p & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1 \end{cases}$
Mean	$p\,$
Median	$\begin{cases} 0 & \text{if } q > p\\ 0.5 & \text{if } q=p\\ 1 & \text{if } q<p \end{cases}$
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,^[1] 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. It is also a special case of the binomial distribution; the Bernoulli distribution is a binomial distribution where n=1.

Properties

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 Bernoulli distribution is a special case of the binomial distribution with $n = 1$.^[2]

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.

Mean

The expected value of a Bernoulli random variable $X$ is

$\operatorname{E}\left(X\right)=p$

This is due to the fact that for a Bernoulli distributed random variable $X$ with $\Pr(X=1)=p$ and $\Pr(X=0)=q$ we find

$\operatorname{E}[X] = \Pr(X=1)\cdot 1 + \Pr(X=0)\cdot 0 = p \cdot 1 + q\cdot 0 = p$

Variance

The variance of a Bernoulli distributed $X$ is

$\operatorname{Var}[X] = pq = p(1-p)$

We first find

$\operatorname{E}[X^2] = \Pr(X=1)\cdot 1^2 + \Pr(X=0)\cdot 0^2 = p \cdot 1^2 + q\cdot 0^2 = p$

From this follows

$\operatorname{Var}[X] = \operatorname{E}[X^2]-\operatorname{E}[X]^2 = p-p^2 = p(1-p) = pq$

Skewness

The skewness is $\frac{q-p}{\sqrt{pq}}=\frac{1-2p}{\sqrt{pq}}$. When we take the standardized Bernoulli distributed random variable $\frac{X-\operatorname{E}[X]}{\sqrt{\operatorname{Var}[X]}}$ we find that this random variable attains $\frac{q}{\sqrt{pq}}$ with probability $p$ and attains $-\frac{p}{\sqrt{pq}}$ with probability $q$. Thus we get

$\begin{align} \gamma_1 &= \operatorname{E} \left[\left(\frac{X-\operatorname{E}[X]}{\sqrt{\operatorname{Var}[X]}}\right)^3\right] \\ &= p \cdot \left(\frac{q}{\sqrt{pq}}\right)^3 + q \cdot \left(-\frac{p}{\sqrt{pq}}\right)^3 \\ &= \frac{1}{\sqrt{pq}^3} \left(pq^3-qp^3\right) \\ &= \frac{pq}{\sqrt{pq}^3} (q-p) \\ &= \frac{q-p}{\sqrt{pq}} \end{align}$

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 $\mathrm{B}(1,p)$.

• The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
• The Beta distribution is the conjugate prior of the Bernoulli distribution.
• The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.
• If Y ~ Bernoulli(0.5), then (2Y-1) has a Rademacher distribution.

See also

• Bernoulli process
• Bernoulli sampling
• Bernoulli trial
• Binary entropy function
• Binomial distribution

Notes

1. James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
2. McCullagh and Nelder (1989), Section 4.2.2.

References

• McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Boca Raton: Chapman and Hall/CRC. ISBN 0-412-31760-5.
• Johnson, N.L., Kotz, S., Kemp A. (1993) Univariate Discrete Distributions (2nd Edition). Wiley. ISBN 0-471-54897-9

External links

• Hazewinkel, Michiel, ed. (2001), "Binomial distribution", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W., "Bernoulli Distribution", MathWorld.
• Interactive graphic: Univariate Distribution Relationships