Bernoulli distributions are used to represent events like coin flips when there is single trial that is either successful or unsuccessful. The Bernoulli distribution is a special case of the Binomial() distribution with n = 1.

Bernoulli(p = 0.5)

Arguments

p

The success probability for the distribution. p can be any value in [0, 1], and defaults to 0.5.

Value

A Bernoulli object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail.

In the following, let \(X\) be a Bernoulli random variable with parameter p = \(p\). Some textbooks also define \(q = 1 - p\), or use \(\pi\) instead of \(p\).

The Bernoulli probability distribution is widely used to model binary variables, such as 'failure' and 'success'. The most typical example is the flip of a coin, when \(p\) is thought as the probability of flipping a head, and \(q = 1 - p\) is the probability of flipping a tail.

Support: \(\{0, 1\}\)

Mean: \(p\)

Variance: \(p \cdot (1 - p) = p \cdot q\)

Probability mass function (p.m.f):

$$ P(X = x) = p^x (1 - p)^{1-x} = p^x q^{1-x} $$

Cumulative distribution function (c.d.f):

$$ P(X \le x) = \left \{ \begin{array}{ll} 0 & x < 0 \\ 1 - p & 0 \leq x < 1 \\ 1 & x \geq 1 \end{array} \right. $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = (1 - p) + p e^t $$

See also

Examples

set.seed(27) X <- Bernoulli(0.7) X
#> Bernoulli distribution (p = 0.7)
random(X, 10)
#> [1] 0 1 0 1 1 1 1 1 1 1
pdf(X, 1)
#> [1] 0.7
log_pdf(X, 1)
#> [1] -0.3566749
cdf(X, 0)
#> [1] 0.3
quantile(X, 0.7)
#> [1] 1
cdf(X, quantile(X, 0.7))
#> [1] 1
quantile(X, cdf(X, 0.7))
#> [1] 0