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):

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

Other discrete distributions: Binomial, Categorical, Geometric, HyperGeometric, Multinomial, NegativeBinomial, Poisson

cdf(X, quantile(X, 0.7))#> [1] 1quantile(X, cdf(X, 0.7))#> [1] 0