distributional (version 0.3.2)

dist_bernoulli: The Bernoulli distribution

Description

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.

Usage

dist_bernoulli(prob)

Arguments

prob

The probability of success on each trial, prob can be any value in [0, 1].

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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):

$$P(X\le x)=\{\begin{array}{ll}0& x<0\\ 1-p& 0\le x<1\\ 1& x\ge 1\end{array}$$

Moment generating function (m.g.f):

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

Examples

dist <- dist_bernoulli(prob = c(0.05, 0.5, 0.3, 0.9, 0.1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)