# Bernoulli: Create a Bernoulli distribution In alexpghayes/distributions: Probability Distributions as S3 Objects

## 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

 `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`.

## 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 π 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 (1 - p)

Probability mass function (p.m.f):

P(X = x) = p^x (1 - p)^(1-x)

Cumulative distribution function (c.d.f):

P(X ≤ x) = (1 - p) 1_{[0, 1)}(x) + 1_{1}(x)

Moment generating function (m.g.f):

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

## Value

A `Bernoulli` object.

## See Also

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```set.seed(27) X <- Bernoulli(0.7) X mean(X) variance(X) skewness(X) kurtosis(X) random(X, 10) pdf(X, 1) log_pdf(X, 1) cdf(X, 0) quantile(X, 0.7) cdf(X, quantile(X, 0.7)) quantile(X, cdf(X, 0.7)) ```

alexpghayes/distributions documentation built on April 8, 2021, 5:55 a.m.