# Bernoulli: Create a Bernoulli distribution In distributions3: Probability Distributions as S3 Objects

 Bernoulli R Documentation

## Create a 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

```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.

Other discrete distributions: `Binomial()`, `Categorical()`, `Geometric()`, `HurdleNegativeBinomial()`, `HurdlePoisson()`, `HyperGeometric()`, `Multinomial()`, `NegativeBinomial()`, `Poisson()`, `ZINegativeBinomial()`, `ZIPoisson()`, `ZTNegativeBinomial()`, `ZTPoisson()`

### Examples

```
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))
```

distributions3 documentation built on Sept. 7, 2022, 5:07 p.m.