Bernoulli {distributions3} | R Documentation |

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)
```

`p` |
The success probability for the distribution. |

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
```

A `Bernoulli`

object.

Other discrete distributions:
`Binomial()`

,
`Categorical()`

,
`Geometric()`

,
`HurdleNegativeBinomial()`

,
`HurdlePoisson()`

,
`HyperGeometric()`

,
`Multinomial()`

,
`NegativeBinomial()`

,
`Poisson()`

,
`ZINegativeBinomial()`

,
`ZIPoisson()`

,
`ZTNegativeBinomial()`

,
`ZTPoisson()`

```
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))
```

[Package *distributions3* version 0.2.1 Index]