Description Usage Arguments Details Value See Also Examples

Binomial distributions are used to represent situations can that can
be thought as the result of *n* Bernoulli experiments (here the
*n* is defined as the `size`

of the experiment). The classical
example is *n* independent coin flips, where each coin flip has
probability `p`

of success. In this case, the individual probability of
flipping heads or tails is given by the Bernoulli(p) distribution,
and the probability of having *x* equal results (*x* heads,
for example), in *n* trials is given by the Binomial(n, p) distribution.
The equation of the Binomial distribution is directly derived from
the equation of the Bernoulli distribution.

1 | ```
Binomial(size, p = 0.5)
``` |

`size` |
The number of trials. Must be an integer greater than or equal
to one. When |

`p` |
The success probability for a given trial. |

The Binomial distribution comes up when you are interested in the portion
of people who do a thing. The Binomial distribution
also comes up in the sign test, sometimes called the Binomial test
(see `stats::binom.test()`

), where you may need the Binomial C.D.F. to
compute p-values.

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 Binomial random variable with parameter
`size`

= *n* and `p`

= *p*. Some textbooks define *q = 1 - p*,
or called *π* instead of *p*.

**Support**: *{0, 1, 2, ..., n}*

**Mean**: *np*

**Variance**: *np (1 - p)*

**Probability mass function (p.m.f)**:

*
P(X = k) = choose(n, k) p^k (1 - p)^(n - k)
*

**Cumulative distribution function (c.d.f)**:

*
P(X ≤ k) = ∑_{i=0}^k choose(n, i) p^i (1 - p)^(n-i)
*

**Moment generating function (m.g.f)**:

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

A `Binomial`

object.

Other discrete distributions: `Bernoulli`

,
`Categorical`

, `Geometric`

,
`HyperGeometric`

, `Multinomial`

,
`NegativeBinomial`

, `Poisson`

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