Binomial: Create a Binomial distribution

In distributions3: Probability Distributions as S3 Objects

Create a Binomial distribution

Description

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.

Usage

Binomial(size, p = 0.5)

Arguments

size	The number of trials. Must be an integer greater than or equal to one. When size = 1L, the Binomial distribution reduces to the bernoulli distribution. Often called n in textbooks.
p	The success probability for a given trial. p can be any value in ⁠[0, 1]⁠, and defaults to 0.5.

Details

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 \pi instead of p.

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

Mean: np

Variance: np \cdot (1 - p) = np \cdot q

Probability mass function (p.m.f):

P(X = k) = {n \choose k} p^k (1 - p)^{n-k}

Cumulative distribution function (c.d.f):

P(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n \choose i} p^i (1 - p)^{n-i}

Moment generating function (m.g.f):

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

Value

A Binomial object.

See Also

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

Examples

set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.