dist_binomial {distributional}

The 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

dist_binomial(size, prob)

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.

prob

The probability of success on each trial, prob can be any value in ⁠[0, 1]⁠.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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.

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

Examples

dist <- dist_binomial(size = 1:5, prob = c(0.05, 0.5, 0.3, 0.9, 0.1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)