dist_binomial {distributional} | R Documentation |
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.
dist_binomial(size, prob)
size |
The number of trials. Must be an integer greater than or equal
to one. When |
prob |
The probability of success on each trial, |
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 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
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)