| Binomial {distributions3} | R Documentation |
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 |
p |
The success probability for a given trial. |
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(),
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))