distributions3 (version 0.2.1)

Binomial: 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)

Value

A Binomial object.

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):

$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$$

Cumulative distribution function (c.d.f):

$$P(X\le k)=\sum _{i=0}^{\lfloor k\rfloor }(\genfrac{}{}{0ex}{}{n}{i})p^i(1-p{)}^{n-i}$$

Moment generating function (m.g.f):

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

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))