The Geometric distribution can be thought of as a generalization of the Bernoulli() distribution where we ask: "if I keep flipping a coin with probability p of heads, what is the probability I need \(k\) flips before I get my first heads?" The Geometric distribution is a special case of Negative Binomial distribution.

Geometric(p = 0.5)

Arguments

p

The success probability for the distribution. p can be any value in [0, 1], and defaults to 0.5.

Value

A Geometric object.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

In the following, let \(X\) be a Geometric random variable with success probability p = \(p\). Note that there are multiple parameterizations of the Geometric distribution.

Support: 0 < p < 1, \(x = 0, 1, \dots\)

Mean: \(\frac{1-p}{p}\)

Variance: \(\frac{1-p}{p^2}\)

Probability mass function (p.m.f):

$$ P(X = x) = p(1-p)^x, $$

Cumulative distribution function (c.d.f):

$$ P(X \le x) = 1 - (1-p)^{x+1} $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = \frac{pe^t}{1 - (1-p)e^t} $$

See also

Examples

set.seed(27) X <- Geometric(0.3) X
#> Geometric distribution (p = 0.3)
random(X, 10)
#> [1] 0 1 9 2 4 6 4 2 3 1
pdf(X, 2)
#> [1] 0.147
log_pdf(X, 2)
#> [1] -1.917323
cdf(X, 4)
#> [1] 0.83193
quantile(X, 0.7)
#> [1] 3