Description

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.

Usage

Arguments

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, …

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 ≤ x) = 1 - (1-p)^{x+1}

Moment generating function (m.g.f):

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

Value

A Geometric object.

See Also

Other discrete distributions: Bernoulli, Binomial, Categorical, HyperGeometric, Multinomial, NegativeBinomial, Poisson

Examples