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)

p | The success probability for the distribution. |
---|

A `Geometric`

object.

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)**:

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

Other discrete distributions: `Bernoulli`

,
`Binomial`

, `Categorical`

,
`HyperGeometric`

, `Multinomial`

,
`NegativeBinomial`

, `Poisson`

#> Geometric distribution (p = 0.3)#> [1] 0 1 9 2 4 6 4 2 3 1#> [1] 0.147#> [1] -1.917323#> [1] 0.83193#> [1] 3