The Geometric distribution can be thought of as a generalization
of the `dist_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.

## Arguments

- prob
probability of success in each trial. `0 < prob <= 1`

.

## Details

We recommend reading this documentation on
https://pkg.mitchelloharawild.com/distributional/, where the math
will render nicely.

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)**:

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

## Examples

```
dist <- dist_geometric(prob = c(0.2, 0.5, 0.8))
dist
#> <distribution[3]>
#> [1] Geometric(0.2) Geometric(0.5) Geometric(0.8)
mean(dist)
#> [1] 4.00 1.00 0.25
variance(dist)
#> [1] 20.0000 2.0000 0.3125
skewness(dist)
#> [1] 2.012461 2.121320 2.683282
kurtosis(dist)
#> [1] 6.05 6.50 9.20
generate(dist, 10)
#> [[1]]
#> [1] 11 6 4 1 0 6 4 2 0 1
#>
#> [[2]]
#> [1] 1 0 2 0 1 2 1 0 0 5
#>
#> [[3]]
#> [1] 0 0 0 0 0 0 0 0 0 0
#>
density(dist, 2)
#> [1] 0.128 0.125 0.032
density(dist, 2, log = TRUE)
#> [1] -2.055725 -2.079442 -3.442019
cdf(dist, 4)
#> [1] 0.67232 0.96875 0.99968
quantile(dist, 0.7)
#> [1] 5 1 0
```