HyperGeometric {distributions3} | R Documentation |

To understand the HyperGeometric distribution, consider a set of
`r`

objects, of which `m`

are of the type I and
`n`

are of the type II. A sample with size `k`

(`k<r`

)
with no replacement is randomly chosen. The number of observed
type I elements observed in this sample is set to be our random
variable `X`

. For example, consider that in a set of 20
car parts, there are 4 that are defective (type I).
If we take a sample of size 5 from those car parts, the
probability of finding 2 that are defective will be given by
the HyperGeometric distribution (needs double checking).

```
HyperGeometric(m, n, k)
```

`m` |
The number of type I elements available. |

`n` |
The number of type II elements available. |

`k` |
The size of the sample taken. |

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 HyperGeometric random variable with
success probability `p`

= `p = m/(m+n)`

.

**Support**: `x \in { \{\max{(0, k-n)}, \dots, \min{(k,m)}}\}`

**Mean**: `\frac{km}{n+m} = kp`

**Variance**: ```
\frac{km(n)(n+m-k)}{(n+m)^2 (n+m-1)} =
kp(1-p)(1 - \frac{k-1}{m+n-1})
```

**Probability mass function (p.m.f)**:

```
P(X = x) = \frac{{m \choose x}{n \choose k-x}}{{m+n \choose k}}
```

**Cumulative distribution function (c.d.f)**:

```
P(X \le k) \approx \Phi\Big(\frac{x - kp}{\sqrt{kp(1-p)}}\Big)
```

**Moment generating function (m.g.f)**:

Not useful.

A `HyperGeometric`

object.

Other discrete distributions:
`Bernoulli()`

,
`Binomial()`

,
`Categorical()`

,
`Geometric()`

,
`HurdleNegativeBinomial()`

,
`HurdlePoisson()`

,
`Multinomial()`

,
`NegativeBinomial()`

,
`Poisson()`

,
`ZINegativeBinomial()`

,
`ZIPoisson()`

,
`ZTNegativeBinomial()`

,
`ZTPoisson()`

```
set.seed(27)
X <- HyperGeometric(4, 5, 8)
X
random(X, 10)
pdf(X, 2)
log_pdf(X, 2)
cdf(X, 4)
quantile(X, 0.7)
```

[Package *distributions3* version 0.2.1 Index]