# BinomialPointProcess

BinomialPointProcess[n,reg]

represents a binomial point process with n points in the region reg.

# Details

• BinomialPointProcess[n,reg] generates points that are uniformly distributed in reg with the total number of points equal to n.
• Typical uses include cases where the total number of points is specified, such as a known number of wireless transceivers in a region, number of fish in an aquarium, etc.
• BinomialPointProcess and PoissonPointProcess both yield a uniform distribution of points over a region. The former uses a deterministic number of points and the latter a random number of points.
• The number of points inside a bounded subregion follows BinomialDistribution[n,p] with p=RegionMeasure[sreg]/RegionMeasure[reg].
• The number of points in disjoint subregions of reg for a binomial point process is not independent.
• For disjoint subregions that satisfy , the joint distribution of the number of points inside the subregions with volume follows MultinomialDistribution[n,{ν1/ν,,νn/ν}], where is the volume of reg.
• BinomialPointProcess[n,reg] is conditioned on the event with a random variable following .
• BinomialPointProcess allows n to be any positive integer and reg any parameter-free region.
• BinomialPointProcess can be used with such functions as RipleyK and RandomPointConfiguration.

# Examples

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## Basic Examples(2)

Sample a BinomialPointProcess:

Sample several realizations:

## Scope(2)

Sample from any valid RegionQ whose RegionEmbeddingDimension is equal to its RegionDimension:

Create a binomial point process on a geo region:

Simulate points on the same region:

## Applications(1)

A shooter at a range shoots 12 bullets at random at a round target of diameter 1 meter. Simulate possible bullet patterns:

## Properties & Relations(4)

The number of points in a BinomialPointProcess is defined by n:

Simulate BinomialPointProcess over a unit disk:

The corresponding PointCountDistribution over a bounded subset:

Compare the histogram of point counts in the subset with the PDF:

Fit a BinomialDistribution to the point counts:

Test goodness of fit:

Test against theoretical distribution:

PointCountDistribution over region covering:

Define a three-set covering:

Point count distribution for the covering:

Compute void probabilities for a binomial point process:

For a rectangle:

Binomial point process is stationarythe intensity is translation invariant:

Point count distribution in a subregion:

Point count distribution in the translated subregion:

Wolfram Research (2020), BinomialPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/BinomialPointProcess.html.

#### Text

Wolfram Research (2020), BinomialPointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/BinomialPointProcess.html.

#### CMS

Wolfram Language. 2020. "BinomialPointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BinomialPointProcess.html.

#### APA

Wolfram Language. (2020). BinomialPointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BinomialPointProcess.html

#### BibTeX

@misc{reference.wolfram_2023_binomialpointprocess, author="Wolfram Research", title="{BinomialPointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/BinomialPointProcess.html}", note=[Accessed: 16-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_binomialpointprocess, organization={Wolfram Research}, title={BinomialPointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/BinomialPointProcess.html}, note=[Accessed: 16-April-2024 ]}