# InhomogeneousPoissonProcess

InhomogeneousPoissonProcess[λ[t],t]

represents an inhomogeneous Poisson process with intensity λ[t] given as a function of t.

# Examples

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## Basic Examples(3)

Simulate an inhomogeneous Poisson process:

Mean and variance functions:

Covariance function:

## Scope(10)

### Basic Uses(5)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for different intensity functions:

Correlation function:

Absolute correlation function:

### Process Slice Properties(5)

Univariate SliceDistribution:

Univariate probability density:

Multi-time slice distribution:

Higher-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Generating functions:

CentralMoment has no closed form for symbolic order:

FactorialMoment and its generating function:

Cumulant and its generating function:

Covariance for a multi-time slice distribution:

Compare with the result from simulation:

## Applications(3)

Simulate the arrival process at a small fast-food restaurant if the hourly arrival rates of customers are given by:

Use linear interpolation to obtain the intensity function for the arrival process:

Define an inhomogeneous Poisson process for the arrivals:

Probability that more than 200 customers visit the restaurant during the day:

Simulate the arrival process for a day:

Use simulation to find the effective hourly mean arrival rate:

Define the square of an inhomogeneous Poisson process:

Simulate the process:

Mean and variance for the process slices:

An inhomogeneous Poisson process with Weibull failure rate intensity is known as Weibull Poisson process:

Sample process trajectories:

Use simulation to find the effective mean intensity rate for a day:

Compare to the mean intensity rate given by the integral of the rate function:

## Properties & Relations(3)

InhomogeneousPoissonProcess is a jump process:

An inhomogeneous Poisson process is not weakly stationary:

The mean function is not constant:

An inhomogeneous Poisson process with constant intensity is a PoissonProcess:

Compare univariate slice distributions:

Multi-slice properties:

## Possible Issues(1)

Some of the simulation methods require a bounded intensity function:

The default "Thinning" method fails:

The inversion method also fails for this intensity function:

The direct method is suitable for an unbounded intensity function, although it may be slow:

## Neat Examples(1)

Simulate paths from an inhomogeneous Poisson process:

Take a slice at 50 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 50:

Wolfram Research (2015), InhomogeneousPoissonProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.

#### Text

Wolfram Research (2015), InhomogeneousPoissonProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.

#### CMS

Wolfram Language. 2015. "InhomogeneousPoissonProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.

#### APA

Wolfram Language. (2015). InhomogeneousPoissonProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html

#### BibTeX

@misc{reference.wolfram_2022_inhomogeneouspoissonprocess, author="Wolfram Research", title="{InhomogeneousPoissonProcess}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html}", note=[Accessed: 27-September-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2022_inhomogeneouspoissonprocess, organization={Wolfram Research}, title={InhomogeneousPoissonProcess}, year={2015}, url={https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html}, note=[Accessed: 27-September-2022 ]}