represents an inhomogeneous Poisson process with intensity λ[t] given as a function of t.
- InhomogeneousPoissonProcess is a continuous-time and discrete-state process.
- InhomogeneousPoissonProcess at time t is the number of events in the interval 0 to t.
- The number of events in the interval 0 to t follows PoissonDistribution with mean .
- The intensity function λ[t] in the definition of InhomogeneousPoissonProcess is assumed to be valid. In particular, it is assumed that it is a continuous, positive-valued function of t.
- InhomogeneousPoissonProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examplesopen allclose all
Basic Examples (3)
Basic Uses (5)
Process Slice Properties (5)
Univariate probability density:
Multi-time slice distribution:
Compute the expectation of an expression:
Calculate the probability of an event:
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:
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:
Mean and variance for the process slices:
An inhomogeneous Poisson process with Weibull failure rate intensity is known as Weibull Poisson process:
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:
Possible Issues (1)
Wolfram Research (2015), InhomogeneousPoissonProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.
Wolfram Language. 2015. "InhomogeneousPoissonProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html.
Wolfram Language. (2015). InhomogeneousPoissonProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InhomogeneousPoissonProcess.html