represents a Poisson process with rate μ.
- PoissonProcess is a continuous-time and discrete-state random process.
- PoissonProcess 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[μ t].
- The times between events are independent and follow ExponentialDistribution[μ].
- PoissonProcess allows μ to be any positive real number.
- PoissonProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examplesopen allclose all
Basic Uses (6)
Process Slice Properties (6)
CentralMoment has no closed form for symbolic order:
FactorialMoment and its generating function:
Cumulant and its generating function:
Inquiries arrive at a recorded messaging device according to a Poisson process rate of 15 inquiries per minute. Find the probability that in a one-minute period, three inquiries arrive during the first 10 seconds and two inquiries arrive during the last 15 seconds:
An insurance company has two types of policies, A and B. Total claims from the company arrive according to a Poisson process at the rate of nine per day. Find the probability that the total claims from the company will be fewer than two on a given day:
A server handles queries that arrive according to a Poisson process with a rate of 10 queries per minute. Find the probability that no queries go unanswered if the server is unavailable for 20 seconds:
The number of failures that occur in a computer network follow a Poisson process. On average, there is a failure after every four hours. Find the probability that the third failure occurs after eight hours:
The failures of a certain machine occur according to a Poisson process with a rate of per week. Find the probability that the machine will have at least one failure during each of the first two weeks considered:
Travelers arrive at a bus station starting at 6am, according to a Poisson process with a rate of one per two minutes. Find the mean and variance for the number of passengers on the first bus to leave after 6am if the bus departures follow an exponential distribution with a mean of 15 minutes:
The number of flaws appearing on a polished mirror surface is a Poisson random variable. For a mirror with an area of 8.54 cm, the probability of no flaws is 0.91. Using the same process, another mirror with an area of 17.50 cm is fabricated. Find the probability of no flaws on the larger mirror:
A light bulb has a lifetime that is exponential with a mean of 200 days. When it burns out, a janitor replaces it immediately. In addition, there is a handyman who comes at times with a Poisson rate of 0.01 and replaces the light bulb as a part of preventive maintenance. Find the mean number of days after which the light bulb is replaced:
At night, vehicles circulate on a certain highway with separate roadways according to a Poisson process with rate 2 per minute in each direction. Due to an accident, traffic must be stopped in one direction. Suppose that 60% of the vehicles are cars, 30% are trucks, and 10% are semitrailers. Suppose also that the length of a car is equal to 5 meters, that of a truck is equal to 10 meters, and that of a semitrailer is 20 meters. Find the time at which there is a 10% probability that the length of the queue is greater than 1 km:
Properties & Relations (10)
PoissonProcess is a jump process:
The time between events in a Poisson process follows an ExponentialDistribution:
InhomogeneousPoissonProcess with constant intensity is a Poisson process:
Parameter mixture distribution of a slice distribution follows GeometricDistribution::
Wolfram Research (2012), PoissonProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/PoissonProcess.html.
Wolfram Language. 2012. "PoissonProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PoissonProcess.html.
Wolfram Language. (2012). PoissonProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PoissonProcess.html