Probability mass function:

Cumulative distribution function:

Mean and variance of a Poisson distribution are the same:

Generate a sample of pseudorandom numbers from a Poisson distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness:

A Poisson distribution becomes more symmetric for large values of μ:

Kurtosis:

Kurtosis of normal distribution is the limiting value for kurtosis of a Poisson distribution:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Closed form for symbolic order:

Cumulant:

All cumulants equal μ:

Hazard function:

Quantile function:

CDF of PoissonDistribution is an example of a right-continuous function:

The average number of accidents in a city is 100 per day. Simulate the daily accidents:

Find the probability that there are 90 or more accidents on a single day:

Find the standard deviation of the number of accidents per day:

The expected number of raindrops falling into a bucket in a 5-second interval is 20. Simulate the raindrop count for each 5-second interval:

Find the probability that exactly 20 raindrops fall into the bucket in 5 seconds:

A radioactive material on average emits 3.2 -particles per second; show the distribution:

Compute the probability that more than 4 -particles are emitted over the next second:

Simulate a typical particle count per second over 10 minutes:

Suppose flaws in plywood occur on average of one flaw per 50 square feet. Simulate the process of finding flaws on a per-square-foot basis:

Find the probability that a 4-foot-by-8-foot sheet will have no flaws:

For a mirror with area 7.54 cm, the probability of no flaws is 0.9100. Using the same polishing process, another mirror with an area of 19.50 cm is fabricated. Assuming a Poisson error process, find the probability of no flaws on the larger mirror. Using the conditions for the smaller mirror and with error distribution PoissonDistribution [λ area]:

The resulting mirror polishing defect distribution is then:

The probability of no errors in the larger mirror:

Typographical errors in a book are occurring randomly according to a Poisson process. On 384 pages, 158 errors are counted. Find the distribution of errors per page where the distribution is of the form PoissonDistribution[λ p], where p is the number of pages:

Find the probability of exactly 0 errors per page:

Find the probability of fewer than 2 errors per page:

Find the probability of 1 or more errors per page:

In modeling an adverse reaction to a medicine, it is determined that on average 2 people per 100000 have the adverse reaction. Assuming a Poisson distribution, find the adverse reaction distribution:

Find the probability that there will be at least 5 adverse reactions when the medicine is administered to 350000 people:

The number of queries arriving in seconds to a call center is Poisson distributed with parameter , where is the average arrival rate in queries per second. Assuming that the average arrival rate is 4 queries per minute, find the probability of getting more than 4 queries in 10 seconds:

Find the probability that there are fewer than 5 queries in 2 minutes:

The number of packet arrivals in seconds at a multiplexer is Poisson distributed with parameter , where is the average arrival rate in packets per second. Find the probability that there are no packet arrivals in seconds:

Find the probability that or fewer packets arrive in seconds:

A data center has 10000 disk drives. Suppose that a disk fails in a given day with probability . Find the probability that there are no failures in a given day:

That distribution is closely approximated with a Poisson distribution with parameter :

Find the probability that there are fewer than 10 failures in 2 days:

Find the number of spare disk drives that should be available so that all failures in a day can be replaced with probability 99.9%:

Plot the cumulative probability that the number of failures is less than :

Find the minimum number for which the probability is greater than 99.9%:

Check the result:

An LCD display has 1920×1080 pixels. A display is accepted if it has 15 or fewer faulty pixels. The probability that a pixel is faulty from production is . Find the proportion of displays that are accepted:

Find the pixel failure rate required to produce 4000×2000 pixel displays and still have an acceptance rate of at least 90%:

Plot the acceptance rate as a function of the pixel failure rate:

Find the maximal acceptable pixel failure rate:

Check the result:

A switchboard receives on average 100 calls per minute. Find what the switchboard capacity should be so that it gets saturated less than once in every 60 minutes:

Find the minimum capacity that satisfies the constraint:

In an optical communication system, transmitted light generates current at the receiver. The number of electrons follows the parametric mixture of a Poisson distribution and another distribution, depending on the type of light. If the source uses coherent laser light of intensity , then the electron count distribution is Poisson:

Which is PoissonDistribution:

If the source uses thermal illumination, then the Poisson parameter follows ExponentialDistribution with parameter and the electron count distribution is:

These two distributions are distinguishable and allow you to determine the type of source:

The sum of Poisson variables is Poisson distributed:

Relationships to other distributions:

PoissonDistribution is a limiting case for BinomialDistribution with :

PoissonConsulDistribution simplifies to Poisson distribution:

Poisson distribution is a limiting case for PolyaAeppliDistribution:

NegativeBinomialDistribution is a mixture of PoissonDistribution and GammaDistribution:

GeometricDistribution is a mixture of PoissonDistribution and GammaDistribution:

The parametric mixture of Poisson distribution and ExponentialDistribution follows GeometricDistribution:

PoissonConsulDistribution is a parameter mixture of BorelTannerDistribution and PoissonDistribution:

The limit of NegativeBinomialDistribution when the mean is fixed is Poisson distribution:

For large μ, the PoissonDistribution can be approximated by NormalDistribution:

PoissonDistribution is not defined when μ is non-positive:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: