examples on poisson distribution

A particularity of the Poisson distribution is that the convolution of m such distributions with parameters a1, ..., am is again a Poisson distribution with parameter a = a1+a2+...+am, and it is the only distribution with this convenient property.

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The number of telephone calls arriving at a switchboard during any specified time interval may have a Poisson distribution, and the number of calls arriving during one time interval may be Quick Facts about: statistically independent
Quick Summary not found for this subjectstatistically independent of the number of calls arriving during any other non-overlapping time interval. This is a one-dimensional Poisson process. In simple models, one may assume a constant average rate of arrival, e.g., λ = 12.3 calls per minute. In that case, the Quick Facts about: expected value
The sum of the values of a random variable divided by the number of valuesexpected value of the number of calls in any time interval is that rate times the amount of time, λt. In messier and more realistic problems, one uses a non-constant rate function λ(t). In that case, the expected value of the number of calls between time a and time b is
The number of bombs falling on a specified area of London in the early days of the Second World War may be a random variable with a Poisson distribution, and the number of bombs falling on two areas of the city that do not overlap may be statistically independent. This is a 2-dimensional Poisson process.
Astonomers may treat the number of stars in a given volume of space as a random variable with a Poisson distribution, and the numbers of stars in any two or more non-overlapping regions as statistically independent. This is a 3-dimensional Poisson process.