Theory

Theory 1 - Poisson variable

Poisson variable

A random variable is Poisson, written , when counts the number of “arrivals” in a fixed “interval.” It is applicable when:

• The arrivals come at a constant average rate .
• The arrivals are independent of each other.

Poisson PMF:

A Poisson variable is comparable with a binomial variable. Both count the occurrences of some “arrivals” over some “space of opportunity.”

• The binomial opportunity is a set of repetitions of a trial.
• The Poisson opportunity is a continuous interval of time.

In the binomial case, success occurs at some rate , since where is the success event.

In the Poisson case, arrivals occur at some rate .

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The Poisson distribution is actually the limit of binomial distributions by taking while remains fixed, so in perfect balance with .

Let and let . Fix and let . Then for any :

For example, let , so , and look at as :

Interpretation - Binomial model of rare events

Let us interpret the assumptions of this limit. For large but small such that remains moderate, the binomial distribution describes a large number of trials, a low probability of success per trial, but a moderate total count of successes.

This setup describes a physical system with a large number of parts that may activate, but each part is unlikely to activate; and yet the number of parts is so large that the total number of arrivals is still moderate.

Theory 2 - Poisson limit of binomial

Extra - Derivation of binomial limit to Poisson

Consider a random variable , and suppose is very large.

Suppose also that is very small, such that is not very large, but the extremes of and counteract each other. (Notice that then will not be large so the normal approximation does not apply.) In this case, the binomial PMF can be approximated using a factor of . Consider the following rearrangement of the binomial PMF:

Since is very large, the factor in brackets is approximately , and since is very small, the last factor of is also approximately 1 (provided we consider small compared to ). So we have:

Write , a moderate number, to find:

Here at last we find , since as . So as :

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Extra - Binomial limit to Poisson and divisibility

Consider a sequence of increasing with decreasing such that is held fixed. For example, let while .

Think of this process as increasing the number of causal agents represented: group the agents together into bunches, and consider the odds that such a bunch activates. (For the call center, a bunch is a group of users; for radioactive decay, a bunch is a unit of mass of Uranium atoms.)

As doubles, we imagine the number of agents per bunch to drop by half. (For the call center, we divide a group in half, so twice as many groups but half the odds of one group making a call; for the Uranium, we divide a chunk of mass in half, getting twice as many portions with half the odds of a decay occurring in each portion.

This process is formally called division of a distribution, and the fact that the Poisson distribution arises as the limit of such division means that it is infinitely divisible.

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Extra - Theorem: Poisson approximation of the binomial

Suppose and . Then:

for any .

In consequence of this theorem, a Poisson distribution may be used to approximate the probabilities of a binomial distribution for large when it is impracticable (even for a computer) to calculate large binomial coefficients.

The theorem shows that the Poisson approximation is appropriate when is a moderate number while is a small number.