Poisson  (Distribution)

Definition of the Poisson distribution

The Poisson distribution is defined as the limit of the B(n, p) binomial distribution under the following conditions :

    * n tends to infinity,

    * p tends to 0,

in such a way that np tends to a positive constant l.

So the Poisson distribution is an alternative to the normal distribution for the purpose of approximating a binomial r.v. for large values of n and small values of p. The normal approximation is appropriate only when p is not too small (or too large), because if it is, then the binomial distribution is strongly skewed.

Probability mass function of the Poisson distribution

The Poisson distribution is discrete. Its support is the infinite set of integers k = 0, 1, 2, ... .

The probability for a realization to be equal to the integer k is :

Note that the Poisson distribution has only one parameter l, whereas the binomial distribution B(n, p) has two. This is of course due to the constraint

lim(np) = l.

Origin of the Poisson distribution

The Poisson distribution is most commonly encountered in settings where random events are occuring throughout time. Such typical settings are :

* Customers entering a shop.

* Telephone callers requesting a connection.

* Computing tasks requesting CPU or I/O service.

* etc...

The Poisson distribution will then often turn out to accurately describe the distribution of the number of events occuring during a time interval T.

In each of these settings, it may be assumed, as a first approximation, that there is a given probability p that one event will occur during a time interval Dt   (the approximation is that we ignore the possibility for more than just one event to occur during Dt). Then the number k of events that will occur during the time interval T = n.Dt  is binomial B(n, p).

As we reduce Dt (and p accordingly), and keep T constant, n increases. Our approximation also becomes more and more accurate (this point can be given a precise meaning within the context of Poisson processes). As Dt tends to 0, the number k of events that will occur during the time period T may be perceived as being drawn from binomial distribution with an infinitely small probability p and an infinitely large number n of "tosses", in such a way that np has a finite value l. This limit distribution is, by definition, the Poisson distribution.

Estimation of the parameter of the Poisson distribution

Sufficient statistic

We show here that the sample mean  is a sufficient statistic for the parameter l of the Poisson distribution.

Efficient estimator

We show here that the sample mean is an efficient estimator of the parameter l of the Poisson distribution.

Exponential family

We show here that the Poisson distribution belongs to the exponential family. This will lead us to another demonstration of the efficiency of the sample mean  as an estimator of l.

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We calculate here the probability mass function of the Poisson distribution as the limit of the binomial distribution for large n and small p with the product np converging to a positive value l.

We first guess its mean and variance by and intuitive argument, then calculate these quantities directly, using nothing but the ordinary definitions of the mean and of the variance.

We further calculate the moment generating function of the Poisson distribution, from which we will derive again the mean and the variance. This approach will turn out to be easier that the direct calculation.

BASIC PROPERTIES OF THE POISSON DISTRIBUTION

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    1) We here show that the sum of two independent Poisson r.v. is also Poisson, and we calculate its parameter. This important property is called "Additivity", and is also encountered with other classical r.v. (e.g. binomial, Gamma).

    2) We also address the reverse problem : how to decompose a Poisson r.v. X into two independent Poisson r.v. X₁ and X₂ in such a way that X₁ + X₂ = X. We show how this can be done by introducing a series of auxillary binomial r.v. Z_n~B(n, p), and calculate the parameters of X₁ and X₂ as functions of p.

    3) We finally show that the joint distribution of a set of Poisson variables conditionally to their sum is a multinomial distribution.

ADDITIVITY OF POISSON R.V.s, SPLITTING A POISSON R.V.

DISTRIBUTION OF POISSON VARIABLES CONDITIONALLY TO THEIR SUM IS MULTINOMIAL

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