Pólya distribution

The Pólya distribution describes a certain type of random experiment and thus belongs to stochastics . It is named after the Hungarian-American mathematician George Pólya . The Pólya distribution is also known as the distribution of contagion because it can be used to characterize the process that a sick person infects others.

The concept of the Pólyas model is not clear: In the literature there are various short descriptions that not only include more or less direct generalizations of the standard experiment, but sometimes even transition from the usual discrete to the continuous case. Nevertheless, the basic principle is always comparable.

Pólya distribution concept

The concept of polya distribution can be demonstrated using a model urn : an urn contains two types of balls, for example red and blue. You choose a ball at random. This ball is put back again. In addition, more balls of the same color are added to the urn (from the outside). This process is carried out several times. The random variable is the number of experiments in which a red ball is drawn when the random process conducts times. Such a random variable is called pólya-distributed. ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle n}$ ${\ displaystyle X_ {n}}$ ${\ displaystyle n}$

distribution

Let the proportions of the balls in the urn be defined as

{\ displaystyle p = {\ frac {a} {N}}, \ quad q = {\ frac {b} {N}}, \ quad r = {\ frac {c} {N}},}

with . ${\ displaystyle N = a + b}$

The probability function of is then ${\ displaystyle X_ {n}}$

{\ displaystyle P (X_ {n} = x) = {n \ choose x} {\ frac {p \ cdot (p + r) \ cdot (p + 2r) \ cdots (p + (x-1) r) \ cdot q \ cdot (q + r) \ cdot (q + 2r) \ cdots (q + (nx-1) r)} {1 \ cdot (1 + r) \ cdot (1 + 2r) \ cdots (1+ ( n-1) r)}}}

for the occurrences of the random variables as ${\ displaystyle X}$

{\ displaystyle x = {\ begin {cases} 0,1, \ dots, n & {\ text {if}} c \ geq 0 \\\ max (0, \, nb), \ dots, \ min (a, \, n) & {\ text {falls}} c = -1 \\\ end {cases}}}

.

For other values of the probability is set equal to zero. ${\ displaystyle x}$

The expectation of is ${\ displaystyle X_ {n}}$

{\ displaystyle \ operatorname {E} (X_ {n}) = n \ cdot p}

and the variance is

{\ displaystyle \ operatorname {Var} (X_ {n}) = p \ cdot q \ cdot n \ cdot {\ frac {1 + r \ cdot n} {1 + r}}}

.

application

Consider two different pathogens A and B , both of which are spreading in the same area. They both reproduce epidemically, but hinder each other. (Similarly, one could also imagine two competing corporations, etc.) If a person comes with one of the pathogens - e.g. B. A - in contact, it remains infected, but becomes immune to the competing pathogen B. Virus A will now reproduce in the new host and distribute its copies in the area (e.g. by sneezing). Assuming that the newly generated pathogens can spread over the entire area quickly enough (e.g. by winds), the probability of the next victim being infected with A increases. For the sake of simplicity, people should be infected one after the other and there should be enough time to mix between the new infections. What is the probability of a certain sequence of infections with A or B ? The problem could also be rewritten:

It's not a virus, but a larger insect that jumps from person to person
The virus does not reproduce but is destroyed by the immune system.
Once in contact with the host, it produces enough antibodies to immediately destroy a whole range of other viruses of the same type (e.g. anti-virus software).

Special cases and generalizations

Special cases of polya distribution

With only the drawn ball is put back into the urn, so you get the binomial distribution with the parameters and . ${\ displaystyle c = 0}$ ${\ displaystyle n}$ ${\ displaystyle p}$

With no ball is replaced, the result is an urn model without replacing. Is thus obtained in a dichotomous population (two kinds of balls) is a hypergeometric distribution with the parameters , , and . ${\ displaystyle c = -1}$ ${\ displaystyle N}$ ${\ displaystyle a}$ ${\ displaystyle n}$

${\ displaystyle c = 1}$ describes the classic constellation of the Pólya distribution.

Generalizations

A person can be infected more than once.
There are more than two different types of pathogen.
The set of possible types of spheres becomes a continuum.

literature

PH Müller (Ed.): Lexicon of Stochastics , Berlin 1991

Web links

Polya urn - Pollyanna - May 1, 2014 (pdf; 244 kB)
Use of several "infections" (pdf; 147 kB)
More than two "pathogen types" (pdf; 2.09 MB)
A continuum of possible types of spheres