Pólya distribution
The probability distribution of a random variable 
taking non-negative integer values 
, 
, given by the formula
 |
 |
where 
and the integers 
, 
, 
, and 
are parameters; or, when 
, give by the equivalent formula
 | (2) |
 |
 |
where the integer 
, the real numbers 
, 
, and 
are parameters. The relation between
and (2) is given by
 |
The mathematical expectation and variance of the Pólya distribution are 
and 
, respectively. Special cases of the Pólya distribution are: 
has a binomial distribution with parameters 
and 
for 
; 
has a hypergeometric distribution for 
with parameters 
, 
and 
. For 
and 
such that 
is constant and 
, the distribution tends to the binomial distribution with parameters 
and 
.
The distribution was considered by G. Pólya (1923) in connection with the so-called Pólya urn scheme. From an urn containing 
black and 
red balls one makes a selection with replacement, subject to the condition that each extracted ball is returned to the urn together with 
balls of the same colour. If 
is the total number of black balls at the 
-th trial, the distribution of 
is given by
or (2). The sequence 
, 
is a discrete Markov process, where the states are defined by the numbers of black balls in the sample at time 
, while the conditional probability of transition from a state 
at time 
to a state 
at time 
equals
 |
(this depends on 
).
By passing to the limit from the Pólya urn scheme one obtains the Pólya process, which is an inhomogeneous Markov process in continuous time and belongs to the class of "pure multiplication" processes. Under the condition that there is only one extraction of a ball in an infinitesimal time 
, one obtains the conditional limit probability of transition from the state 
to state the 
during the time 
for 
when 
and 
, as
 |
On transition from the Pólya urn scheme to the Pólya process one obtains an important limit form for the Pólya distribution. That is, the probability 
of remaining in the state 
at time 
is
 |
 |
This limit distribution is the negative binomial distribution with parameters 
and 
(the corresponding mathematical expectation is 
, while the variance is 
).
The urn model and the Pólya process, in which the Pólya distribution and the limit form of it arise, are models with an after effect (extracting a ball of a particular colour from the urn increases the probability of extracting a ball of the same colour in a subsequent trial).
As 
tends to zero, the Pólya process goes over to a Poisson process, while the Pólya distribution for 
has as limit the Poisson distribution with parameter 
.
References
| [1] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1968) |
Comments
References
| [a1] | N.L. Johnson, S. Kotz, "Urn models and their application" , Wiley (1977) |
Pólya distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P%C3%B3lya_distribution&oldid=25927