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Difference between revisions of "Pólya distribution"

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The probability distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735401.png" /> taking non-negative integer values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735403.png" />, given by the formula
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735404.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735405.png" /></td> </tr></table>
+
The probability distribution of a random variable  $  X _ {n} $
 +
taking non-negative integer values  $  k $,
 +
0 \leq  k \leq  n $,
 +
given by the formula
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735406.png" /> and the integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p0735409.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354010.png" /> are parameters; or, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354011.png" />, give by the equivalent formula
+
$$
 +
{\mathsf P} \{ X _ {n} = k \} =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
= \
 +
\left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right )
 +
\frac{( b; c) _ {k-} 1 ( r; c) _ {n-} k- 1 }{( b+ r; c) _ {n-} 1 }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354013.png" /></td> </tr></table>
+
where  $  ( b; c) _ {k-} 1 = b( b+ c) \dots [ b+( k- 1) c] $
 +
and the integers  $  n > 0 $,
 +
$  b > 0 $,
 +
$  r > 0 $,
 +
and  $  c \geq  - 1 $
 +
are parameters; or, when  $  c > 0 $,
 +
give by the equivalent formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354014.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
{\mathsf P} \{ X _ {n} = k \} =
 +
$$
  
where the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354015.png" />, the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354018.png" /> are parameters. The relation between
+
$$
 +
= \
 +
\left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right )
 +
\frac{( p;  \gamma ) _ {k-} 1 ( q;  \gamma ) _ {n-} k- 1 }{( 1;  \gamma ) _ {n-} 1 }
 +
=
 +
$$
 +
 
 +
$$
 +
= \
 +
 
 +
\frac{\left ( \begin{array}{c}
 +
( p/ \gamma )+ k- 1 \\
 +
k
 +
\end{array}
 +
\right )
 +
\left ( \begin{array}{c}
 +
( q/ \gamma )+ n- k- 1 \\
 +
n- k
 +
\end{array}
 +
\right ) }{
 +
\left ( \begin{array}{c}
 +
( 1/ \gamma )+ n- 1 \\
 +
n
 +
\end{array}
 +
\right ) }
 +
,
 +
$$
 +
 
 +
where the integer  $  n > 0 $,  
 +
the real numbers $  0 < p < 1 $,  
 +
$  q = 1- p $,  
 +
and $  \gamma > 0 $
 +
are parameters. The relation between
  
 
and (2) is given by
 
and (2) is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354019.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{b}{b+}
 +
r ,\ \
 +
=
 +
\frac{r}{b+}
 +
r ,\ \
 +
\gamma  =
 +
\frac{c}{b+}
 +
r .
 +
$$
  
The mathematical expectation and variance of the Pólya distribution are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354021.png" />, respectively. Special cases of the Pólya distribution are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354022.png" /> has a [[Binomial distribution|binomial distribution]] with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354024.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354025.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354026.png" /> has a [[Hypergeometric distribution|hypergeometric distribution]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354027.png" /> with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354030.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354033.png" /> is constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354034.png" />, the distribution tends to the binomial distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354036.png" />.
+
The mathematical expectation and variance of the Pólya distribution are $  {\mathsf E} X _ {n} = np $
 +
and $  {\mathsf D} X _ {n} = npq( 1 + \gamma n)/( 1 + \gamma ) $,  
 +
respectively. Special cases of the Pólya distribution are: $  X _ {n} $
 +
has a [[Binomial distribution|binomial distribution]] with parameters $  n $
 +
and p $
 +
for $  c = 0 $;  
 +
$  X _ {n} $
 +
has a [[Hypergeometric distribution|hypergeometric distribution]] for $  s = - 1 $
 +
with parameters $  M = b $,  
 +
$  N = b+ r $
 +
and $  n $.  
 +
For $  b \rightarrow \infty $
 +
and $  r \rightarrow \infty $
 +
such that $  p = b/( b+ r) $
 +
is constant and $  \gamma = c/( b+ r) \rightarrow 0 $,  
 +
the distribution tends to the binomial distribution with parameters $  n $
 +
and p $.
  
The distribution was considered by G. Pólya (1923) in connection with the so-called Pólya urn scheme. From an urn containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354037.png" /> black and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354038.png" /> red balls one makes a selection with replacement, subject to the condition that each extracted ball is returned to the urn together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354039.png" /> balls of the same colour. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354040.png" /> is the total number of black balls at the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354041.png" />-th trial, the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354042.png" /> is given by
+
The distribution was considered by G. Pólya (1923) in connection with the so-called Pólya urn scheme. From an urn containing $  b $
 +
black and $  r $
 +
red balls one makes a selection with replacement, subject to the condition that each extracted ball is returned to the urn together with $  c $
 +
balls of the same colour. If $  X _ {n} $
 +
is the total number of black balls at the $  n $-
 +
th trial, the distribution of $  X _ {n} $
 +
is given by
  
or (2). The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354044.png" /> is a discrete [[Markov process|Markov process]], where the states are defined by the numbers of black balls in the sample at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354045.png" />, while the conditional probability of transition from a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354046.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354047.png" /> to a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354048.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354049.png" /> equals
+
or (2). The sequence $  X _ {n} $,
 +
$  n = 1, 2 \dots $
 +
is a discrete [[Markov process|Markov process]], where the states are defined by the numbers of black balls in the sample at time $  n $,  
 +
while the conditional probability of transition from a state $  k $
 +
at time $  n $
 +
to a state $  k+ 1 $
 +
at time $  n+ 1 $
 +
equals
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354050.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ X _ {n+} 1 = k+ 1 \mid  X _ {k} = k \}  = \
  
(this depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354051.png" />).
+
\frac{b + kc }{b + r + nc }
 +
  =
 +
\frac{p + k \gamma }{1 + n \gamma }
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354052.png" />, one obtains the conditional limit probability of transition from the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354053.png" /> to state the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354054.png" /> during the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354056.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354058.png" />, as
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354059.png" /></td> </tr></table>
+
(this depends on  $  n $).
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354060.png" /> of remaining in the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354061.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354062.png" /> is
+
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  $  \Delta t $,  
 +
one obtains the conditional limit probability of transition from the state $  k $
 +
to state the  $  k+ 1 $
 +
during the time $  \Delta t $
 +
for  $  n \rightarrow \infty $
 +
when  $  np \rightarrow t $
 +
and  $  n \gamma \rightarrow \alpha t $,
 +
as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354063.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ X ( t+ \Delta t) = k + 1 \mid  X( t) = k \}  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354064.png" /></td> </tr></table>
+
\frac{1 + \alpha k }{1 + \alpha t }
 +
\Delta t + o( \Delta t).
 +
$$
  
This limit distribution is the [[Negative binomial distribution|negative binomial distribution]] with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354066.png" /> (the corresponding mathematical expectation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354067.png" />, while the variance is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354068.png" />).
+
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  $  {\mathsf P} _ {k} ( t) $
 +
of remaining in the state  $  k $
 +
at time  $  t $
 +
is
 +
 
 +
$$
 +
{\mathsf P} _ {n} ( t)  = \
 +
\left ( \begin{array}{c}
 +
{( 1 / \alpha ) + n- 1 } \\
 +
n
 +
\end{array}
 +
\right )
 +
\left (
 +
\frac{\alpha t }{1 + \alpha t }
 +
\right )  ^ {n}
 +
\left (
 +
\frac{1}{1 + \alpha t }
 +
\right ) ^ {1/ \alpha }
 +
$$
 +
 
 +
$$
 +
( {\mathsf P} _ {0} ( 0)  =  1).
 +
$$
 +
 
 +
This limit distribution is the [[Negative binomial distribution|negative binomial distribution]] with parameters $  1/ \alpha $
 +
and $  1/( 1+ \alpha t) $(
 +
the corresponding mathematical expectation is $  t $,  
 +
while the variance is $  t( 1 + \alpha t) $).
  
 
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).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354069.png" /> tends to zero, the Pólya process goes over to a [[Poisson process|Poisson process]], while the Pólya distribution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354070.png" /> has as limit the [[Poisson distribution|Poisson distribution]] with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073540/p07354071.png" />.
+
As $  \alpha $
 +
tends to zero, the Pólya process goes over to a [[Poisson process|Poisson process]], while the Pólya distribution for $  \alpha \rightarrow 0 $
 +
has as limit the [[Poisson distribution|Poisson distribution]] with parameter $  t $.
  
 
====References====
 
====References====
Line 49: Line 196:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Urn models and their application" , Wiley  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Urn models and their application" , Wiley  (1977)</TD></TR></table>

Latest revision as of 08:08, 6 June 2020


The probability distribution of a random variable $ X _ {n} $ taking non-negative integer values $ k $, $ 0 \leq k \leq n $, given by the formula

$$ {\mathsf P} \{ X _ {n} = k \} = $$

$$ = \ \left ( \begin{array}{c} n \\ k \end{array} \right ) \frac{( b; c) _ {k-} 1 ( r; c) _ {n-} k- 1 }{( b+ r; c) _ {n-} 1 } , $$

where $ ( b; c) _ {k-} 1 = b( b+ c) \dots [ b+( k- 1) c] $ and the integers $ n > 0 $, $ b > 0 $, $ r > 0 $, and $ c \geq - 1 $ are parameters; or, when $ c > 0 $, give by the equivalent formula

$$ \tag{2 } {\mathsf P} \{ X _ {n} = k \} = $$

$$ = \ \left ( \begin{array}{c} n \\ k \end{array} \right ) \frac{( p; \gamma ) _ {k-} 1 ( q; \gamma ) _ {n-} k- 1 }{( 1; \gamma ) _ {n-} 1 } = $$

$$ = \ \frac{\left ( \begin{array}{c} ( p/ \gamma )+ k- 1 \\ k \end{array} \right ) \left ( \begin{array}{c} ( q/ \gamma )+ n- k- 1 \\ n- k \end{array} \right ) }{ \left ( \begin{array}{c} ( 1/ \gamma )+ n- 1 \\ n \end{array} \right ) } , $$

where the integer $ n > 0 $, the real numbers $ 0 < p < 1 $, $ q = 1- p $, and $ \gamma > 0 $ are parameters. The relation between

and (2) is given by

$$ p = \frac{b}{b+} r ,\ \ q = \frac{r}{b+} r ,\ \ \gamma = \frac{c}{b+} r . $$

The mathematical expectation and variance of the Pólya distribution are $ {\mathsf E} X _ {n} = np $ and $ {\mathsf D} X _ {n} = npq( 1 + \gamma n)/( 1 + \gamma ) $, respectively. Special cases of the Pólya distribution are: $ X _ {n} $ has a binomial distribution with parameters $ n $ and $ p $ for $ c = 0 $; $ X _ {n} $ has a hypergeometric distribution for $ s = - 1 $ with parameters $ M = b $, $ N = b+ r $ and $ n $. For $ b \rightarrow \infty $ and $ r \rightarrow \infty $ such that $ p = b/( b+ r) $ is constant and $ \gamma = c/( b+ r) \rightarrow 0 $, the distribution tends to the binomial distribution with parameters $ n $ and $ p $.

The distribution was considered by G. Pólya (1923) in connection with the so-called Pólya urn scheme. From an urn containing $ b $ black and $ r $ red balls one makes a selection with replacement, subject to the condition that each extracted ball is returned to the urn together with $ c $ balls of the same colour. If $ X _ {n} $ is the total number of black balls at the $ n $- th trial, the distribution of $ X _ {n} $ is given by

or (2). The sequence $ X _ {n} $, $ n = 1, 2 \dots $ is a discrete Markov process, where the states are defined by the numbers of black balls in the sample at time $ n $, while the conditional probability of transition from a state $ k $ at time $ n $ to a state $ k+ 1 $ at time $ n+ 1 $ equals

$$ {\mathsf P} \{ X _ {n+} 1 = k+ 1 \mid X _ {k} = k \} = \ \frac{b + kc }{b + r + nc } = \frac{p + k \gamma }{1 + n \gamma } $$

(this depends on $ n $).

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 $ \Delta t $, one obtains the conditional limit probability of transition from the state $ k $ to state the $ k+ 1 $ during the time $ \Delta t $ for $ n \rightarrow \infty $ when $ np \rightarrow t $ and $ n \gamma \rightarrow \alpha t $, as

$$ {\mathsf P} \{ X ( t+ \Delta t) = k + 1 \mid X( t) = k \} = \ \frac{1 + \alpha k }{1 + \alpha t } \Delta t + o( \Delta t). $$

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 $ {\mathsf P} _ {k} ( t) $ of remaining in the state $ k $ at time $ t $ is

$$ {\mathsf P} _ {n} ( t) = \ \left ( \begin{array}{c} {( 1 / \alpha ) + n- 1 } \\ n \end{array} \right ) \left ( \frac{\alpha t }{1 + \alpha t } \right ) ^ {n} \left ( \frac{1}{1 + \alpha t } \right ) ^ {1/ \alpha } $$

$$ ( {\mathsf P} _ {0} ( 0) = 1). $$

This limit distribution is the negative binomial distribution with parameters $ 1/ \alpha $ and $ 1/( 1+ \alpha t) $( the corresponding mathematical expectation is $ t $, while the variance is $ t( 1 + \alpha t) $).

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 $ \alpha $ tends to zero, the Pólya process goes over to a Poisson process, while the Pólya distribution for $ \alpha \rightarrow 0 $ has as limit the Poisson distribution with parameter $ t $.

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)
How to Cite This Entry:
Pólya distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P%C3%B3lya_distribution&oldid=48357
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article