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A discrete [[Probability distribution|probability distribution]] of a random variable  $  X $
taking non-negative integer values  $  k = 0, 1 \dots $
in accordance with the formula

$$ 
{\mathsf P} \{ X = k \}  = \ 
\left ( \begin{array}{c}
r+ k- 1 \\
 r- 1 
\end{array}
 \right ) p  ^ {r} ( 1- p)  ^ {k} ,
$$

where  $  0 < p < 1 $
and the integers  $  r > 0 $
are parameters.

The generating function and characteristic function of a Pascal distribution are

$$ 
P( z)  =  p  ^ {r} ( 1- qz)  ^ {-} r
$$

and

$$ 
f( t)  =  p  ^ {r} ( 1- qe  ^ {it} )  ^ {-} r ,\ \ 
q = 1- p.
$$

The mathematical expectation and the variance are  $  rq/p $
and  $  rq/p  ^ {2} $, 
respectively.

The Pascal distribution with parameters  $  r $
and  $  p $
arises naturally in the scheme of the Bernoulli trial (cf. [[Bernoulli trials|Bernoulli trials]]) with probability of  "success"   $  p $
and of  "failure"   $  1- p $, 
as the distribution of the number of failures up to the occurrence of the  $  r $-
th success. For  $  r= 1 $
a Pascal distribution is the same as the [[Geometric distribution|geometric distribution]] with parameter  $  p $, 
and for  $  r > 1 $
it is the same as the distribution of the sum of independent random variables having an identical geometric distribution with parameter  $  p $. 
Accordingly, the sum of independent random variables  $  X _ {1} \dots X _ {n} $
having Pascal distributions with parameters  $  p $
and  $  r _ {1} \dots r _ {n} $, 
respectively, has the Pascal distribution with parameters  $  p $
and  $  r _ {1} + \dots + r _ {n} $.

The distribution function of a Pascal distribution for  $  k = 0, 1 \dots $
is given by the formula

$$ 
F( k)  =  
\frac{1}{B( r, k+ 1) }
 \int\limits _ { 0 } ^ { p }  x  ^ {r-} 1 ( 1- x)  ^ {k}  dx,
$$

where on the right-hand side there stands the value of the [[Beta-distribution|beta-distribution]] function at the point  $  p $(
here  $  B( r, k+ 1) $
is the beta-function). Using this relation one can define  $  F( k) $
for all  $  r > 0 $. 
In this generalized sense a Pascal distribution is called a [[Negative binomial distribution|negative binomial distribution]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''' , Wiley  (1957)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,   S. Kotz,   "Distributions in statistics: discrete distributions" , Houghton Mifflin  (1970)</TD></TR></table>