Pascal distribution

A discrete 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) 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 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 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.

References

[1]	W. Feller, "An introduction to probability theory and its applications", 1 , Wiley (1957)

References

[a1]	N.L. Johnson, S. Kotz, "Distributions in statistics: discrete distributions" , Houghton Mifflin (1970)