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)