Also known as the Pascal Distribution and Pólya Distribution. The probability of
successes and failures in trials, and success on the th trial is
|
(1) |
where is a Binomial Coefficient. Let
The Characteristic Function is given by
|
(4) |
and the Moment-Generating Function by
|
(5) |
but, since
,
The Moments about zero
are therefore
(Beyer 1987, p. 487, apparently gives the Mean incorrectly.)
The Moments about the mean are
The Mean, Variance, Skewness and Kurtosis are then
which can also be written
The first Cumulant is
|
(24) |
and subsequent Cumulants are given by the recurrence relation
|
(25) |
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 533, 1987.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992.
© 1996-9 Eric W. Weisstein
1999-05-25