The beta-geometric distribution has the following probability density function:
with , , and B denoting the two shape parameters and the complete beta function, respectively. See the documentation for the BETA command for a description of the complete beta function.
Dataplot computes the cumulative distribution function using a recurrence relation given by Hesselager. Hesselager gives the recurrence relation as:
Converting this to the parameterization above yields
where <x> is a number, parameter, or variable containing non-negative integer values;
<alpha> is a number, parameter, or variable that specifies the first shape parameter;
<beta> is a number, parameter, or variable that specifies the second shape parameter;
<y> is a variable or a parameter (depending on what <x> is) where the computed beta-geometric cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BGECDF(X,2.1,4)
PLOT BGECDF(X,ALPHA,BETA) FOR X = 1 1 20
We will refer to the first parameterization as the unshifted parameterization and the second parameterization as the shifted parameterization.
To specify the shifted parameterization (i.e., starting at x = 0), enter the command
To reset the unshifted parameterization (i.e., starting at x = 1), enter the command
This distribution is also sometimes given with and reversed. In this case, the probability mass functions become
Irwin developed the Waring distribution based on the Waring expansion. The probability mass function for the Waring distribution is
The Waring distribution can be computed with the shifted form of the beta-geometric distribution with the following change in parameters:
= c - a
If a = 1, then the Waring distribution reduces to the Yule distribution.
You can compute the Waring (and Yule) distributions using the BGECDF routine with the above re-parameterization or you can use the WARCDF or YULCDF routines directly (enter HELP WARCDF or HELP YULCDF for details).
Sudhir R. Paul (2004), "Applications of the Beta Distribution" in "Handbook of the Beta Distribution", edited by Gupta and Nadarajah, Marcel-Dekker, pp. 431-436.
J. O. Irwin (1963), "The Place of Mathematics in Medical and Biological Statistics", Journal of the Royal Statistical Society, Series A, 126, pp. 1-44.
Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, chapter 6.
XLIMITS 0 50 XTIC OFFSET 0.5 0.5 LINE BLANK SPIKE ON SPIKE THICKNESS 0.3 . TITLE CASE ASIS LABEL CASE ASIS X1LABEL Number of Successes Y1LABEL Probability . MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 2 . TITLE Alpha = 0.5, Beta = 0.5 PLOT BGECDF(X,0.5,0.5) FOR X = 0 1 50 . TITLE Alpha = 3, Beta = 0.5 PLOT BGECDF(X,3.0,0.5) FOR X = 0 1 50 . TITLE Alpha = 0.5, Beta = 3 PLOT BGECDF(X,0.5,3.0) FOR X = 0 1 50 . TITLE Alpha = 3, Beta = 3 PLOT BGECDF(X,3.0,3.0) FOR X = 0 1 50 . END OF MULTIPLOT . CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT Beta-Geometric Cumulative Distribution Functions
Date created: 8/23/2006