with and m denoting the shape parameters. The case where m = 1 reduces to the geometric distribution. For this reason, the Consul distribution is also referred to as the generalized geometric distribution.
The mean and variance of the Consul distribution are:
The Consul distribution is also parameterized in terms of its mean, , which results in the following probability mass function:
can be expressed in terms of :
For this parameterization, the mean and variance are:
Dataplot supports both parameterizations (see the Note section below).
Dataplot computes the cumulative distribution function using the following recurrence relation given on page 320 of Consul and Famoye:
where . If the parameterization is used, it is converted to the parameterization and the above recurrence relation is applied.
where <x> is a positive integer variable, number, or parameter;
<shape> is a number, parameter, or variable that specifies the valuie of theta (or mu);
<m> 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 Consul cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET Y = CONCDF(X,0.3,1.6)
PLOT CONCDF(X,0.3,1.6) FOR X = 1 1 20
To restore the THETA parameterization, enter the command
Famoye (1997), "Generalized Geometric and Some of its Applications", Journal of Mathematical Sciences, 8, pp. 1-13.
Consul (1990), "New Class of Location-Parameter Discrete Probability Distributions and Their Chaacterizations", Communications in Statistics--Theory and Methods, 19, pp. 4653-4666.
set consul definition theta title size 3 tic label size 3 label size 3 legend size 3 height 3 x1label displacement 12 y1label displacement 15 . multiplot corner coordinates 0 0 100 95 multiplot scale factor 2 label case asis title case asis case asis tic offset units screen tic offset 3 3 title displacement 2 y1label Probability x1label X . ylimits 0 1 major ytic mark number 6 minor ytic mark number 3 xlimits 0 20 line blank spike on . multiplot 2 2 . title Theta = 0.3, M = 1.8 plot concdf(x,0.3,1.8) for x = 1 1 20 . title Theta = 0.5, M = 1.5 plot concdf(x,0.5,1.5) for x = 1 1 20 . title Theta = 0.7, M = 1.2 plot concdf(x,0.7,1.2) for x = 1 1 20 . title Theta = 0.9, M = 1.1 plot concdf(x,0.9,1.1) for x = 1 1 20 . end of multiplot . justification center move 50 97 text Cumulative Distribution Functions for Consul
Date created: 8/23/2006