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Dataplot Vol 2 Vol 1

PAPCDF

Name:
    PAPCDF (LET)
Type:
    Library Function
Purpose:
    Compute the Polya-Aeppli cumulative distribution function.
Description:
    The formula for the Polya-Aeppli probability mass function is

      p(x;theta,p) = EXP(-theta) x = 0; = EXP(-theta)*p**x*SUM[j=1 to x][(x-1 j-1)*(theta*(1-p)/p)**j/j!] x = 1, ...; 0 < p < 1; theta > 0

    with theta and p denoting the shape parameters.

    The cumulative distribution function is computed using the following recurrence relation (from page 379 of Johnson, Kemp, and Kotz)

      p(x+1;theta,p) = (theta*(1-p)/(x+1))*SUM[j=0 TO x][(x+1-j)*p**(x-j)* p(x;theta,p)]
Syntax:
    LET <y> = PAPCDF(<x>,<theta>,<p>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a non-negative integer variable, number, or parameter;
                <theta> is a positive number or parameter that specifies the first shape parameter;
                <p> is a positive number or parameter that specifies the second shape parameter;
                <y> is a variable or a parameter where the computed Polya-Aeppli cdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = PAPCDF(3,3,0.5)
    LET Y = PAPCDF(X1,2,0.3)
    PLOT PAPCDF(X,2,0.3) FOR X = 0 1 20
Default:
    None
Synonyms:
    None
Related Commands:
    PAPPDF = Compute the Polya-Aeppli probability mass function.
    PAPPPF = Compute the Polya-Aeppli percent point function.
    LPOPDF = Compute the Lagrange-Poisson percent point function.
    BTAPDF = Compute the Borel-Tanner probability mass function.
    LOSPDF = Compute the lost games probability mass function.
    POIPDF = Compute the Poisson probability mass function.
    HERPDF = Compute the Hermite probability mass function.
    BINPDF = Compute the binomial probability mass function.
    NBPDF = Compute the negative binomial probability mass function.
    GEOPDF = Compute the geometric probability mass function.
    INTEGER FREQUENCY TABLE = Generate a frequency table at integer values with unequal bins.
    COMBINE FREQUENCY TABLE = Convert an equal width frequency table to an unequal width frequency table.
    KS PLOT = Generate a minimum chi-square plot.
    MAXIMUM LIKELIHOOD = Perform maximum likelihood estimation for a distribution.
References:
    Douglas (1980), "Analysis with Standard Contagious Distributions", International Co-operative Publishing House, Fairland, MD.

    Evans (1953), "Experimental Evidence Concerning Contagious Distributions in Ecology", Biometrika, 40, pp. 186-211.

    Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 378-382.

Applications:
    Distributional Modeling
Implementation Date:
    2006/6
Program: 
     
title size 3
tic label size 3
label size 3
legend size 3
height 3
multiplot scale factor 1.5
x1label displacement 12
y1label displacement 17
.
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 = 0.5, P = 0.5
plot papcdf(x,0.5,0.5) for x = 1 1 20
.
title Theta = 1, P = 0.5
plot papcdf(x,1,0.5) for x = 1 1 20
.
title Theta = 2.5, P = 0.5
plot papcdf(x,2.5,0.5) for x = 1 1 20
.
title Theta = 5, P = 0.5
plot papcdf(x,5,0.5) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Cumulative Distribution for Polya-Aeppli
    plot generated by sample program

Date created: 6/20/2006
Last updated: 6/20/2006
Please email comments on this WWW page to alan.heckert@nist.gov.