MATPDF

MATPDF (LET)

Library Function

Compute the classical matching probability mass function.

The classical matching distribution has the following probability mass function:

with k a non-negative integer denoting the number of items parameter.

Given k items numbered 1, 2, ..., k that are arranged in a random order, the classical matching distribution is the distribution of the number of items were the original numbering and the random ordering are the same.

Feller (see References below) formulates this as the problem where we have two matching decks of cards and we want to determine the probability of matching X cards in the two decks.

LET <y> = MATPDF(<x>,<k>)             <SUBSET/EXCEPT/FOR qualification>
where <x> is a variable, a number, or a parameter containing values between 0 and <k>;
            <k> is a number or parameter that defines the upper limit of the matching distribution;
            <y> is a variable or a parameter (depending on what <x> is) where the computed pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = MATPDF(3,20)
LET Y = MATPDF(X,100)

For a number of commands utilizing the matching distribution, it is convenient to bin the data. There are two basic ways of binning the data.
1. For some commands (histograms, maximum likelihood estimation), bins with equal size widths are required. This can be accomplished with the following commands:
   LET AMIN = MINIMUM Y
   LET AMAX = MAXIMUM Y
   LET AMIN2 = AMIN - 0.5
   LET AMAX2 = AMAX + 0.5
   CLASS MINIMUM AMIN2
   CLASS MAXIMUM AMAX2
   CLASS WIDTH 1
   LET Y2 X2 = BINNED
   

2. For some commands, unequal width bins may be helpful. In particular, for the chi-square goodness of fit, it is typically recommended that the minimum class frequency be at least 5. In this case, it may be helpful to combine small frequencies in the tails. Unequal class width bins can be created with the commands
   LET MINSIZE = <value>
   LET Y3 XLOW XHIGH = INTEGER FREQUENCY TABLE Y
   

   If you already have equal width bins data, you can use the commands

   LET MINSIZE = <value>
   LET Y3 XLOW XHIGH = COMBINE FREQUENCY TABLE Y2 X2
   

   The MINSIZE parameter defines the minimum class frequency. The default value is 5.

You can generate matching random numbers, probability plots, and chi-square goodness of fit tests with the following commands:
LET N = VALUE
LET K = <value>
LET Y = MATCHING RANDOM NUMBERS FOR I = 1 1 N


MATCHING PROBABILITY PLOT Y
MATCHING PROBABILITY PLOT Y2 X2
MATCHING PROBABILITY PLOT Y3 XLOW XHIGH


MATCHING CHI-SQUARE GOODNESS OF FIT Y
MATCHING CHI-SQUARE GOODNESS OF FIT Y2 X2
MATCHING CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH


Dataplot does not provide any explicit parameter estimation methods. It is assummed that the number of objects is a known quantity. We can then apply goodness of fit tests (i.e., the probability plot or the chi-square goodness of fit) to see if the classical matching distribution is an appropriate distribution.

For sufficiently large values of k, the classical matching distribution can be accurately approximated with a Poisson distribution with = 1. Dataplot computes MATPDF from the above definition for values of k < 20. For values of k ≥ 20, Dataplot computes MATPDF using the Poisson pdf with = 1.

None

None

MATCDF	= Compute the matching cumulative distribution function.
MATPPF	= Compute the matching percent point function.
POIPDF	= Compute the Poisson probability density function.
LCTPDF	= Compute the leads in coin tossing probability mass function.
DISPDF	= Compute the discrete uniform probability mass function.
LOSPDF	= Compute the lost games probability mass function.
ARSPDF	= Compute the arcsine probability density function.
BETPDF	= Compute the beta probability density function.
UNIPDF	= Compute the uniform probability mass function.

Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 409-410.

Feller (1957), "Introduction to Probability Theory", Third Edition, John Wiley and Sons, pp. 107-109.

Distributional Modeling

2006/6

 
TITLE CASE ASIS
TITLE Matching Probability Mass Function CR() ...
      (N = 50)
LABEL CASE ASIS
Y1LABEL Probability Mass
X1LABEL X
LINE BLANK
SPIKE ON
TIC OFFSET UNITS SCREEN
TIC OFFSET 3 3
PLOT MATPDF(X,50) FOR X = 0 1 50