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Dataplot Vol 2 Vol 1

MAXPDF

Name:
    MAXPDF (LET)
Type:
    Library Function
Purpose:
    Compute the Maxwell-Boltzman probability density function.
Description:
    The Maxwell-Boltzman distribution has the following probability density function:

      f(x;mu,sigma)=SQRT(2/PI)*(1/sigma**2)**(3/2)*(x-mu)**2*
 EXP(-(x-mu)**2/(2*sigma**2))   x >= mu; sigma > 0
    with mu and sigma denoting the location and scale parameters, respectively.

    The case where mu = 0 and sigma = 1 is referred to as the standard Maxwell-Boltzman distribution.

    The Maxwell-Boltzman distribution is equivalent to the chi distribution with 3 degrees of freedom.

    The Maxwell-Boltzman distribution is sometimes parameterized using

      a = 1/sigma**2

    In scientific applications, the sigma parameter is typically parameterized in a way that has physical meaning.

Syntax:
    LET <y> = MAXPDF(<x>,<loc>,<sigma>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable or a parameter;
                <loc> is an optional number or parameter that specifies the value of the location parameter;
                <sigma> is an optional number or parameter that specifies the value of the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Maxwell-Boltzman pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    If <loc> and <sigma> are omitted, they default to 0 and 1, respectively.

Examples:
    LET Y = MAXPDF(3)
    LET Y = MAXPDF(3,0,0.3)
    LET Y = MAXPDF(X1,MU,SIGMA)
    PLOT MAXPDF(X,0,SIGMA) FOR X = 0 0.01 5
Note:
    To generate Maxwell random numbers, probability plots, and goodness of fit tests, enter the following commands

      LET Y = MAXWELL RANDOM NUMBERS FOR I = 1 1 N
      MAXWELL PROBABILITY PLOT Y
      MAXWELL PROBABILITY PLOT Y X
      MAXWELL PROBABILITY PLOT Y XLOW XHIGH
      MAXWELL KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
      MAXWELL CHI-SQUARE GOODNESS OF FIT Y X
      MAXWELL CHI-SQUARE GOODNESS OF FIT Y XLOW XHIGH

    You can use the probability plot to estimate mu and sigma

      MAXWELL PROBABILITY PLOT Y
      LET SIGMA = PPA1
      LET MU = PPA0

    You can obtain a maximum likelihood estimate of sigma with the command

      MAXWELL MAXIMUM LIKELIHOOD Y

    This command will generate an estimate of sigma using 0 as the estimate of location and an estimate of sigma using the minimum of the data as an estimate of location. If the data minimum is negative, then both cases will use the data minimum as the estimate of location (i.e., the estimate of sigma will be the same). If you have a different estimate of locaiton, enter the command

      LET MAXWLOC = <value>

    before the MAXWELL MAXIMUM LIKELIHOOD command. This will be used in place of the data minimum estimate of location. Uncertainty estimates can be obtained using the DISTRIBUTIONAL BOOTSTRAP command

Default:
    None
Synonyms:
    None
Related Commands:
    MAXCDF = Compute the Maxwell cumulative distribution function.
    MAXPPF = Compute the Maxwell percent point function.
    CHPDF = Compute the chi probability density function.
    RAYPDF = Compute the Rayleigh probability density function.
    WEIPDF = Compute the Weibull probability density function.
    NORPDF = Compute the normal probability density function.
    LGNPDF = Compute the lognormal probability density function.
Reference:
    Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions: Volume I", Second Edition, Wiley, p. 451.
Applications:
    Distributional Modeling, Statistical Physics
Implementation Date:
    7/2004
    2/2008: Corrected sigma to be a scale parameter rather than a shape parameter
Program:
    Y1LABEL Probability
    X1LABEL X
    LABEL CASE ASIS
    TITLE CASE ASIS
    TITLE Maxwell Probability Density
    PLOT MAXPDF(X) FOR X = 0  0.01  5
        
    plot generated by sample program

Date created: 7/28/2004
Last updated: 7/28/2004
Please email comments on this WWW page to alan.heckert@nist.gov.