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Dataplot Vol 2 Vol 1

MAXCDF

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

      F(x;sigma)=2*GAMMAI(3/2,0.5*(1/sigma**2)*x**2)/SQRT(PI) x >= 0; sigma > 0

    with sigma denoting the scale parameter and GAMMAI denoting the incomplete gamma function.

    If sigma = 1, the Maxwell-Boltzman distribution is equivalent to the standard chi distribution with 3 degrees of freedom.

    Note that sigma is essentially a scale parameter. However, it is not strictly a scale parameter in the sense that the following relationship does not hold:

      f(x;scale) = (1/scale)*f(x/scale;1)

    The (1/sigma^2)^(3/2) term would have to be 1/sigma^2 for this relationship to hold (that is, there is an extra SQRT(1/sigma^2) term).

    The Maxwell-Boltzman distribution is sometimes parameterized using

      a

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

      The Maxwell-Boltzman distribution can be generalized with location and scale parameters in the usual way. However, the scale parameter is not typically used since sigma behaves much like a scale parameter already.

    Syntax:
      LET <y> = MAXCDF(<x>,<sigma>,<loc>,<scale>)
                              <SUBSET/EXCEPT/FOR qualification>
      where <x> is a variable or a parameter;
                  <sigma> is an optional number or parameter that specifies the value of the shape parameter;
                  <loc> is an optional number or parameter that specifies the value of the location parameter;
                  <scale> is an optional positive 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 cdf value is stored;
      and where the <SUBSET/EXCEPT/FOR qualification> is optional.

      If <sigma> is omitted, it defaults to 1. The location and scale parameters are optional.

    Examples:
      LET Y = MAXCDF(3)
      LET Y = MAXCDF(3,0.3)
      LET Y = MAXCDF(X1,SIGMA,MU)
      PLOT MAXCDF(X,SIGMA) FOR X = 0 0.01 5
    Default:
      None
    Synonyms:
      None
    Related Commands:
      MAXPDF = Compute the Maxwell probability density 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:
      "Continuous Univariate Distributions: Volume I", Second Edition, Johnson, Kotz, and Balakrishnan, Wiley, 1994, p. 451.
    Applications:
      Distributional Modeling, Statistical Physics
    Implementation Date:
      7/2004
    Program: 
       
Y1LABEL Probability
X1LABEL X
LABEL CASE ASIS
TITLE CASE ASIS
TITLE Maxwell Cumulative Distribution
PLOT MAXCDF(X,1) 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.