MAXPDF

MAXPDF (LET)

Library Function

Compute the Maxwell-Boltzman probability density function.

The Maxwell-Boltzman distribution has the following probability density function:
with and denoting the location and scale parameters, respectively.

The case where = 0 and = 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

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

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.

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

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 and

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


You can obtain a maximum likelihood estimate of with the command

MAXWELL MAXIMUM LIKELIHOOD Y


This command will generate an estimate of using 0 as the estimate of location and an estimate of 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

None

None

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.

Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions: Volume I", Second Edition, Wiley, p. 451.

Distributional Modeling, Statistical Physics

7/2004
2/2008: Corrected to be a scale parameter rather than a shape parameter

Y1LABEL Probability
X1LABEL X
LABEL CASE ASIS
TITLE CASE ASIS
TITLE Maxwell Probability Density
PLOT MAXPDF(X) FOR X = 0  0.01  5