HERCDF

HERCDF (LET)

Library Function

Compute the Hermite cumulative distribution function.

If X₁ and X₂ are independent Poisson random variables with shape parameters and (1/2)², respectively, then X₁ + 2X₂ follows a Hermite distribution with shape parameters and .

Some sources in the literature use the parameterization

a = a₁ =
b = a₂ = 0.5²

The shape parameters and can be expressed in terms of a₁ and a₂ as





The probability mass function for the Hermite distribution is:



where



with denoting the modified Hermite polynomial:

with [n/2] denoting the integer part of (n/2).

The first few terms of the Hermite probability mass function are:













A general recuurence relation is:

For x < 26, Dataplot uses the above recurrence relation to compute the probabilities. For x > 25, Dataplot uses an asymptotic formula due to Patel (see Reference section below) to compute the probabilities.

LET <y> = HERCDF(<x>,<alpha>,<beta>)             <SUBSET/EXCEPT/FOR qualification>
where <x> is a non-negative integer variable, number, or parameter;
            <alpha> is a number or parameter that specifies the first shape parameter;
            <beta> is a number or parameter that specifies the second shape parameter;
            <y> is a variable or a parameter (depending on what <x> is) where the computed Hermite cdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = HERCDF(3,0.5,2)
LET X2 = HERCDF(X1,ALPHA,BETA)
PLOT HERCDF(X,0.8,1.4) FOR X = 0 1 20

None

None

HERPDF	= Compute the Hermite probability density function.
HERPPF	= Compute the Hermite percent point function.
POIPDF	= Compute the Poisson cumulative distribution function.
BINPDF	= Compute the binomial probability density function.
NBPDF	= Compute the negative binomial probability density function.
GEOPDF	= Compute the geometric probability density function.

"Discrete Univariate Distributions" Second Edition, Johnson, and Kotz, and Kemp, Wiley, 1992, pp. 357-364.

"An Asymptotic Expression for Cumulative Sum of Probabilities of the Hermite Distribution", Y. C. Patel, Communications in Statistics--Theory and Methods, 14, pp. 2233-2241.

"Some Properties of the Hermite Distribution", Kemp and Kemp, Biometrika (1965), 52, 3 and 4, P. 381.

"Even Point Estimation and Moment Estimation in Hermite Distributions", Y. C. Patel, Biometrics, 32, December, 1976, pp. 865-873.

Distributional Modeling

2004/4

 
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 100
XTIC OFFSET 0.5 0.5
LINE BLANK
SPIKE ON
TITLE AUTOMATIC
X1LABEL X
Y1LABEL PROBABILITY
X1LABEL DISPLACEMENT 12
Y1LABEL DISPLACEMENT 12
TITLE SIZE 3
PLOT HERCDF(X,0.5,2) FOR X = 0 1 50
PLOT HERCDF(X,2,0.5) FOR X = 0 1 50
PLOT HERCDF(X,0.5,0.5) FOR X = 0 1 50
PLOT HERCDF(X,2,2) FOR X = 0 1 50
END OF MULTIPLOT