V135 Gamma or Chi-Square Random Numbers

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V135 Gamma or Chi-Square Random Numbers

Routine ID: V135
Author(s): F. James, K.S. Kölbig	Library: MATHLIB
Submitter:	Submitted: 15.10.1994
Language: Fortran	Revised:

Function subprogram RANGAM generates a positive random number x according to the gamma distribution with parameter p>0, i.e., according to the density

$P(t < x < t+dt) = {1 Γ(p)} t$ ^p-1e^-tdt.

A special case is the $χ$ ² -distribution with N degrees of freedom

$χ$ ²(t < 2x < t+dt) = {12^N Γ({12}N)} t^{12}N-1 e^-{12}t dt.

Structure:

FUNCTION subprogram
User Entry Names: RNGAMA
External References: RANLUX (V115), RNORMX (V120)

Usage:

In any arithmetic expression,

    RNGAMA(P)

has the value of a gamma-distributed random number, where

P > 0

is of type REAL. The value of P may vary from call to call without influencing the efficiency.

Method:

For integral values of $p ≤15$ , the logarithm of the product of p uniform random numbers is used. For any value of p > 15, the Wilson-Hilferty approximation (a transformed normal distribution) is used. For all other p, Johnk's algorithm is used.

Notes:

The routine is fast for small integer values of p, and for p > 15, (one Gaussian random number and one square root, plus a few multiplications). Non-integral values of p < 15 are rather slow.

Examples:

    CHI2 = 2*RNGAMA(0.5*N)

sets CHI2 to a random number distributed as

χ

² with N degrees of freedom.

•

V136

Janne Saarela
Mon Apr 3 15:06:23 METDST 1995