NBRAN Subroutine

nbran(3f) - [M_datapac:RANDOM] generate negative binomial random numbers

   SUBROUTINE NBRAN(N,P,Npar,Istart,X)

    INTEGER,intent(in)        :: N
    REAL(kind=wp),intent(in)  :: P
    INTEGER,intent(in)        :: Npar
    INTEGER,intent(inout)     :: Istart
    REAL(kind=wp),intent(out) :: X(:)

NBRAN(3f) generates a random sample of size N from the negative
binomial distribution with precision 'Bernoulli probability'
parameter = P, and integer 'number of successes in Bernoulli trials'
parameter = NPAR. The negative binomial distribution used herein has
mean = NPAR*(1-P)/P and standard deviation = sqrt(NPAR*(1-P)/(P*P))).

This distribution is defined for all non-negative integer X-- X = 0,
1, 2, ... .

This distribution has the probability function

    f(X) = c(NPAR+X-1,NPAR) * P**NPAR * (1-P)**X

Where c(NPAR+X-1,NPAR) is the combinatorial function equaling the
number of combinations of NPAR+X-1 items taken NPAR at a time.

The negative binomial distribution is the distribution of the number
of failures before obtaining NPAR successes in an indefinite sequence
of Bernoulli (0,1) trials where the probability of success in a precision
trial = P.

N       The desired integer number of random numbers to be generated.

P       The value of the 'Bernoulli probability' parameter for the
        negative binomial distribution. P Should be between
        0.0 (exclusively) and 1.0 (exclusively).

NPAR    The integer value of the 'number of successes in Bernoulli
        trials' parameter. NPAR should be a positive integer.

ISTART  An integer flag code which (if set to 0) will start the
        generator over and hence produce the same random sample
        over and over again upon successive calls to this subroutine
        within a run; or (if set to some integer value not equal to 0,
        like, say, 1) will allow the generator to continue from where
        it stopped and hence produce different random samples upon
        successive calls to this subroutine within a run.

X       A vector (of dimension at least N) into which the generated
        random sample of size N from the negative binomial distribution
        will be placed.

program demo_nbran
use m_datapac, only : nbran, plott, label, plotxt, sort
implicit none
integer,parameter :: n=400
real              :: p
integer           :: Npar
integer           :: Istart
real              :: x(n)
   call label('nbran')
   p=0.4
   Npar=3
   istart=12345
   call nbran(N,P,Npar,Istart,X)
   call plotxt(x,n)
   call sort(x,n,x) ! sort to show distribution
   call plotxt(x,n)
end program demo_nbran

The original DATAPAC library was written by James Filliben of the
Statistical Engineering Division, National Institute of Standards
and Technology.

John Urban, 2022.05.31

CC0-1.0

norcdf(3f) - [M_datapac:CUMULATIVE_DISTRIBUTION] compute the normal cumulative
distribution function

   SUBROUTINE NORCDF(X,Cdf)

    REAL(kind=wp),intent(out) :: Cdf
    REAL(kind=wp),intent(in)  :: X

NORCDF(3f) computes the cumulative distribution function value for the
normal (Gaussian) distribution with mean = 0 and standard deviation
= 1.

This distribution is defined for all X and has the probability
density function

    f(X) = (1/sqrt(2*pi))*exp(-X*X/2)

X      The value at which the cumulative distribution function is to
       be evaluated.

CDF    The cumulative distribution function value.

program demo_norcdf
!@(#) line plotter graph of cumulative distribution function
use M_datapac, only : norcdf, plott, label
implicit none
real,allocatable  :: x(:), y(:)
integer           :: i
   call label('norcdf')
   x=[(real(i),i=-100,100,1)]
   if(allocated(y))deallocate(y)
   allocate(y(size(x)))
   do i=1,size(x)
      call norcdf(x(i)/10.0,y(i))
   enddo
   call plott(x,y,size(x))
end program demo_norcdf

 The following is a plot of Y(I) (vertically) versus X(I) (horizontally)
                   I-----------I-----------I-----------I-----------I
  0.1000000E+03 -                                                  X
  0.9166666E+02 I                                                  X
  0.8333334E+02 I                                                  X
  0.7500000E+02 I                                                  X
  0.6666667E+02 I                                                  X
  0.5833334E+02 I                                                  X
  0.5000000E+02 -                                                  X
  0.4166667E+02 I                                                  X
  0.3333334E+02 I                                                  X
  0.2500000E+02 I                                                 XX
  0.1666667E+02 I                                             XXXXX
  0.8333336E+01 I                                   X XX XXXXX
  0.0000000E+00 -                   XX X X X X X XX
 -0.8333328E+01 I        XXXXX XX X
 -0.1666666E+02 I   XXXXX
 -0.2499999E+02 I  XX
 -0.3333333E+02 I  X
 -0.4166666E+02 I  X
 -0.5000000E+02 -  X
 -0.5833333E+02 I  X
 -0.6666666E+02 I  X
 -0.7500000E+02 I  X
 -0.8333333E+02 I  X
 -0.9166666E+02 I  X
 -0.1000000E+03 -  X
                   I-----------I-----------I-----------I-----------I
            0.0000E+00  0.2500E+00  0.5000E+00  0.7500E+00  0.1000E+01

The original DATAPAC library was written by James Filliben of the
Statistical Engineering Division, National Institute of Standards
and Technology.

John Urban, 2022.05.31

CC0-1.0