#! /usr/bin/env python
#
def poisson_cdf ( x, a ):

#*****************************************************************************80
#
## POISSON_CDF evaluates the Poisson CDF.
#
#  Discussion:
#
#    CDF(X,A) is the probability that the number of events observed
#    in a unit time period will be no greater than X, given that the
#    expected number of events in a unit time period is A.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer X, the argument of the CDF.
#    0 <= X.
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real CDF, the value of the CDF.
#
  import numpy as np

  if ( x < 0 ):

    cdf = 0.0

  else:

    next = np.exp ( - a )
    sum2 = next

    for i in range ( 1, x + 1 ):
      last = next
      next = last * a / float ( i )
      sum2 = sum2 + next

    cdf = sum2

  return cdf

def poisson_cdf_inv ( cdf, a ):

#*****************************************************************************80
#
## POISSON_CDF_INV inverts the Poisson CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real CDF, a value of the CDF.
#    0 <= CDF < 1.
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, integer X, the corresponding argument.
#
  import numpy as np

  xmax = 100
#
#  Now simply start at X = 0, and find the first value for which
#  CDF(X-1) <= CDF <= CDF(X).
#
  sum2 = 0.0

  for i in range ( 0, xmax + 1 ):

    sumold = sum2

    if ( i == 0 ):
      next = np.exp ( - a )
      sum2 = next
    else:
      last = next
      next = last * a / float ( i )
      sum2 = sum2 + next

    if ( sumold <= cdf and cdf <= sum2 ):
      x = i
      return x

  print ( '' )
  print ( 'POISSON_CDF_INV - Warning!' )
  print ( '  Exceeded XMAX = %d' % ( xmax ) )

  x = xmax

  return x

def poisson_cdf_test ( ):

#*****************************************************************************80
#
## POISSON_CDF_TEST tests POISSON_CDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'POISSON_CDF_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  POISSON_CDF evaluates the Poisson CDF,' )
  print ( '  POISSON_CDF_INV inverts the Poisson CDF.' )
  print ( '  POISSON_PDF evaluates the Poisson PDF.' )

  a = 10.0

  check = poisson_check ( a )

  if ( not check ):
    print ( '' )
    print ( 'POISSON_CDF_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  print ( '' )
  print ( '  PDF parameter A =  %14g' % ( a ) )

  seed = 123456789

  print ( '' )
  print ( '       X            PDF           CDF            CDF_INV' )
  print ( '' )

  for i in range ( 0, 10 ):

    x, seed = poisson_sample ( a, seed )

    pdf = poisson_pdf ( x, a )

    cdf = poisson_cdf ( x, a )

    x2 = poisson_cdf_inv ( cdf, a )

    print ( '  %14d  %14g  %14g  %14d' % ( x, pdf, cdf, x2 ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'POISSON_CDF_TEST' )
  print ( '  Normal end of execution.' )
  return

def poisson_check ( a ):

#*****************************************************************************80
#
## POISSON_CHECK checks the parameter of the Poisson PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    19 September 2004
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
  check = True

  if ( a <= 0.0 ):
    print ( '' )
    print ( 'POISSON_CHECK - Fatal error!' )
    print ( '  A <= 0.' )
    check = False

  return check

def poisson_mean ( a ):

#*****************************************************************************80
#
## POISSON_MEAN returns the mean of the Poisson PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real MEAN, the mean of the PDF.
#
  mean = a

  return mean

def poisson_pdf ( x, a ):

#*****************************************************************************80
#
## POISSON_PDF evaluates the Poisson PDF.
#
#  Formula:
#
#    PDF(X)(A) = EXP ( - A ) * A^X / X!
#
#  Discussion:
#
#    PDF(X)(A) is the probability that the number of events observed
#    in a unit time period will be X, given the expected number
#    of events in a unit time.
#
#    The parameter A is the expected number of events per unit time.
#
#    The Poisson PDF is a discrete version of the Exponential PDF.
#
#    The time interval between two Poisson events is a random
#    variable with the Exponential PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer X, the argument of the PDF.
#    0 <= X
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real PDF, the value of the PDF.
#
  import numpy as np
  from r8_factorial import r8_factorial

  if ( x < 0 ):
    pdf = 0.0
  else:
    pdf = np.exp ( - a ) * a ** x / r8_factorial ( x )

  return pdf

def poisson_sample ( a, seed ):

#*****************************************************************************80
#
## POISSON_SAMPLE samples the Poisson PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Input, integer SEED, a seed for the random number generator.
#
#    Output, integer X, a sample of the PDF.
#
#    Output, integer SEED, an updated seed for the random number generator.
#
  from r8_uniform_01 import r8_uniform_01

  cdf, seed = r8_uniform_01 ( seed )

  x = poisson_cdf_inv ( cdf, a )

  return x, seed

def poisson_sample_test ( ):

#*****************************************************************************80
#
## POISSON_SAMPLE_TEST tests POISSON_SAMPLE.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
  import numpy as np
  import platform
  from i4vec_max import i4vec_max
  from i4vec_mean import i4vec_mean
  from i4vec_min import i4vec_min
  from i4vec_variance import i4vec_variance

  nsample = 1000
  seed = 123456789

  print ( '' )
  print ( 'POISSON_SAMPLE_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  POISSON_MEAN computes the Poisson mean' )
  print ( '  POISSON_SAMPLE samples the Poisson distribution' )
  print ( '  POISSON_VARIANCE computes the Poisson variance.' )

  a = 10.0

  check = poisson_check ( a )

  if ( not check ):
    print ( '' )
    print ( 'POISSON_SAMPLE_TEST - Fatal error!' )
    print ( '  The parameters are not legal.' )
    return

  mean = poisson_mean ( a )
  variance = poisson_variance ( a )

  print ( '' )
  print ( '  PDF parameter A =             %14g' % ( a ) )
  print ( '  PDF mean =                    %14g' % ( mean ) )
  print ( '  PDF variance =                %14g' % ( variance ) )

  x = np.zeros ( nsample )
  for i in range ( 0, nsample ):
    x[i], seed = poisson_sample ( a, seed )

  mean = i4vec_mean ( nsample, x )
  variance = i4vec_variance ( nsample, x )
  xmax = i4vec_max ( nsample, x )
  xmin = i4vec_min ( nsample, x )

  print ( '' )
  print ( '  Sample size =     %6d' % ( nsample ) )
  print ( '  Sample mean =     %14g' % ( mean ) )
  print ( '  Sample variance = %14g' % ( variance ) )
  print ( '  Sample maximum =  %6d' % ( xmax ) )
  print ( '  Sample minimum =  %6d' % ( xmin ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'POISSON_SAMPLE_TEST' )
  print ( '  Normal end of execution.' )
  return

def poisson_variance ( a ):

#*****************************************************************************80
#
## POISSON_VARIANCE returns the variance of the Poisson PDF.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    27 March 2016
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, real A, the parameter of the PDF.
#    0.0 < A.
#
#    Output, real VARIANCE, the variance of the PDF.
#
  variance = a

  return variance

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( ) 
  poisson_cdf_test ( )
  poisson_sample_test ( )
  timestamp ( )