def normal_01_moment ( order ):
#*****************************************************************************80
#
## NORMAL_01_MOMENT evaluates the moments of the Normal 01 PDF.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 03 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
#
# Output, real VALUE, the value of the moment.
#
from r8_factorial2 import r8_factorial2
if ( ( order % 2 ) == 0 ):
value = r8_factorial2 ( order - 1 )
else:
value = 0.0
return value
def normal_ms_moment(order, mu, sigma):
#*****************************************************************************80
#
## NORMAL_MS_MOMENT evaluates a moment of the Normal MS distribution.
#
# Discussion:
#
# The formula was posted by John D Cook.
#
# Order Moment
# ----- ------
# 0 1
# 1 mu
# 2 mu ** 2 + sigma ** 2
# 3 mu ** 3 + 3 mu sigma ** 2
# 4 mu ** 4 + 6 mu ** 2 sigma ** 2 + 3 sigma ** 4
# 5 mu ** 5 + 10 mu ** 3 sigma ** 2 + 15 mu sigma ** 4
# 6 mu ** 6 + 15 mu ** 4 sigma ** 2 + 45 mu ** 2 sigma ** 4 + 15 sigma ** 6
# 7 mu ** 7 + 21 mu ** 5 sigma ** 2 + 105 mu ** 3 sigma ** 4 + 105 mu sigma ** 6
# 8 mu ** 8 + 28 mu ** 6 sigma ** 2 + 210 mu ** 4 sigma ** 4 + 420 mu ** 2 sigma ** 6 + 105 sigma ** 8
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 05 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
#
# Input, real MU, the mean of the distribution.
#
# Input, real SIGMA, the standard deviation of the distribution.
#
# Output, real VALUE, the value of the moment.
#
from r8_choose import r8_choose
from r8_factorial2 import r8_factorial2
j_hi = (order // 2)
value = 0.0
for j in range(0, j_hi + 1):
value = value \
+ r8_choose ( order, 2 * j ) \
* r8_factorial2 ( 2 * j - 1 ) \
* mu ** ( order - 2 * j ) * sigma ** ( 2 * j )
return value
def normal_ms_moment ( order, mu, sigma ):
#*****************************************************************************80
#
## NORMAL_MS_MOMENT evaluates the moments of the Normal MS distribution.
#
# Discussion:
#
# The formula was posted by John D Cook.
#
# Order Moment
# ----- ------
# 0 1
# 1 mu
# 2 mu ** 2 + sigma ** 2
# 3 mu ** 3 + 3 mu sigma ** 2
# 4 mu ** 4 + 6 mu ** 2 sigma ** 2 + 3 sigma ** 4
# 5 mu ** 5 + 10 mu ** 3 sigma ** 2 + 15 mu sigma ** 4
# 6 mu ** 6 + 15 mu ** 4 sigma ** 2 + 45 mu ** 2 sigma ** 4 + 15 sigma ** 6
# 7 mu ** 7 + 21 mu ** 5 sigma ** 2 + 105 mu ** 3 sigma ** 4 + 105 mu sigma ** 6
# 8 mu ** 8 + 28 mu ** 6 sigma ** 2 + 210 mu ** 4 sigma ** 4 + 420 mu ** 2 sigma ** 6 + 105 sigma ** 8
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 05 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
#
# Input, real MU, the mean of the distribution.
#
# Input, real SIGMA, the standard deviation of the distribution.
#
# Output, real VALUE, the value of the moment.
#
from r8_choose import r8_choose
from r8_factorial2 import r8_factorial2
j_hi = ( order // 2 )
value = 0.0
for j in range ( 0, j_hi + 1 ):
value = value \
+ r8_choose ( order, 2 * j ) \
* r8_factorial2 ( 2 * j - 1 ) \
* mu ** ( order - 2 * j ) * sigma ** ( 2 * j )
return value
def normal_ms_moment_central(order, mu, sigma):
#*****************************************************************************80
#
## NORMAL_MS_MOMENT_CENTRAL evaluates central moments of the Normal MS distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 05 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
#
# Input, real MU, the mean of the distribution.
#
# Input, real SIGMA, the standard deviation of the distribution.
#
# Output, real VALUE, the value of the central moment.
#
from r8_factorial2 import r8_factorial2
if ((order % 2) == 0):
value = r8_factorial2(order - 1) * sigma**order
else:
value = 0.0
return value
def normal_ms_moment_central ( order, mu, sigma ):
#*****************************************************************************80
#
## NORMAL_MS_MOMENT_CENTRAL evaluates central moments of the Normal MS distribution.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 05 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer ORDER, the order of the moment.
#
# Input, real MU, the mean of the distribution.
#
# Input, real SIGMA, the standard deviation of the distribution.
#
# Output, real VALUE, the value of the central moment.
#
from r8_factorial2 import r8_factorial2
if ( ( order % 2 ) == 0 ):
value = r8_factorial2 ( order - 1 ) * sigma ** order
else:
value = 0.0
return value