How to integrate a simple normal distribution in python ?

To integrate a simple normal distribution in python, a solution is to use quad, example:

from scipy.integrate import quad

import matplotlib.pyplot as plt
import scipy.stats
import numpy as np

#----------------------------------------------------------------------------------------#
# Normal Distribution

x_min = 0.0
x_max = 16.0

mean = 8.0 
std = 3.0

x = np.linspace(x_min, x_max, 100)

y = scipy.stats.norm.pdf(x,mean,std)

plt.plot(x,y, color='black')

#----------------------------------------------------------------------------------------#
# integration between x1 and x1

def normal_distribution_function(x):
    value = scipy.stats.norm.pdf(x,mean,std)
    return value

x1 = mean + std
x2 = mean + 2.0 * std

res, err = quad(normal_distribution_function, x1, x2)

print('Normal Distribution (mean,std):',mean,std)
print('Integration bewteen {} and {} --> '.format(x1,x2),res)

#----------------------------------------------------------------------------------------#
# plot integration surface

ptx = np.linspace(x1, x2, 10)
pty = scipy.stats.norm.pdf(ptx,mean,std)

plt.fill_between(ptx, pty, color='#0b559f', alpha='1.0')

#----------------------------------------------------------------------------------------#

plt.grid()

plt.xlim(x_min,x_max)
plt.ylim(0,0.25)

plt.title('How to integrate a normal distribution in python ?',fontsize=10)

plt.xlabel('x')
plt.ylabel('Normal Distribution')

plt.savefig("integrate_normal_distribution.png")
plt.show()

with a mean and standard deviation (std) of 8.0 and 3.0 respectively, the integration between 1 * std and 2 * std
returns:

>>> Normal Distribution (mean,std): 8.0 3.0
>>> Integration bewteen 11.0 and 14.0 -->  0.13590512198327787

It is possible to integrate a function that takes several parameters with quad in python, example of syntax for a function f that takes two arguments: arg1 and arg2:

quad( f, x_min, x_max, args=(arg1,arg2,))

Example of code using quad with a function that takes multiple arguments:

from scipy.integrate import quad

import matplotlib.pyplot as plt
import scipy.stats
import numpy as np


def normal_distribution_function(x,mean,std):
    value = scipy.stats.norm.pdf(x,mean,std)
    return value

x_min = 0.0
x_max = 30.0

mean = 15.0 
std = 4.0


ptx = np.linspace(x_min, x_max, 100)
pty = scipy.stats.norm.pdf(ptx,mean,std)

plt.plot(ptx,pty, color='gray')

plt.fill_between(ptx, pty, color='#e1b1b4', alpha='1.0')

plt.grid()

plt.title('How to integrate a function that takes parameteres in python ?', fontsize=10)
plt.xlabel('x', fontsize=8)
plt.ylabel('Probability Density Function', fontsize=8)

res, err = quad(normal_distribution_function, x_min, x_max, args=(mean,std,))

print(res)

plt.savefig("integrate_function_takes_parameters.png")
plt.show()

References