Total time in the system: Suppose the time spent in each point i in the path follows a normal distribution N(μi, σi), ∀i = 1, · · · , m. Then answer the following question: What is the probability the product would take in the travel through path a total time between an interval [a, b]?

from scipy.stats import norm

import matplotlib.pyplot as plt

import numpy as np

#creating an array of values between

#1 to 9 with a difference of 0.1

x_1 = np.arange(1, 9, 0.1)

mean_1 = 5

std_1 = 1

y_1 = norm.pdf(x_1, loc = mean_1, scale = std_1)

plt.plot(x_1,y_1,'r-',x_1, y_1,'bo')

plt.xlabel('X')

plt.ylabel('P(X)')

plt.title('PDF for the First process')

plt.show()

from scipy.stats import norm

import matplotlib.pyplot as plt

import numpy as np

# Plotting both distributions.

plt.plot(x_1,y_1,'b-')

plt.plot(x_2,y_2,'g-')

# Creating the joint distribution.

x_3 = np.arange(1, 25, 0.2)

mean_3 = mean_2 + mean_1

std_3 = std_1**2 + std_2**2

y_3 = norm.pdf(x_3, loc = mean_3, scale = std_3)

plt.plot(x_1,y_1,'-b',label='Process 1')

plt.plot(x_2,y_2,'-g',label='Process 2')

plt.plot(x_3,y_3,'r-',label='Both')

plt.grid()

plt.xlabel('X')

plt.ylabel('P(X)')

plt.title('PDFs for all and joint processes')

plt.legend()

plt.show()