About

Subject Area	Mathematical/Numerical Methods
Level	First Year
Available Implementations	Mathematica and Python
Learning Objectives	Students will be able to: Generate normally distributed random numbers (Exercise 1) Plot histograms. Calculate mean, median, and standard deviation for a distribution. Generate a new distribution from previously generated random numbers. (Exercise 3) Compare the analytical (calculus) approach to the Monte Carlo approach (Exercises 2 and 3)
Time to Complete	30 min

1. Produce a large set of numbers that obey a normal distribution with a mean of 5.4 and a standard deviation of 0.2. Many computer programming languages have a weighted random number built in. You can also build your own using the Box Muller transformation that takes two uniformly distributed random numbers between 0 and 1 and returns two normally distributed random numbers with a mean of zero and a standard deviation of 1. These can then be transformed to match the requested mean and standard deviation. Plot a histogram of the large data set, confirming that the peak and width are what you expected.

2. Determine the histogram for the speed example above by using the calculus error propagation approach.

   a) Assuming that all values are distributed according to a normal distribution, the calculus approach is given by

   σf=(∂f∂xσx)2+(∂f∂yσy)2−−−−−−−−−−−−−−−−−−√(2)$$\begin{array}{}\text{(2)}& {\sigma }_f=\sqrt{{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }x}{\sigma }_x\right)}^2+{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }y}{\sigma }_y\right)}^2}\end{array}$$
   Show that in the case of the speed calculation, this becomes :
   σv=xt(σxx)2+(σyy)2−−−−−−−−−−−−−−√(3)$$\begin{array}{}\text{(3)}& {\sigma }_v=\frac{x}{t}\sqrt{{\left(\frac{{\sigma }_x}{x}\right)}^2+{\left(\frac{{\sigma }_y}{y}\right)}^2}\end{array}$$

   b) Using Eq. 2, determine the error for the speed example from Exercise 1. 3. Now use the Monte Carlo method. Generate several hundreds or thousands of values for both position and time according to their respective distributions. Calculate the speed that corresponds to each of these values, and plot a histogram of speeds. Compare with the results of exercise 2.

3. Produce several (hundreds or thousands) of both position and time estimates according to their respective distributions and calculate their associated speeds. Then plot the histogram of those speeds and compare with (2).

4. Determine the mean, median, and standard deviation of the histrogram for (3) and compare with 2.
5. Extend the Monte Carlo approach to lab data of your own.

#!/usr/bin/env python

'''

montecarlo.py

Eric Ayars

June 2016

Python solutions to PICUP "Monte Carlo error propagation"

exercise set.

'''

from pylab import *

import random

#########################

# Exercise 1

#########################

# Generate the gaussian distribution

N = 1000 # How many to generate

mu = 5.4 # mean

sigma = 0.2 # sigma

dist = array([random.gauss(mu, sigma) for j in range(N)])

# Plot a histogram of the distribution

hist(dist, 30)

show()

#########################

# Exercise 3

#########################

# characteristits of d and t

avg_d = 5.4

sigma_d = 0.2

avg_t = 6.2

sigma_t = 1.5

N = 1000 # How many to generate

# Generate distribution of distances

d = array([random.gauss(avg_d, sigma_d) for j in range(N)])

t = array([random.gauss(avg_t, sigma_t) for j in range(N)])

# Since d and t are numpy arrays, we can just divide, the

# resulting index-wise multiplication will generate another

# numpy array of speeds.

v = d/t

# plot histogram of velocity

hist(v, 30)

show()

#########################

# Exercise 4

#########################

# report average and sigma

print('mean speed = %0.3f' % mean(v))

print('median speed = %0.3f' % median(v))

print('standard deviation = %0.3f' % std(v))

 

Translations

Code	Language		Translator	Run

Software Requirements

	Android	iOS	Windows	MacOS
with best with	Chrome	Chrome	Chrome	Chrome
support full-screen?	Yes. Chrome/Opera No. Firefox/ Samsung Internet	Not yet	Yes	Yes
cannot work on	some mobile browser that don't understand JavaScript such as.....		cannot work on Internet Explorer 9 and below

 

Credits

Fremont Teng; Loo Kang Wee; based on codes by Andy Runquist

end faq