MAT 4830Mathematical Modeling

04Monte Carlo Integrations

http://myhome.spu.edu/lauw

Preview Look at how to use random numbers to

evaluate some integrals First application of Monte Carlo Methods Keep in mind that there are “better”

numerical methods to evaluate integrals No programs will be provided for the

example below

Example 0Suppose we want to estimate the value of the integral 3

1

0

xe dx

Example 0Since the function is non-negative over the interval [0,1], the value of the integral is the same as the area under the graph

31

0

xe dx

Example 0We are going to estimate the area by random numbers.

31

0

xe dx

failure

success

Example 03

1

0

xe dx

failure

number of success Area of the boxnumber of trails

Area

success

Example 0In general, the no. of trials needs to be large.

31

0

xe dx

failure

number of success Area of the boxnumber of trails

Area

success

Example 03

1

0

xe dx

success

failure

Maple Commands

Activate random number generator for various probability distributions.

Can be used before the program (as shown) or within the program.

Maple Commands

Generate random numbers for uniform distributions.

Example 0What are the disadvantages?

Example 0What are the disadvantages? 1.2.3.4.

Monte Carlo Methods Statistical simulation methods Method that utilizes sequences of

random numbers to perform the simulations

Classwork Individual* (Each of you need to think

through the process) Absolutely no communications. Fact: some of you will be faster than

some of the other which is normal!

ClassworkWrite a program to estimate the value of the integral

31

0

xe dx

Hint:3

1

0

xe dx

failure

success

Input n

Hint:3

1

0

xe dx

failure

success

Input n

Repeat the following n times. 1. Generate a random point (x,y) inside the box. 2. Decide if the point is under the graph. Keep track of the number of success.

Hint:3

1

0

xe dx

failure

success

Input n

Repeat the following n times. 1. Generate a random point (x,y) inside the box. 2. Decide if the point is under the graph. Keep track of the number of success.

Compute the estimated area.

Output the estimated area.

HW Problem 1Write a program to estimate the value of the integral

2

0

5sin xdx

HW Problem 2Write a program to estimate the value of the integral

1 1

0 0

xye dxdy

HW Problem 3Design an experiment using Monte Carlo Integration to estimate the value of