1. Monté Carlo Simulation  Understand the concept of Monté Carlo Simulation  Learn how to use Monté Carlo Simulation to make good decisions  Learn how to use Monté Carlo Simulation for estimating complex integrals

2. What is Monte Carlo Simulation ?  Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations.  Monte Carlo algorithm is often a numerical Monte Carlo method used to find solutions to mathematical problems (which may have many variables) that cannot easily be solved, (e.g. integral calculus, or other numerical methods)

3. What is Monte Carlo Simulation ?  A Monte Carlo simulation is a statistical simulation technique that provides approximate solutions to problems expressed mathematically. It utilizes a sequence of random numbers to perform the simulation.  This technique can be used in different domains:  complex integral computations,  economics,  making decisions in specific complex problems, …

4. General Algorithm of Monte Carlo Simulation  In general, Monte Carlo Simulation is roughly composed of five steps: 1.Set up probability distributions: what is the probability distribution that will be considered in the simulation 2.Build cumulative probability distributions 3.Establish an interval of random numbers for each variable 4.Generate random numbers: only accept numbers that satisfies a given condition. 5.Simulate trials

5. Examples  Example 1 : using Monte Carlo simulation for the analysis of real systems  Example 2: using Monte Carlo simulation to evaluate an integral.

6. Example 1. HERFY Cake Shop

7. Probability of Demand (1)(2)(3)(4) Demand for Tires Frequency Probability of Occurrence Cumulative Probability 010 10/200 =.05.05 120 20/200 =.10.15 240 40/200 =.20.35 360 60/200 =.30.65 440 40/200 =.20.85 530 30/ 200 =.15 1.00 200 days 200/200 = 1.00

8. Assignment of Random Numbers Daily Demand Probability Cumulative Probability Interval of Random Numbers 0.05.05 01 through 05 1.10.15 06 through 15 2.20.35 16 through 35 3.30.65 36 through 65 4.20.85 66 through 85 5.151.00 86 through 00  Table F.3

9. Table of Random Numbers 5250605205 3727806934 8245533355 6981693209 9866373077 9674064808 3330638845 5059571484 8867020284 9060948377  Table F.4

10. Simulation Example 1 Select random numbers from Table F.3 DayNumberRandomNumberSimulated Daily Demand 1523 2373 3824 4694 5985 6965 7332 8503 9885 10905 39Total 3.9 Average

11. Simulation Example 1 DayNumberRandomNumberSimulated Daily Demand 1523 2373 3824 4694 5985 6965 7332 8503 9885 10905 39Total 3.9 Average  Expected demand  = ∑ (probability of i units) x (demand of i units)  =(.05)(0) + (.10)(1) + (.20)(2) + (.30)(3) + (.20)(4) + (.15)(5)  =0 +.1 +.4 +.9 +.8 +.75  =2.95 tires  5  i =1

12. Set up probability distributions

13. Step 1: Set up the probability distribution for cake sales. Using historical data HERFY Shop determined that 5% of the time 0 cakes were demanded, 10% of the time 1 cake was demanded, etc… P(1) = 10%

14. Step 2: Build a Cumulative Probability Distribution 15% of the time the demand was 0 or 1 cake P(0) = 5% + P(1) = 10%

15. Example 2. Computation of Integrals

16.  The Monte Carlo method can be used to numerically approximate the value of an integral  Pick n randomly distributed points x 1, x 2, …, x n in the interval [a,b]  Determine the average value of the function  Compute the approximation to the integral  An estimate for the error is Where