Monté Carlo Simulation  Understand the concept of Monté Carlo Simulation  Learn how to use Monté Carlo Simulation to make good decisions  Learn how.

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Presentation transcript:

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

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)

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, …

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

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

Example 1. HERFY Cake Shop

Probability of Demand (1)(2)(3)(4) Demand for Tires Frequency Probability of Occurrence Cumulative Probability /200 = /200 = /200 = /200 = /200 = / 200 = days 200/200 = 1.00

Assignment of Random Numbers Daily Demand Probability Cumulative Probability Interval of Random Numbers through through through through through through 00  Table F.3

Table of Random Numbers  Table F.4

Simulation Example 1 Select random numbers from Table F.3 DayNumberRandomNumberSimulated Daily Demand Total 3.9 Average

Simulation Example 1 DayNumberRandomNumberSimulated Daily Demand Total 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)  =  =2.95 tires  5  i =1

Set up probability distributions

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%

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

Example 2. Computation of Integrals

 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