MAT 4830 Mathematical Modeling 04 Monte Carlo Integrations

Slides:

Advertisements

Similar presentations

Probabilistic models Jouni Tuomisto THL. Outline Deterministic models with probabilistic parameters Hierarchical Bayesian models Bayesian belief nets.

Advertisements

Simulating Single server queuing models. Consider the following sequence of activities that each customer undergoes: 1.Customer arrives 2.Customer waits.

Combining Monte Carlo Estimators If I have many MC estimators, with/without various variance reduction techniques, which should I choose?

1 Overview of Simulation When do we prefer to develop simulation model over an analytic model? When not all the underlying assumptions set for analytic.

11 Simulation. 22 Overview of Simulation – When do we prefer to develop simulation model over an analytic model? When not all the underlying assumptions.

E(X 2 ) = Var (X) = E(X 2 ) – [E(X)] 2 E(X) = The Mean and Variance of a Continuous Random Variable In order to calculate the mean or expected value of.

Monte Carlo Methods and Statistical Physics

Simulation Operations -- Prof. Juran.

Monte Carlo Simulation of the Craps Dice Game Sencer Koç.

CS433 Modeling and Simulation Lecture 11 Monté Carlo Simulation Dr. Anis Koubâa 05 Jan 2009 Al-Imam Mohammad Ibn.

Graduate School of Information Sciences, Tohoku University

1 Statistical Tests of Returns to Scale Using DEA Rajiv D. Banker Hsihui Chang Shih-Chi Chang.

Module F: Simulation. Introduction What: Simulation Where: To duplicate the features, appearance, and characteristics of a real system Why: To estimate.

Computational statistics 2009 Random walk. Computational statistics 2009 Random walk with absorbing barrier.

Steps of a sound simulation study

Introduction to Simulation. What is simulation? A simulation is the imitation of the operation of a real-world system over time. It involves the generation.

Analysis of Simulation Input.. Simulation Machine n Simulation can be considered as an Engine with input and output as follows: Simulation Engine Input.

Chapter 14 Simulation. Monte Carlo Process Statistical Analysis of Simulation Results Verification of the Simulation Model Computer Simulation with Excel.

Monté Carlo Simulation MGS 3100 – Chapter 9. Simulation Defined A computer-based model used to run experiments on a real system.  Typically done on a.

Monte Carlo Simulation 1.  Simulations where random values are used but the explicit passage of time is not modeled Static simulation  Introduction.

Computer Simulation A Laboratory to Evaluate “What-if” Questions.

Problem 1 Given a high-resolution computer image of a map of an irregularly shaped lake with several islands, determine the water surface area. Assume.

Simulation of Random Walk How do we investigate this numerically? Choose the step length to be a=1 Use a computer to generate random numbers r i uniformly.

Exercises of computational methods in finance

Simulating the value of Asian Options Vladimir Kozak.

1. Normal Approximation 1. 2 Suppose we perform a sequence of n binomial trials with probability of success p and probability of failure q = 1 - p and.

Monte Carlo Simulation and Personal Finance Jacob Foley.

Monte Carlo Simulation CWR 6536 Stochastic Subsurface Hydrology.

1 Theoretical Physics Experimental Physics Equipment, Observation Gambling: Cards, Dice Fast PCs Random- number generators Monte- Carlo methods Experimental.

The AIE Monte Carlo Tool The AIE Monte Carlo tool is an Excel spreadsheet and a set of supporting macros. It is the main tool used in AIE analysis of a.

Chapter 14 Monte Carlo Simulation Introduction Find several parameters Parameter follow the specific probability distribution Generate parameter.

Probabilistic Mechanism Analysis. Outline Uncertainty in mechanisms Why consider uncertainty Basics of uncertainty Probabilistic mechanism analysis Examples.

Chapter 10 Introduction to Simulation Modeling Monte Carlo Simulation.

Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.

Sampling Methods  Sampling refers to how observations are “selected” from a probability distribution when the simulation is run. 1.

Module 1: Statistical Issues in Micro simulation Paul Sousa.

Modular 11 Ch 7.1 to 7.2 Part I. Ch 7.1 Uniform and Normal Distribution Recall: Discrete random variable probability distribution For a continued random.

McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 1 Simulation.

SUPPLEMENT TO CHAPTER NINETEEN Irwin/McGraw-Hill © The McGraw-Hill Companies, Inc., 1999 SIMULATION 19S-1 Chapter 19 Supplement Simulation.

MAT 4830 Mathematical Modeling 05 Mean Time Between Failures

14.3 Simulation Techniques and the Monte Carlo Method simulation technique A simulation technique uses a probability experiment to mimic a real-life situation.

Lecture 19 Nov10, 2010 Discrete event simulation (Ross) discrete and continuous distributions computationally generating random variable following various.

Crystal Ball: Risk Analysis  Risk analysis uses analytical decision models or Monte Carlo simulation models based on the probability distributions to.

Monte Carlo Methods So far we have discussed Monte Carlo methods based on a uniform distribution of random numbers on the interval [0,1] p(x) = 1 0  x.

Outline of Chapter 9: Using Simulation to Solve Decision Problems Real world decisions are often too complex to be analyzed effectively using influence.

Simulation is the process of studying the behavior of a real system by using a model that replicates the behavior of the system under different scenarios.

4. Numerical Integration. Standard Quadrature We can find numerical value of a definite integral by the definition: where points x i are uniformly spaced.

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

Selecting Input Probability Distribution. Simulation Machine Simulation can be considered as an Engine with input and output as follows: Simulation Engine.

MAT 4830 Mathematical Modeling

FIN 614: Financial Management Larry Schrenk, Instructor.

Monte Carlo Process Risk Analysis for Water Resources Planning and Management Institute for Water Resources 2008.

Simulation is the process of studying the behavior of a real system by using a model that replicates the system under different scenarios. A simulation.

Machine Design Under Uncertainty. Outline Uncertainty in mechanical components Why consider uncertainty Basics of uncertainty Uncertainty analysis for.

Computer simulation Sep. 9, QUIZ 2 Determine whether the following experiments have discrete or continuous out comes A fair die is tossed and the.

1 1 Slide Simulation Professor Ahmadi. 2 2 Slide Simulation Chapter Outline n Computer Simulation n Simulation Modeling n Random Variables and Pseudo-Random.

1 1 Slide © 2004 Thomson/South-Western Simulation n Simulation is one of the most frequently employed management science techniques. n It is typically.

From Wikipedia: “Parametric statistics is a branch of statistics that assumes (that) data come from a type of probability distribution and makes inferences.

Week 111 Some facts about Power Series Consider the power series with non-negative coefficients a k. If converges for any positive value of t, say for.

1 Impact of Sample Estimate Rounding on Accuracy ERCOT Load Profiling Department May 22, 2007.

Lecture 3 Types of Probability Distributions Dr Peter Wheale.

Project Proposal Monte Carlo Simulation of Photon Migration.

G. Cowan Lectures on Statistical Data Analysis Lecture 5 page 1 Statistical Data Analysis: Lecture 5 1Probability, Bayes’ theorem 2Random variables and.

Module 9.4 Random Numbers from Various Distributions -MC requires the use of unbiased random numbers.

Bootstrapping James G. Anderson, Ph.D. Purdue University.

Problem 1: Service System Capacity CustomersServed Customers Queue Server Problem: Can a server taking an average of x time units meet the demand? Solution.

Probabilistic Slope Stability Analysis with the

Computer Simulation Henry C. Co Technology and Operations Management,

Monte Carlo Simulation Managing uncertainty in complex environments.

Chapter 14 Monte Carlo Simulation

Presentation transcript:

MAT 4830 Mathematical Modeling 04 Monte Carlo Integrations

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 0 Suppose we want to estimate the value of the integral

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

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

Example 0

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

Example 0

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 0 What are the disadvantages?

Example 0 What are the disadvantages?

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!

Classwork Write a program to estimate the value of the integral

Hint: Input n

Hint: 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: 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 1 Write a program to estimate the value of the integral

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

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