The document discusses various methods for generating continuous random variables in simulations, including the inverse transform method and acceptance-rejection method. It provides examples of how to generate random variables from important distributions like the exponential, normal, Poisson, and nonhomogeneous Poisson distributions. The agenda includes an introduction, overview of methods, generating specific distributions, summary, and exercises in R to apply the methods.
Modular Monolith - a Practical Alternative to Microservices @ Devoxx UK 2024
Simulation - Generating Continuous Random Variables
1. Chair of Information Systems IV (ERIS)
Institute for Enterprise Systems (InES)
16 April 2013, 10.15 am – 11.15 am
Martin Kretzer
Phone: +49 621 181 3276
E-Mail: kretzer@es.uni-mannheim.de
Generating Continuous Random Variables
(IS 802 “Simulation”, Section 3)
2. Agenda 2
Agenda
1 Introduction
2 Selected Methods
3 Generating Important Distributions
4 Summary
5 Exercises (Simulations using R)
Generating Continuous Random Variables
3. Why “Generation of Continuous Random Variables”? 3
Simulations are able to address magnifold questions, e.g.:
How many customers will we have?
How long does it take to handle arriving customers?
Will queues develop?
Problem: Simulations require random variables
Uniformly distributed random variables are often not enough, but can be used for generation of further distributions
Uniformly distributed random variables can be generated, e.g., modulo function (section 2)
Example: Customers in a store arrive randomly, between 10 and 20 minutes apart, normally / exponentially
distributed on this interval.
Goal: Generate random variables
Discrete random variables (section 2)
Continuous random variables (univariate) (section 3)
Each method for generating a discrete random variable has its analogue in the continuous case!
Multivariate normal distributed variables (section 4)
Generating Continuous Random Variables
Discrete Variables Continuous Variables
Finite set of values
Example: population
0, 1, 2, 3, …
4. Learning Outcomes 4
Generating Continuous Random Variables
Generate random variables using the
Inverse-Transform and Acceptance-Rejection
Method
Develop algorithms for simulating
Exponential, Normal, Poisson and
Nonhomogeneous Poisson distributions
Perform simulations using R
10. Agenda 10
Agenda
1 Introduction
2 Selected Methods
3 Generating Important Distributions
3.1 Exponential Distribution
3.2 Normal Distribution
3.3 Poisson Distribution
3.4 Nonhomogeneous Poisson Distribution
4 Summary
5 Exercises (Simulations using R)
Generating Continuous Random Variables
18. Comparison: Homogeneous and Nonhomogeneous Poisson Process 18
Generating Continuous Random Variables
Nonhomogeneous Poisson Process(Homogeneous) Poisson Process
Previously: Now:
19. Nonhomogeneous Poisson Process (1/2) 19
Generating Continuous Random Variables
simulation of a
Poisson process
randomly
counting events
21. Agenda 21
Agenda
1 Introduction
2 Selected Methods
3 Generating Important Distributions
4 Summary
5 Exercises (Simulations using R)
Generating Continuous Random Variables
22.
Summary 22
Generating Continuous Random Variables
Distribution
Inverse Transform
Method
Acceptance-Rejection
Method
Exponential X
Normal X
(Homogeneous)
Poisson
X
Nonhomogeneous
Poisson
X
39. Bibliography 39
ETH Zürich. 2013. The Uniform Distribution, ETH Zürich, Zürich, CH (available online at
http://stat.ethz.ch/R-manual/R-devel/library/stats/html/Uniform.html ).
Haugh, M. 2010. “Generating Random Variables and Stochastic Processes,” Columbia
University, US (available online at http://www.columbia.edu/~mh2078/MCS_ Generate_
RVars .pdf ).
The R Core Team 2013. “R: A Language and Environment for Statistical Computing.
Reference Index. Version 3.0.0 (2013-04-03),” (part of the R download).
R-Project 2013. “Probability Distributions,” (available online at http://cran.r-
project.org/web/views/Distributions.html ).
Ross, S. M. 2013. Simulation, 5th ed., San Diego, CA: Academic Press.
Sigman, K. 2009. “Inverse Transform Method,” Columbia University, US (available online at
http://www.columbia.edu/~ks20/4404-13-Spring/4404-Notes-ITM.pdf ).
Valdez, E. A. 2008. “Generating Continuous Random Variables,” University of Connecticut,
US (available at http://www.math.uconn.edu/~valdez/math276s08/Math276-Week45.pdf ).
Venables, W. N., Smith, D. M., and the R Core Team. 2013. “An Introduction to R. Notes on
R: A Programming Environment for Data Analysis and Graphics Version 3.0.0 (2013-04-
03),” (part of the R download).
Generating Continuous Random Variables
Editor's Notes
Why do we need the exponential distribution?Exponential distribution is useful for generation of a Poisson random variableThe exponential distribution describes the time between events in a Poisson process, i.e., a process in which events occur continuously and independently at a constant average rate (= lambda).Times between events are independent exponentials with rate L (section 2.9 in the book)