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lecturenotes04 - MS&E 223 Simulation Peter J. Haas Lecture...

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MS&E 223 Lecture Notes #4 Simulation Input Distributions Peter J. Haas Spring Quarter 2009-10 Input Distributions Ref : Chapter 6 in Law and Kelton To specify a simulation model for a discrete-event system, we need to define the distributions of the clock-setting “input sequences” to the model. Examples: (1) interarrival sequences (2) processing time sequences for a production system (3) interest rate sequence for a financial model As a simplifying assumption, it is often assumed that the input sequences are i.i.d. Even having made this (major!) assumption, this leaves us with a couple of questions to be answered: 1. What type of distribution should we use (e.g. gamma or Weibull)? 2. Once we’ve settled on the type (or “family”) of probability distribution, what parameter values should we use in our simulation (e.g. if we decide to use a gamma distribution, what values for the scale parameter λ and shape parameter α should we use)? Often, real-world historical data is available to guide us in making these decisions. However, knowledge of some probability theory can also help us here, particularly in answering the first question. 1. Theoretical Justification for Normal Random Variables Suppose that the quantity X that we are trying to model can be expressed as a sum of other random variables as follows: X = Y 1 + . .. + Y n . In great generality, various versions of the central limit theorem assert that, for n large, X D = N( μ , σ 2 ) where μ = E(X) and σ 2 = var(X). Note that the Y i ’s need not be identically distributed, nor do they need to be independent of one another. (They need to be “not too dependent” and “not too different”.) Moral of the Story : If X is the sum of a large number of other random quantities, then X can be approximately modeled as a normal random variable. 2. Theoretical Justification for Log-Normal Random Variables Suppose that the quantity X can be expressed as a product of other random variables as follows: X = Z 1 Z 2 ⋅⋅⋅ Z n . Page 1 of 8
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MS&E 223 Lecture Notes #4 Simulation Input Distributions Peter J. Haas Spring Quarter 2009-10 Then, it follows that log X = Y 1 + Y 2 + . .. + Y n where Y i = log Z i . It follows that for large n, log X is approximately normally distributed with mean μ and variance σ 2 . Hence, X D = exp [N( μ , σ 2 )] . In other words, X is approximately lognormally distributed. Moral of the Story : If X is the product of a large number of other random quantities, then X can be approximately modeled as a log-normal random variable. Example: If X is the net worth of some asset at time n and W 0 is the initial worth, with G i being the (ratio) change in the value of the asset in period i, then X = W 0 G 1 G 2 ... G n . Evidently, X is approximately log-normally distributed. This explains why lognormal random variables often arise in financial asset modelling.
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This note was uploaded on 05/06/2010 for the course MSE 223 taught by Professor Unknown during the Spring '09 term at Stanford.

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lecturenotes04 - MS&E 223 Simulation Peter J. Haas Lecture...

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