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Unformatted text preview: Simulation Modeling and Analysis (ORIE 4580/5580/5581)
Week 7: Input Modeling and Parameter Estimation (10/05/10 - 10/07/10) 1 Announcement and Agenda • • • HW 4 is available on Blackboard and it is due by 11am on 10/7 HW 5 will be available on Thursday (10/7), and it will be due by 11am on 10/14 I will be traveling on Thursday (10/7). The lecture will be covered by Fan Zhu. 2 Model Speciﬁcation • THREE things to consider when choosing the family of distribution • • • • • • • Theoretical justiﬁcation: Using theoretical properties of each family of distribution to justify its use in modeling different situations Histogram and Bar Plots: Assessing the “rough shape” of the probability density function Q-Q Plots: Assessing the cumulative distribution function Normal Log-normal Poisson Process Weibull • Theoretical Properties of Commonly Used Random Variables 3 When should we use Normal RVs? • Assume that the quantity X that we try to model can be express as • • • • • X = Y1 + Y2 + ⋅⋅⋅+ Yn If Y1,Y2, ...Yn are “approximately” i.i.d., then the Central Limit Theorem tells us that X is approximately normal when n is large! • Example: Modeling total value of the home foreclosures received by a mortage company in a year Yi = default amount by the ith customer X = Y1 + Y2 + ⋅⋅⋅+ Yn Caveat: When might this fail? Recent examples? • Key Takeaway: If X is the sum of a large number of independent random quantities, then X can be approximately modeled as a normal RV. 4 Log-Normal Random Variables • • A random variable X is log-normally distributed if ln X has a normal distribution. Suppose that X = Y1×Y2×⋅⋅⋅×Yn where Y1,Y2,...,Yn are i.i.d. random variables • • • • • By CLT, ln X = ln Y1 + ln Y2 + ⋅⋅⋅ + ln Yn is approximately normal So, X has approximately log-normal distribution. Example: Let Wn denote the worth of an asset (e.g. stock price, housing price) at the end of period n, and let Gn denote the total return in period n. Then, W1 = W0×G1, W2 = W0×G1×G2, ....., Wn...
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