Week07-Lecture (20101005)

Week07-Lecture (20101005) - Simulation Modeling and...

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1 Simulation Modeling and Analysis (ORIE 4580/5580/5581) Week 7: Input Modeling and Parameter Estimation (10/05/10 - 10/07/10)
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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
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Model Specifcation THREE things to consider when choosing the Family oF distribution Theoretical justifcation: 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 Theoretical Properties oF Commonly Used Random Variables Normal Log-normal Poisson Process Weibull 3
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When should we use Normal RVs? Assume that the quantity X that we try to model can be express as X = Y 1 + Y 2 + ⋅⋅⋅ + Y n If Y 1 , Y 2 , . .. Y n 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 Y i = default amount by the i th customer X = Y 1 + Y 2 + ⋅⋅⋅ + Y n 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
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Log-Normal Random Variables A random variable X is log-normally distributed if ln X has a normal distribution. Suppose that X = Y 1 × Y 2 × ⋅⋅⋅ × Y n where Y 1 , Y 2 ,..., Y n are i.i.d. random variables By CLT, ln X = ln Y 1 + ln Y 2 + ⋅⋅⋅ + ln Y n is approximately normal So, X has approximately log-normal distribution. Example: Let W n denote the worth of an asset (e.g. stock price, housing price) at the end of period n, and let G n denote the total return in period n. Then, W 1 = W 0 × G 1 , W 2 = W 0 × G 1 × G 2 , . ...., W n = W 0 × G 1 × G 2 × ⋅⋅⋅ × G n This is one of the reason why log-normal random variables appear frequently in Fnancial asset modeling! Key Takeaway: If X is a product of a large number of independent random quantities, then X can e approximately modeled as a log-normal random variable. 5
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Poisson Processes 6 Customer 2 Total Arrivals Customer 1 The arrivals can be expressed as a superposition of n independent sources. In this case, each source corresponds to a customer. Customer 3 . . . Palm-Khintchine Theorem: As n →∞ , the arrival process approaches the Poisson process! Consider the arrivals to Amazon.com over a period of 1 month Other Examples: call center, restaurant, grocery stores, etc.
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Week07-Lecture (20101005) - Simulation Modeling and...

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