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Unformatted text preview: = W0×G1× G2 ×⋅⋅⋅× Gn This is one of the reason why lognormal random variables appear frequently in ﬁnancial 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 lognormal random variable. 5 • Poisson Processes
Consider the arrivals to Amazon.com over a period of 1 month
Total Arrivals Customer 1 Customer 2 Customer 3 . . . independent The arrivals can be expressed as a superposition of n sources. In this case, each source corresponds to a customer. • • PalmKhintchine Theorem: As n →∞, the arrival process approaches the Poisson process!
Other Examples: call center, restaurant, grocery stores, etc. Key Takeaway: Poisson process works well in modeling arrival processes Note: The PalmKhintchine Theorem assumes that each source is stationary (that is, no timeofday or seasonality effects). If each source exhibits timeofday effects, then the superpositions will approach nonstationary Poisson process! 6 • Weibull Random Variables
Consider serial and parallel systems consisting of n components: • • • In a serial system, if one component breaks, the entire system fails. In a parallel system, the system fail when ALL components break. Let Y1,Y2, ...,Yn denote the lifetime of each component, and let L denote the life time of the system L = max{ Y1,Y2,....,Yn} L = min{ Y1,Y2,....,Yn} Under certain conditions, if the lifetimes of the components are i.i.d., then L will approximately have a Weibull distribution when n is large! Key Takeaway: The lifetime of a complicated system can be approximated by a Weibull distribution! 7 Histograms and Bar Plots • • • In addition to theoretical justiﬁcations, we also need do a “visual” check to assess the ﬁt of a particular family of distribution. Histogram of the data indcates the shape of the probability density function (p.d.f.) from which the data have been sampled. • •
• By looking at the histogram, we can often “guess” the family of distribution...
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This note was uploaded on 10/26/2010 for the course OR&IE 5580 at Cornell University (Engineering School).
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