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Unformatted text preview: 1 Exponential Distribution & Poisson Process Memorylessness & other exponential distribution properties; Poisson process and compound P.P.’s 2 Exponential Distribution: Basic Facts Pr{ T ≤ t ) = ∫ t λ e – λ u d u = 1 – e – λ t ( t ≥ 0) f ( t ) = { λ e – λ t , t ≥ 0, t < 0 PDF CDF Mean Variance E [ T ] = 1 λ Var [ T ] = 1 λ 2 3 Key Property: Memorylessness • Reliability: Amount of time a component has been in service has no effect on the amount of time until it fails • Interevent times: Amount of time since the last event contains no information about the amount of time until the next event • Service times: Amount of remaining service time is independent of the amount of service time elapsed so far Pr{ T a + b  T b } = Pr{ T a } a , b ≥ Memoryless Property 4 Properties of Exponential Distribution If X 1 and X 2 are independent exponential r.v.’s with parameters (rate) l 1 and l 2 respectively, then P(x1< x2) = l 1/( l 1+ l 2) That is, the probability X1 occurs before X2 is l 1/( l 1+ l 2) Minimum of Two Exponentials: If X 1 , X 2 , …, X n are independent exponential r.v.’s where X n has parameter (rate) l i , then min( X 1 , X 2 , …, X n ) is exponential with parameter (rate) l 1 + l 2 + … + l n Competing Exponentials: 5 Properties of Exponential RV. The probability of 1 event happening in the next ∆ t is Pr{ T ≤ ∆ t ) = 1  e – λ ∆ t = λ ∆ t When ∆ t is small , (– λ ∆ t )n...
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This note was uploaded on 01/06/2011 for the course CHE 6270 taught by Professor Dr.dmitry during the Spring '10 term at University of Florida.
 Spring '10
 Dr.Dmitry

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