Poisson (ISE 484) - 1 Exponential Distribution...

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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 e u d u = 1 e t ( t 0) f ( t ) = { e t , t 0, t < 0 PDF CDF Mean Variance E [ T ] = 1 Var [ T ] = 1 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 Inter-event 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 t = t When t is small , ( t )n...
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Poisson (ISE 484) - 1 Exponential Distribution...

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