expo - Spring 2010 Pstat160b H-O #3 Exponential...

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Spring 2010 Pstat160b H-O #3 Exponential Distribution summary Notation: We denote the exponential distribution with parameter λ > 0 by E ( λ ). Interpretation: Exponential random variables will often be interpreted as the time an event occur, or as a ’random clock’. Basic properties : p.d.f f ( x ) = λe - λx x 0 C.d.f F ( x ) = 1 - e - λx x 0 Tail probability P ( X x ) = 1 - F ( x ) = e - λx x 0 M.G.F. M ( t ) = λ λ - t t 6 = λ mean E ( X ) = 1 λ variance Var( X ) = 1 λ 2 Further properties : Memoryless property: P ( X > x + y | X > y ) = P ( X > x ) x,y > 0 . Hazard rate (or juste “rate”): For a continuous random variable T , the probability that T occurs between t and t + h , given it doesn’t occur before t is given by the density r ( t ) called hazard rate function which is r ( t ) = f ( t ) 1 - F ( t ) r ( t ) = λ for exponential E ( λ )only So we can interpret the parameter λ as the “infinitesimal probability that something is going to happen next” or the clock to ring, or the rate at which it happens (say customer arrival/unit time), that is P ( X [0 , 0 + dx ]) = P ( X [ x,x + dx ] | X > x ) = λdx (note how the memoryless property relates to constant hazard rate).
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This note was uploaded on 01/10/2011 for the course STAT PStat 160b taught by Professor Bonnet during the Spring '10 term at UCSB.

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expo - Spring 2010 Pstat160b H-O #3 Exponential...

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