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Unformatted text preview: 10.5 Poisson Process 343 Figure 10.2 Sample Function of a Counting Process Figure 10.2 represents a sample of a counting process. The first event occurs at time t 1 , and subsequent events occur at times t 2 , t 3 , and t 4 . Thus, the number of events that occur in the interval [ , t 4 ] is four. 10.5.2 Independent Increment Processes A counting process is defined to be an independent increment process if the num ber of events that occur in disjoint time intervals is an independent random vari able. For example, in Figure 10.2, consider the two nonoverlapping (i.e., disjoint) time intervals [ , t 1 ] and [ t 2 , t 4 ] . If the number of events occurring in one inter val is independent of the number of events that occur in the other, then the process is an independent increment process. Thus, X ( t ) is an independent in crement process if for every set of time instants t = < t 1 < t 2 < ··· < t n , the increments X ( t 1 ) − X ( t ), X ( t 2 ) − X ( t 1 ),..., X ( t n ) − X ( t n − 1 ) are mutually inde pendent random variables. 10.5.3 Stationary Increments A counting process X ( t ) is defined to possess stationary increments if for every set of time instants t = < t 1 < t 2 < ··· < t n , the increments X ( t 1 ) − X ( t ), X ( t 2 ) − X ( t 1 ),..., X ( t n ) − X ( t n − 1 ) are identically distributed. In general, the mean of an independent increment process X ( t ) with stationary increments has the form E [ X ( t ) ] = mt where the constant m is the value of the mean at time t = 1. That is, m = E [ X ( 1 ) ] . Similarly, the variance of an independent increment process X ( t ) with stationary increments has the form Var [ X ( t ) ] = σ 2 t where the constant σ 2 is the value of the variance at time t = 1; that is, σ 2 = Var [ X ( 1 ) ] . 344 Chapter 10 Some Models of Random Processes 10.5.4 Definitions of a Poisson Process There are two ways to define a Poisson process. The first definition of the process is that it is a counting process X ( t ) in which the number of events in any interval of length t has a Poisson distribution with mean λ t . Thus, for all s , t > 0, P [ X ( s + t ) − X ( s ) = n ] = (λ t ) n n ! e − λ t n = , 1 , 2 ,... The second way to define the Poisson process X ( t ) is that it is a counting process with stationary and independent increments such that for a rate λ > 0 the follow ing conditions hold: 1. P [ X ( t + Delta1 t ) − X ( t ) = 1 ] = λDelta1 t + o (Delta1 t ) , which means that the probability of one event within a small time interval Delta1 t is approximately λDelta1 t , where o (Delta1 t ) is a function of Delta1 t that goes to zero faster than Delta1 t does. That is, lim Delta1 t → o (Delta1 t ) Delta1 t = 2. P [ X ( t + Delta1 t ) − ( X ( t ) ≥ 2 ) ] = o (Delta1 t ) , which means that the probability of two or more events within a small time interval Delta1 t is o (Delta1 t ) ....
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This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
 Fall '07
 Carlton
 Counting

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