Chapt 10b - 10.5 Poisson Process 343 Figure 10.2 Sample...

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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 [ 0 , 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 [ 0 , 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 0 = 0 < t 1 < t 2 < ··· < t n , the increments X ( t 1 ) X ( t 0 ), 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 0 = 0 < t 1 < t 2 < ··· < t n , the increments X ( t 1 ) X ( t 0 ), 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 ) ] .
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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 ] = t ) n n ! e λ t n = 0 , 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 0 o (Delta1 t ) Delta1 t = 0 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 ) . 3. P [ X ( t + Delta1 t ) X ( t ) = 0 ] = 1 λDelta1 t + o (Delta1 t ) . These three properties enable us to derive the PMF of the number of events in a time interval of length t as follows: P [ X ( t + Delta1 t ) = n ] = P [ X ( t ) = n ] P [ X (Delta1 t ) = 0 ] + P [ X ( t ) = n 1 ] P [ X (Delta1 t ) = 1 ] = P [ X ( t ) = n ] ( 1 λDelta1 t ) + P [ X ( t ) = n 1 ] λDelta1 t P [ X ( t + Delta1 t ) = n ] − P [ X ( t ) = n ] = − λ P [ X ( t ) = n ] Delta1 t + λ P [ X ( t ) = n 1 ] Delta1 t P [ X ( t + Delta1 t ) = n ] − P [ X ( t ) = n ] Delta1 t = − λ P [ X ( t ) = n ] + λ P [ X ( t ) = n
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