The interarrival process gives us an alternate definition of a Bernoulli

The interarrival process gives us an alternate

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The interarrival process gives us an alternate definition of a Bernoulli process: Start with an IID Geom( p ) process T 1 , T 2 , . . . . Record the arrival of an event at time T 1 , T 1 + T 2 , T 1 + T 2 + T 3 ,. . . EE 178: Random Processes Page 7 – 14
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Arrival time process: The sequence of r.v.s Y 1 , Y 2 , . . . is denoted by the arrival time process. From its relationship to the interarrival time process Y 1 = T 1 , Y k = k i =1 T i , we can easily find the mean and variance of Y k for any k E( Y k ) = E parenleftBigg k summationdisplay i =1 T i parenrightBigg = k summationdisplay i =1 E( T i ) = k × 1 p Var( Y k ) = Var parenleftBigg k summationdisplay i =1 T i parenrightBigg = k summationdisplay i =1 Var( T i ) = k × 1 p p 2 Note that, Y 1 , Y 2 , . . . is not an IID process It is also not difficult to show that the pmf of Y k is p Y k ( n ) = parenleftbigg n 1 k 1 parenrightbigg p k (1 p ) n k for n = k, k + 1 , k + 2 , . . . , which is called the Pascal pmf of order k EE 178: Random Processes Page 7 – 15 Example: In each minute of a basketball game, Alicia commits a foul independently with probability p and no foul with probability 1 p . She stops playing if she commits her sixth foul or plays a total of 30 minutes. What is the pmf of of Alicia’s playing time? Solution: We model the foul events as a Bernoulli process with parameter p Let Z be the time Alicia plays. Then Z = min { Y 6 , 30 } The pmf of Y 6 is p Y 6 ( n ) = parenleftbigg n 1 5 parenrightbigg p 6 (1 p ) n 6 , n = 6 , 7 , . . . Thus the pmf of Z is p Z ( z ) = ( z 1 5 ) p 6 (1 p ) z 6 , for z = 6 , 7 , . . . , 29 1 29 z =6 p Z ( z ) , for z = 30 0 , otherwise EE 178: Random Processes Page 7 – 16
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Markov Processes A discrete-time random process X 0 , X 1 , X 2 , . . . , where the X n s are discrete-valued r.v.s, is said to be a Markov process if for all n 0 and all ( x 0 , x 1 , x 2 , . . . , x n , x n +1 ) P { X n +1 = x n +1 | X n = x n , . . . , X 0 = x 0 } = P { X n +1 = x n +1 | X n = x n } , i.e., the past , X n 1 , . . . , X 0 , and the future , X n +1 , are conditionally independent given the present X n A similar definition for continuous-valued Markov processes can be provided in terms of pdfs Examples: Any IID process is Markov The Binomial counting process is Markov EE 178: Random Processes Page 7 – 17 Markov Chains A discrete-time Markov process X 0 , X 1 , X 2 , . . . is called a Markov chain if For all n 0 , X n ∈ S , where S is a finite set called the state space . We often assume that S ∈ { 1 , 2 , . . . , m } For n 0 and i, j ∈ S P { X n +1 = j | X n = i } = p ij , independent of n So, a Markov chain is specified by a transition probability matrix P = p 11 p 12 · · · p 1 m p 21 p 22 · · · p 2 m . . . . . . . . . . . . p m 1 p m 2 · · · p mm Clearly m j =1 p ij = 1 , for all i , i.e., the sum of any row is 1 EE 178: Random Processes Page 7 – 18
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By the Markov property, for all n 0 and all states P { X n +1 = j | X n = i, X n 1 = i n 1 , . . . , X 0 = i 0 } = P { X n +1 = j | X n = i } = p ij Markov chains arise in many real world applications: Computer networks Computer system reliability Machine learning Pattern recognition Physics Biology Economics Linguistics EE 178: Random Processes
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