SPLecture10 - Lecture 10 Mariana Olvera-Cravioto Columbia...

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Lecture 10 Mariana Olvera-Cravioto Columbia University [email protected] February 25th, 2015 IEOR 4106, Intro to OR: Stochastic Models Lecture 10 1/20
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Thinning of a Poisson process I Suppose that { N ( t ) : t 0 } is a Poisson process with rate λ . I Suppose each of the events counted by N ( t ) is either of Type I or Type II , with probabilities p and 1 - p , respectively, independently of any other events. I Let N 1 ( t ) be the number of events of Type I that occur during the interval [0 , t ] , and let N 2 ( t ) be the number of events of Type II that occur during the interval [0 , t ] . I What type of processes are N 1 ( t ) and N 2 ( t ) ? IEOR 4106, Intro to OR: Stochastic Models Lecture 10 2/20
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Thinning of a Poisson process I Suppose that { N ( t ) : t 0 } is a Poisson process with rate λ . I Suppose each of the events counted by N ( t ) is either of Type I or Type II , with probabilities p and 1 - p , respectively, independently of any other events. I Let N 1 ( t ) be the number of events of Type I that occur during the interval [0 , t ] , and let N 2 ( t ) be the number of events of Type II that occur during the interval [0 , t ] . I What type of processes are N 1 ( t ) and N 2 ( t ) ? I Are they independent of each other? IEOR 4106, Intro to OR: Stochastic Models Lecture 10 2/20
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Thinning of a Poisson process... continued I Clearly, N i ( t ) is still a counting process for i = 1 , 2 . IEOR 4106, Intro to OR: Stochastic Models Lecture 10 3/20
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Thinning of a Poisson process... continued I Clearly, N i ( t ) is still a counting process for i = 1 , 2 . I Also, it satisfies, for i = 1 , 2 : I N i (0) = 0 I It has independent increments (inherited from N ( t ) ) IEOR 4106, Intro to OR: Stochastic Models Lecture 10 3/20
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Thinning of a Poisson process... continued I Clearly, N i ( t ) is still a counting process for i = 1 , 2 . I Also, it satisfies, for i = 1 , 2 : I N i (0) = 0 I It has independent increments (inherited from N ( t ) ) I Let’s take a look at the distribution of N i ( t + s ) - N i ( s ) .... IEOR 4106, Intro to OR: Stochastic Models Lecture 10 3/20
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Thinning of a Poisson process... continued I Clearly, N i ( t ) is still a counting process for i = 1 , 2 . I Also, it satisfies, for i = 1 , 2 : I N i (0) = 0 I It has independent increments (inherited from N ( t ) ) I Let’s take a look at the distribution of N i ( t + s ) - N i ( s ) .... P ( N i ( t + s ) - N i ( s ) = k ) = X n =0 P ( N i ( t + s ) - N i ( s ) = k | N ( t + s ) - N ( s ) = n ) · P ( N ( t + s ) - N ( s ) = n ) = X n = k P ( N i ( t + s ) - N i ( s ) = k | N ( t + s ) - N ( s ) = n ) · e - λt ( λt ) n n ! , since N ( t + s ) - N ( s ) is a Poisson r.v. with mean λt . IEOR 4106, Intro to OR: Stochastic Models Lecture 10 3/20
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Thinning of a Poisson process... continued I Now note that given that there were n events (of any type) in the interval [ s, s + t ] , then the number of Type I (rest. Type II ) events is Binomial with parameters ( n, p ) (resp. ( n, 1 - p ) ). I It follows that P ( N 1 ( t + s ) - N 1 ( s ) = k | N ( t + s ) - N ( s ) = n ) = n k p k (1 - p ) n - k . I We then have IEOR 4106, Intro to OR: Stochastic Models Lecture 10 4/20
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Thinning of a Poisson process... continued I Now note that given that there were n events (of any type) in the interval [ s, s + t ] , then the number of Type I (rest. Type II ) events is Binomial with parameters ( n, p ) (resp. ( n, 1 - p ) ).
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