lecture16 - ECE 5510: Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 16 Today: Poisson Processes: (1) Indep. Increments, (2) Expo- nential Interarrivals Exam 2 is Tue. Nov. 10. Today’s material is the last new material covered in Exam 2. Tue (Nov. 3) is a of, but Appl. Assignment 4 is due Tue Nov. 3 at midnight. Thu (Nov 5) is a review class, and HW 7 is due beFore the start oF class. Show up with questions. 1 Poisson Process The leFt hand side oF this table covers discrete-time Bernoulli and Binomial R.P.s, which we have covered. We also mentioned the Geometric pmF in the ±rst part oF this course. Now, we are covering the right-hand column, which answer the same questions but For continuous-time R.P.s. Discrete Time Continuous-Time What is this counting process called? “Bernoulli” “Poisson” How long until my ±rst ar- rival/success? Geometric p.m.F. Exponential p.d.F. AFter a set amount oF time, how many arrivals/successes have I had? Binomial p.m.F. Poisson p.m.F. 1.1 Last Time This is the marginal pmF oF Y n during a Binomial counting process: P Y n ( k n ) = p n k n P p k n (1 - p ) n - k n 1.2 Independent Increments Property In the Binomial process, Y n , we derived the pmF by assuming that we had independent Bernoulli trials at each trial i . In the Poisson process,
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ECE 5510 Fall 2009 2 If we consider any two non-overlapping intervals, they are in- dependent. For example, consider the number of arrivals in
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lecture16 - ECE 5510: Random Processes Lecture Notes Fall...

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