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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 off, 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 first 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 first 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 ) = parenleftbigg n k n parenrightbigg 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.
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