# 14poisson - EECS 501 BERNOULLI AND POISSON PROCESSES Fall...

This preview shows pages 1–2. Sign up to view the full content.

EECS 501 BERNOULLI AND POISSON PROCESSES Fall 2001 DEF: Bernoulli random process x ( n ) is a discrete-time 1-sided iidrp with: x ( n ) = 1 success or arrival with prob. p 0 failure or nonarrival with 1 - p p x ( n ) ( X ) = p for X = 1 1 - p for X = 0 Note: Kolmogorov: p x ( i 1 ) ...x ( i N ) ( X 1 . . . X N ) = Q N i =1 p x ( n ) ( X i ). Bernoulli rvs. Question pmf name pmf formula E [ k ] σ 2 k Pr [ k successes in N trials ] Binomial ( N k ) p k (1 - p ) N - k Np Np (1 - p ) [ #trials until next success ] Geometric (1 - p ) k - 1 p, k 1 1 /p (1 - p ) /p 2 [ #trials until r th success ] Pascal ( k - 1 r - 1 ) p r (1 - p ) k - r r/p r (1 - p ) /p 2 Note: “Until” means “up to and including ” in the above. pmf ranges omitted. Binomial: Pr[ k successes in any closed interval of length N - 1 ( N points)] Binomial: =sum of N independent Bernoulli rvs.: z-xform=((1 - p ) + pz ) N . Geometric: 1 st -order interarrival time=#trials from last success to next success. Geometric: Let A=next success on k th trial and B j =no successes on last j trials. Memoryless: Pr [ A | B j ] = P r [ AB j ] P r [ B j ] = P r [ B k + j - 1 ] p P r [ B j ] = (1 - p ) k + j - 1 p (1 - p ) j = (1 - p ) k - 1 p k =1 , 2 ...

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern