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

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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 ...
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