6.041 / 6.431 13. Bernoulli Process

LECTURE 13 The Bernoulli process • Readings: Section 6.1 Lecture outline • Definition of Bernoulli process • Random processes • Basic properties of Bernoulli process • Distribution of interarrival times • The time of the kth success • Merging and splitting The Bernoulli process • A sequence of independent Bernoulli trials • At each trial, i: – P(success) = P(Xi =1)= p – P(failure) = P(Xi =0)=1− p • Examples:– Sequence of lottery wins/losses – Sequence of ups and downs of the Dow Jones– Arrivals (each second) to a bank – Arrivals (at each time slot) to server Random processes • First view: sequence of random variables X1,X2,... • E[Xt]= • Var(Xt)= • Second view: what is the right sample space? • P(Xt = 1 for all t)= • Random processes we will study: – Bernoulli process (memoryless, discrete time) – Poisson process (memoryless, continuous time) – Markov chains (with memory/dependence across time) Number of successes S in n time slots • P(S = k)= • E[S]= • Var(S)= Interarrival times • T1: number of trials until first success – P(T1 = t)= – Memoryless property – E[T1]= – Var(T1)= • If you buy a lottery ticket every day, what is the distribution of the length of the first string of losing days? Time of the kth arrival • Given that first arrival was at time t i.e., T1 = t: additional time, T2, until next arrival – has the same (geometric) distribution – independent of T1 • Yk: number of trials to kth success – E[Yk]= – Var(Yk)= – P(Yk = t)= Splitting of a Bernoulli Process (using independent coin flips) time time time Original process q 1 q yields Bernoulli processes Merging of Indep. Bernoulli Processes time time time Merged process: Bernoulli (p+ qpq) Bernoulli (p) Bernoulli (q) yields a Bernoulli process (collisions are counted as one arrival) Poisson approximation to binomial • Number of arrivals in n slots is binomial n! pS(k)= n ·kp (1)(n −k)!k!−p−k , for k ≥0 • Interesting to think of: n →∞ with λ = np constant n! pS(k)= p k(1 k p)n−(n −k)!k! ·−n(n = −1) ···(n −k +1) λk k! · nk·�λ 1 − nn �−k n = n −1 n ·n −k +1 n · λk n · k! ·�λ 1 − nn �−k • For any fixed k ≥0, lim (1 −nk λλ/n)n−= e −, so: →∞ λklim p(k)= n→∞Sλe − , k =1, 2,... k! MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.