Question

(8) Answer the following in the context of a Bernoulli process X, indexed by n 1,2,3,.. (a) Give an example of a stopping time. (b) Give an example of a random time that is not a stopping time. (c) If Ti and T2 are stopping times, is min(Ti, T2) a stopping time? (d) Describe the 'fresh start' and 'memoryless' properties of Xn with respect to a stopping time.

Answer 1

(a) Observe that if we take T inf{n E N[X = 1}, then {T = t} = {X1 0, Х2 — 0, ..., Х,-1 3D 0, Х, determined by the value of sequence upto t) 1, hence T is a stopping time ( As it can be Let Tm supn NX 1, then observe that we need to know the (b) future values of Xn. Hence it is not a stopping time (c) Yes, if T,T2 depend only on the past stopping times, then so is min{T, T2} as it will also are SO (d) Observe that V n E N, Xn is independent of the other Xi's. Hence after any random time T, this process will continue giving results the starting. Hence it has the 'Fresh start' property Oberve that the waiting times follow geometric distribution. And we already know that they have 'memoryless' property i.e. P(Ti s t|T; 2 s) = P(T 2 t), where T is the waiting time for the ith one in the process. Hence this process is 'memoryless as it would have from