Consider a sequence of independent tosses of a coin and let P{head} be the probability of a head on any toss. Let A be the hypothesis that P{head} = a and let B be the hypothesis that P{head} =

b, 0 <

a, b < 1 Let X, denote the outcome of the ith toss and let Zn P{XX A} P{X,..,X|B} Show that if B is true, then:

(a) Z, is a martingale, and

(b) lim,..., Z, exists with probability 1

(c) If ba, what is lim,... Z?6.25. Let Zn, n 1, be a sequence of random variables such that Z = 1 and given Z, Z-1, Z is a Poisson random variable with mean Z-1, n> 1 What can we say about Z, for n large?