In Exercise 30.14 (and again in Exercise 31.23) we considered the sample space .N; P /

Chapter 32, Problem 32.16

(choose chapter or problem)

In Exercise 30.14 (and again in Exercise 31.23) we considered the sample space .N; P / property of the geometric distribution on N. If we think of P .k/ as the probability of, say, a disaster happening in the k th time interval, then P .Ak / is the probability that the disaster occurs at time k or later. The conditional P .AkCj jAj / asks: given that the disaster did not occur before time j , what is the probability it occurs k time units or more after time j . The result is that these two probabilities are the same. That is, we forget that the disaster did not yet occur and restart the countdown to doom anew. in which P .k/ D ark (where r D 1 a). In this sample space, define the event Ak D fn 2 N W n kg. That is, Ak is the event that the randomly chosen natural number is k or larger. Please do: a. Calculate P .Ak/. b. Calculate P .AkCj jAj /. c. You should observe that your answers to (a) and (b) are the same. This is a special feature of the geometric probability distribution for N. Prove that if .N; P / has the property that P .Ak/ D P .AkCj jAj / for all k; j 2 N, then P must be a geometric distribution. That is, prove that there is a real number a such that P .k/ D a.1 a/k for all k 2 N.