Consider a Bernoulli process with parameter p=1/3. Let T1 be the time of the first success and let T1+T2 be the time of the second success. We are told that the results of the two slots that follow the first success are failures, so that XT1+1=XT1+2=0. What is the conditional expectation of the second interarrival time, T2, given this information? (Recall that the expectation of a geometric random variable with parameter p is equal to 1/p.)
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After time 𝑇1 , we have two failures, and these are part of the interarrival time 𝑇2 . Given this information, the process starts fresh at time 𝑇1+3 and the number of trials from time 𝑇1+3 onwards until the next success is geometric with parameter 1/3 , and has an expected value of 3. Therefore, the conditional expectation of 𝑇2 , given the information we were given, is 2+3=5 .
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