During an academic year, the admissions office of a small

Chapter , Problem 13

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During an academic year, the admissions office of a small college receives student applications at a Poisson rate of 5 per day. It is a policy of this college to double its student recruitment efforts if no applications arrive for two consecutive business days. Find the expected number of business days until a time when the college needs to double its recruitment efforts. Do the admission officers need to worry about this policy at all? Hint: Let X1 be the time until the first application arrives. Let X2 be the time between the first and second applications, and so forth. Let N be the first integer for which X1 2, X2 2, . . . , XN 2, XN+1 > 2. The time that the admissions office has to wait before doubling its student recruitment efforts is SN+1 = X1 + X2 + + XN+1. Find SN+1 by conditioning on N.