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RecitationOne - Rensselaer Electrical Computer and Systems...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 1 Wednesday January 27, 2010 Review of Relevant Mathematics ref. Appendix A in Stark and Woods (S&W) text 1. sum of Geometric Series and related (a) Consider the formula: X n =0 a n = 1 1 a Under what conditions on a is the formula correct? Why? (b) More generally, we have, for n 2 > n 1 : n 2 X n = n 1 a n = a n 1 a n 2 +1 1 a , which holds for any value of a , except a = 1 . Find this result by using the formula aS = S + a n 2 +1 a n 1 , where S , P n 2 n = n 1 a n . Show why is this formula true for the geometric series. (c) Now de fi ne a generating function as: G ( z ) , X n =0 a n z n . 1. Show that G (1) = P n =0 a n = 1 1 a whenever | a | < 1 . 2. Show that G 0 (1) = P n =0 na n = a (1 a ) 2 whenever | a | < 1 . This result will be used for fi nding the average value of the so-called geometric random variable. 3. How can we fi nd P n =0 n 2 a n ?
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