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on a job of any kind, it was our duty to do it in the best possible way, no matter how tired of it we might become It is still almost impossible for me to leave any task unfinished, even though I may wish I had never begun 57 When Myrtle was four, the family moved to Chester in Southern Illinois so her father could get a more stable job as a guard in the state penitentiary, overlooking the Mississippi A big [jail] break was frustrated by father s men who refused to join in because he had been so square with them, she wrote late in her 84-year-long life The pride I felt when people spoke about the way Papa s men saved the day has stayed with me 58 Unfortunately, it was also the prison that gave Myrtle her most painful memory, too, that of her father dying of lung fever, brought on by the fine dust from the quarries he guarded while the prisoners crushed rock He was only 38, his young daughter just 7 Through Nellie s strength and warmth, however, the family stuck together The Nortons retained their sense of fun and adventure, too By August, attendance [at the 1893 Chicago Exposition] was averaging two hundred thousand a month, Myrtle recalled Down in our part of the state farmers were mortgaging their farms, teachers taking out their savings, merchants borrowing on their inventories, and some bankers even foreclosing mortgages in order to have the money to take the family to the Fair Mother certainly scraped the bottom of the barrel when she decided that [my brother] Paul and I were to go to Chicago 59 The trio rode on the Ferris Wheel, listened to John Phillip Sousa s famous band and most exciting of all drank ice water In those days, Myrtle explained, ice was not taken for granted Mr Drake, the founder of the Drake Hotel, had honored Columbus by donating a fountain of ice water kept cold by three tons of ice every day It was a great Fair! What makes a fair great if not the mind stretching of the young and the fun that outlasts the decades 60 After Myrtle graduated from high school in 1898 the same year Walgreen shipped off to Cuba her brother, Paul, decided he wanted to go to Chicago to become a registered pharmacist He thought he would be striking out on his own, but Nellie Norton would hear none of it.

Example C2 Success runs Another illustration of the usefulness of the generating function approach is the analysis of success runs in independent Bernoulli trials How do we compute the probability that in n independent Bernoulli trials with success probability p there is some sequence of s consecutive successes For xed s, denote this probability by n Pn for n 0 The probability Pn can be written as Pn = j =0 pj for n = 0, 1, , where the probability pj is de ned as pj = the probability that for the rst time a sequence of s consecutive successes occurs at the j th trial

Note that {pj , j = 0, 1, } is a probability distribution with j =0 pj = 1 s For j > s, we have the recursion Obviously pj = 0 for j < s and ps = p s

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pk 1 (1 p)pj k ,

MAP SSIN [CCBS]

j = s + 1, s + 2,

To prove this, x j > s and denote by A the event that a sequence of s consecutive successes occurs for the rst time at the j th trial The event A can only occur

if one of the mutually exclusive events B1 , , Bs occurs, where Bk is the event that each of the rst k 1 trials have success as outcome but the kth trial does not Noting that P (A) = pj , P (Bk ) = pk 1 (1 p) and P (A | Bk ) = pj k , the recursion follows by applying the law of conditional probabilities As an alternative to the recursion scheme, the probabilities pj can also be numerically obtained by numerical inversion of the generating function Multiplying both sides of the above recursion for pj by zj and summing over j , it follows that the generating function P (z) = j =0 pj zj satis es P (z) = ps zs + = ps zs +

zk pk 1 (1 p)

.

MAP ISD VLR profile SS-CSI[ ECT, MPTY CD]

= ps zs + P (z) This gives P (z) = 1

(C5)

Hence an explicit expression has been obtained for the generating function P (z) of the unknown probabilities {pj } Using this expression the unknown probabilities pj can also be numerically obtained by applying the discrete Fast Fourier Transform method from Appendix D A simple but extremely useful method to compute pj for large j is to use an asymptotic expansion This approach will be discussed below in a general setting To do so, some basic concepts from complex function theory are needed such as the concept of an analytic function In a nutshell, a function on a domain in the complex plane is called analytic when the function is differentiable in nitely often on that domain A fundamental theorem from complex function theory states that a function f (z) is analytic in the complex region |z| < R if and only if f (z) allows for the power series representation f (z) = fn zn for n=0 |z| < R Asymptotic expansion

Suppose that the generating function P (z) = j =0 pj zj of an (unknown) probability distribution {pj , j = 0, 1, } has the form

MAP SSIN[ECT, MPTY, CD]

P (z) =

N (z) D(z)

(C6)

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