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a hard-working father When her mother, Nellie, knew she was soon to give birth to Myrtle, she gathered her belongings, plus some fresh flowers, preserves, and handmade aprons for gifts, and traveled by horse-drawn wagon to stay with her mother 40 miles away in Carbondale then came right home as soon as she could to tend to her husband and farm That was mid-America in the late 1870s, Myrtle wrote, good neighbors, hospitality, but stand on your own feet 54 Her mother certainly embodied those traits and did so with flair Baking was an art at which no one excelled my mother Things always hummed when she was around 55 Although they were a quintessential farm family, the Nortons had more than a little history and culture Myrtle s great-great-grandfather Ichabod Norton was killed by the British for making bullets for the Revolutionary army Her grandfather Kennedy fought nobly in the Civil War, and her aunts included a professional artist, a piano teacher, and a University of Chicago alum This might not seem like anything out of the ordinary today; but in the era of Stanley and Livingstone, the Boer War, and the assassination of President Garfield a half century before the Walton years such a refined background for a rural family was most uncommon Myrtle s favorite memories, however, were simple ones of family life In my childhood we had a good time at home, she wrote Her father played the violin and would serenade her mother with an eponymous tune, My Gal Nell Papa s pride of mother was very forthright, she said in her book What a wonderful atmosphere for a child to grow up in He always said that the day my mother brought home a daughter was the first time it ever struck him that when a man had a wife, a son, a daughter, and a farm there wasn t much left to wish for 56 It is not difficult when you read Myrtle s descriptions of the virtues her parents stressed to see how those morals were passed on to her son Chuck and to subsequent generations of Walgreens, providing the timeless basic values on which the entire company is based to this day Her father, Myrtle wrote, took his responsibilities as trusts, and he had a keen sense of honor about obligations He made us feel that if we took.

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The variable z is usually taken as a real-valued variable, but in certain applications it may be convenient to treat z as a complex-valued variable It is easily veri ed that the probability distribution {pk , k = 0, 1, } can be recovered analytically from the compressed function P (z) by pk = 1 d k P (z) k! dzk , k = 0, 1, (C1)

The result (C1) shows that a discrete probability distribution is uniquely determined by its generating function Also, the moments of the probability distribution {pk }

kpk = P (1)

k(k 1)pk = P (1)

.

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In general the relation (C1) is only of theoretical value It is often possible to obtain an explicit expression for P (z) when the probabilities pk are unknown (eg from a difference equation for the pk ) In Appendix D we discuss the discrete Fast Fourier Transform algorithm to recover the pk numerically when an explicit expression for P (z) is available Usually it is not possible to analytically recover the pk from (C1) A useful probabilistic interpretation can be given to P (z) If the random variable N is distributed according to {pk }, then P (z) = E(zN ) (C2)

A direct consequence of this relation is that the generating function of the convolution of two discrete probability distributions is the product of the generating functions of these two probability distributions More speci cally, suppose that the random variable N = X + Y , where X and Y are two independent discrete random variables with respective probability distributions {ak , k = 0, 1, } and {bk , k = 0, 1, } Let pk = P {N = k}, k = 0, 1, Then the generating function P (z) of the distribution {pk } is given by P (z) = A(z)B(z), (C3)

where A(z) and B(z) are the generating functions of the probability distributions {ak } and {bk } This follows from E(zX+Y ) = E(zX )E(zY ) In practice it is usually faster to compute the pj by applying the discrete Fast Fourier Transform method j rather than using the convolution formula pj = k=0 aj k bk for j 0 Example C1 The coupon-collecting problem Suppose there are r different types of coupons and each time we obtain a coupon it is equally likely to be any one of the r types How do we compute the probability distribution of the number of coupons we need to collect for a complete set of coupons Denote this number by the random variable X The random variable X can be written as X = Y1 + + Yr , where Yi is the number of additional coupons that need to be collected to increase the number of different coupons in the collection from i 1 to i The random variables Y1 , , Yr are independent of each other and Yi has a geometric distribution

.

Pi (z) =

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