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and he felt he found one 60 miles northeast in Dixon, Illinois Walgreen s older half brother Edwin was attending classes there at the five-year-old Northern Illinois Normal School and told his father, This was the place When the Walgreens moved there in 1887, Dixon still had muddy roads and wooden sidewalks, but its location on the Rock River guaranteed continued growth Established in 1830 when John Dixon set up a ferry service there, the spot became known as Fort Dixon during the Black Hawk War of 1832, which started when Chief Black Hawk roused the local Potowotami and Winnebago tribes to take back the land The war drew hundreds of Union troops, including a host of future famous Americans, among them Jefferson Davis and Abraham Lincoln the presidents involved in the Civil War The outnumbered and overpowered tribesmen stood little chance; and after just a few months of fighting, they were forced to surrender The Rock River was a beautiful country, Chief Black Hawk said I loved it and I fought for it It is now yours Keep it as we did 3 (Chicagoans honored the warrior a century later when they named their NHL hockey team the Blackhawks) The Black Hawk War literally put the tiny settlement of Dixon on the map and, having introduced the soldiers from around the country to the area, served to spread the word about the area s appeal Lincoln returned to Dixon s Nachusa Hotel, where he had boarded during the war, for business trips and campaign stops decades later (A century later, another US president, Ronald Reagan, grew up in Dixon, serving as a lifeguard on the Rock River and as a caddy for Charles Walgreen Sr on the Dixon Country Club golf course, which Charles Sr saved from bankruptcy in the 1930s by purchasing 100 memberships Like Lincoln, Reagan returned as an honored guest while campaigning for president, staying at the Walgreen family s Hazelwood estate) Dixon looked pretty good to Carl Walgreen, so he sold his two farms in Rio to set up a real estate office in downtown Dixon in 1887, when Charles was 14 The community might not have been much by today s standards, but it must have been exhilarating to young Charles, moving.

(E10)

Here P (n) (a) denotes the nth derivative of P (x) at x = a with P (0) (a) = P (a) It is usually not possible to give an explicit expression for the inverse of a given Laplace transform In those situations the unknown function f (x) may be

VPLMN HLR Gr (MAP) SGSN Iu-ps RNC Iu-bis Node-B Uu UE Uu UE UE Iu-bis Node-B Uu Uu UE Gb Gp (GTP) Gn (GTP) BSC A-bis BTS Um UE

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obtained by numerical inversion of its Laplace transform f (s) Numerical inversion methods that perform well for probability functions f (x) are discussed in Appendix F Example E1 The Erlang distribution Suppose that X1 , , Xn are independent random variables having a common exponential distribution with mean 1/ Then X1 + + Xn has the probability density n x n 1 e x , x 0, (n 1)! that is, X1 + + Xn is Erlang (n, ) distributed To prove this, note that the Laplace transform of the probability density fn (x) of X1 + + Xn is given by

fn (s) = E[e s(X1 + +Xn ) ]

sx e x 0 e fn (s) =

Using (E10), the inversion of fn (s) shows that fn (x) is indeed given by the Erlang (n, ) density

.

Example E2 The renewal function Consider a renewal process for which the probability distribution function B(x) of the interoccurrence times of the events has a probability density b(x) The renewal function M(x) is de ned by

.

M(x) =

Bn (x),

(E11)

where Bn (x) is the probability distribution function of X1 + +Xn That is, Bn (x) is the n-fold convolution of B(x) with itself The distribution function Bn (x) has a probability density bn (x) Since bn (x) is the density of X1 + + Xn ,

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