Mathematical Techniques of Applied Probability. Discrete by Jeffrey J. Hunter, J. William Schmidt

By Jeffrey J. Hunter, J. William Schmidt

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V. with parameter A, (i = 1 , 2 , . . , ) . v. with parameter -h · · + A^. ,fc) and Π Pi{s) = e x p | ( . Z >ie)(5 - 1 ) | . 2: η tickets are drawn one at a time (with replacement) from a set of tickets numbered respectively 1 , 2 , . . , a. Under the assumption that at each draw each ticket is equally likely to be selected, what is the probability that the total of the numbers on the tickets drawn is fc? Let Xi be the number on the ith ticket drawn; thus S„ = -h · · + X« is the total of the numbers drawn.

G.. Buck (1956, p. 174), Rudin (1953, p. 143), or Hyslop (1959, pp. 110-115)]. • Note that under condition (a) either the double summations both converge to the same finite sum or else they are both properly divergent. Under conditions (b) and (c) the double summations converge to the same finite sum. 5: Dominated Convergence Theorem. If {a^J} is a double sequence (i = 1 , 2 , . . = 1, 2 , . . , ) such that \aij\ < bj for all ij where Yj=i bj is convergent and such that for all j = 1 , 2 , . .

5: If {c„} = {a„} * {b„},Y^=o a„ converges, a„>0, lim,,^^ b„ exists. Then / c„ = lim n-» 00 * Proof: 00 \ lim b„. „ -b\< n>K. Therefore, for n> K, Κ Cn-ab^ η X - fc) + ε/3α for 00 X - fc) - Σ α^. Let Β = max{|fcfc - fc|, /c = 0 , 1 , . . ,X; and ε/3α}. Then forn > Χ kn - flfc| < Β Σ fc = 0 and + ^ Σ :>¿2jk = #:+l ^n-fc + \b\ Σ «m- m = n+l 48 2 Now we can find an GENERATING FUNCTIONS such that for all n> Also, I: Σ k=0 FL ^n-k = Σ m=n-k 00 ^ Σ m=n-k «M» and thus we can find an Ν2 such that for all π > ÍV2 + Κ Σ m = n-k «M

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