. The Bell System technical journal . of P and for any value of c can then be determined by (1).In this way values of a were calculated for every integral value of cfrom 1 to 101 and for eleven particular values of P: , ,, , , , , , , , These resultsare presented in Table II. The numerical values of the coefficients<2i to Qi, corresponding to the particular values of P used in Table II,are given in Table VII. From the information given in Table II, two sets of curves weredrawn, Figs. 1 and 2, the first for each integral value of c in th
. The Bell System technical journal . of P and for any value of c can then be determined by (1).In this way values of a were calculated for every integral value of cfrom 1 to 101 and for eleven particular values of P: , ,, , , , , , , , These resultsare presented in Table II. The numerical values of the coefficients<2i to Qi, corresponding to the particular values of P used in Table II,are given in Table VII. From the information given in Table II, two sets of curves weredrawn, Figs. 1 and 2, the first for each integral value of c in the rangea = 0 to a = 15 and P = to P = , and the second forevery fifth integral value of c in the range a = 0 to a = 200 and the samerange of P. From these curves any one of the variables (P, c, a)may be found corresponding to assigned values of the other two, sub-ject to the practical condition that c is to be an integer. Proof The well-known expressions for the summation of Poissons ex-ponential binomial limit are:. e~ada. (3) PROBABILITY CURVES 103 The series expansion (1) is determined by equating the integrandsof (2) and (3), -^T ac ~ xe ~ ada = —-j= e ~ iPdt, r(c) V2t (4) and solving for positive values of a with the condition that / = — °°when a = 0. Let c = 1_ b2 a=LQi Q = eL) r(c) V27 = i(62g)-1/62eA/) R=-—ro \t2 + M. (5) Substituting these values (5) in equation (4),L = beR, where L is written for dL/dt. (6) 30 QO CO 00 Let L = Vlsi!, M= ]T Msb*, R= V^& , Q=^QJ>*, (7) where the coefficients are polynomials in / (constants in the case ofthe series for M). Upon substituting these series expansions for thefunctions in the last equality of (5) and equating coefficients of likepowers of b, we obtain 0 =()o-£o-l, 0 =Qi-Llt Ro = Q2-L2-ht2 + M0< R1=QS-L3+Ml, R2^Qi-Li-hM2, R„ = Qn+2-Ln+2 + Mn, (w = l, 2, 3 . . ). (8) From (5) we obtain Q0 = eLo, and then L0-\-\ = eLo, the only realsolution of which is Lo = 0, and therefore, Q0—l- 104 BELL SYS
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