300 BELL SYSTEM TECHNICAL JOURNAL 



Probability that I{t) lies in range (/, / + dl) 



00 



= X^ (Probability of exactly K arrivals) X 

 A-=o 



(Probability that if there are exactly 

 K arrivals, I K{t) lies in (/, / + dl)). 

 Denoting the last probabihty in the summation by PK(I)dI, using notation 

 introduced earlier, and cancelling out the factor dl gives 



P{I) = Z P{K)Pk{I) (1.4-2) 



We shall compute Pk{I) by the method of "characteristic functions" from 

 the definition 



lK{t) = t. Fit - h) (1.3-1) 



of IxiO- The method will be used in its simplest form: the probabihty that 

 the sum 



xi + X2 -\- • ' • + Xk 

 of K independent random variables lies between X and X + dX is 



. -+00 K 



dX — / e"*"^" n (average value of e"*") dti (1.4-3) 



27r J-» fc=i 



The average value of e'"^*", i.e., the characteristic function of the distribution 

 of Xk , is obtained by averaging over the values of Xk . Although this is the 

 simplest form of the method it is also the least general in that the integral 

 does not converge for some important cases. The distribution which gives 

 a probability of | that rr^ = — 1 and ^ that Xk = +1 is an example of such a 

 case. However, we may still use (1.4-3) formally in such cases by employ- 

 ing the relation 



6-'°" du = 27r5(a) (1.4-4) 



£ 



where 5(a) is zero except at a = where it is infinite and its integral from 

 = — e to a = +e is unity where e > 0. 

 When we identify Xk with F{t — ti) we see that the average value of 



tzi.u , 



e is 



1 r^ 



^ exp [inF{t - k)\ dh 



1 JQ 



3 The essentials of this method are due to Laplace. A few remarks on its history are 

 given by E. C. Molina, Bull. Amsr. Math. Soc, 36 (1930), pp. 369-392. An account of 

 the method may be found in any one of several texts on probability theory. We mention 

 "Random Variables and Probability Distributions," by H. CramSr, Camb. Tract in 

 Math and Math. Phys. No. 36 (1937), Chap. IV. Also "Introduction to Mathematical 

 ProbabiUty," by J. V. Uspensky, McGraw-Hill (1937), pages 240, 264, and 271-278. 



