Probability Curves Showing Poisson's 

 Exponential Summation 



By GEORGE A. CAMPBELL 



IN many important practical operations the constant probability 

 of an event happening in a single trial is extremely small, but the 

 number of trials is so large that the event may actually occur a suffi- 

 cient number of times to become a matter of importance. The curves 

 of Figs. 1 and 2 show the probability P of such an event happening at 

 least c times in a number of trials for which the average number of 

 occurrences is a. The probability range shown is from 0.000001 to 

 0.999999 and the average extends from to 15 in Fig. 1 and to 200 

 in Fig. 2. An open scale is obtained at both ends, even when the 

 probability approaches to within one part in a million of the limits 

 and 1, by employing an ordinate scale corresponding to the normal 

 probability integral. 



In the practical use of these curves the first question which arises 

 is — What number of trials is necessary to make the curves applicable? 

 In practice an infinite number of trials, which is the case for which 

 the curves are drawn, can never be attained; and if we had absolutely 

 no knowledge of the relation between the probabilities for an infinite 

 number and a finite number of trials, the curves would have a theo- 

 retical interest only. We do, however, know in a general way when a 

 finite number of trials approximates to the limiting case; the more 

 complete and precise our knowledge on this point, the more generally 

 useful the curves will become. Without attempting to go into the 

 question exhaustively, which would require most careful analysis, a 

 general answer will be found to the question as to the number of trials 

 required by plotting the simple functions (a/c) c , h(c — a — l), and 

 hc(c-a-\). 



The characteristic of all probability curves when ;/ is either finite 

 or infinite, is shown by Fig. 3, where P(c,n,a) denotes the probability 

 of an event happening at least c times in n trials when the average 

 number of occurrences is a. Any curve P(c,n,a) is contained be- 

 tween the ordinates at a—0 and a = n and is asymptotic to these 

 ordinates; it cuts P = y 2 between a = c— 1 and c — 0.3. Thus as n 

 decreases from infinity to c, the central portion of the c curve changes 

 but little, but the curve is confined to the narrowing band to the 

 left of a — n and becomes steeper. On reducing n to c— 1 the c curve 

 disappears entirely, since c cases cannot occur; the number of trials 



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