Probability Curves Showing Poisson's Exponential Summation

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 0 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 0 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 theoretical 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) , hic -- a -- l), and Ar(c-a-l). The characteristic of all probability curves when n is either finite or infinite, is shown by Fig.