POISSON'S PROBABILITY SUMMATION 607 



is infinitely small. The term "uniform" applies, of course, not to 

 the results of the trials (or samples) but to the essential conditions 

 under which they are obtained, and "independent" is used with the 

 meaning that the result of one trial (or sample) does not afTect the 

 occurrence of the event in any other trial (or sample). The first 

 and third assumptions, translated into exact mathematical language, 

 define a particular kind of probability function, which can be de- 

 rived by taking the limit, as n becomes infinite and pn remains finite, 

 of the point binomial (p + q)" for the probability of any number of 

 occurrences of a given event in a group of n independent, uniform 

 trials, when the probability that the event occur in a single trial is p. 

 The second assumption is required in order that we may pass from 

 the abstract idea of a probability function to the concrete idea of a 

 frequency-distribution. 



Throughout this discussion the summation form of the frequency- 

 distribution, giving the probability of at least c occurrences, is used 

 rather than the individual term form, giving the probability of ex- 

 actly c occurrences. One reason for the use of the summation form is 

 its more direct applicability to many practical problems in which 

 the chance of exceeding a certain limit, rather than the chance of 

 obtaining any one particular value, is of practical importance. Sec- 

 ondly, as Fig. 3a shows, the individual term form gives in general 

 two possible values of c for any pair of values of a and P, whereas the 

 summation form is single-valued and introduces no such ambiguity. 



Fig. 3 also calls attention to some of the outstanding characteristics 

 of the Poisson distribution, its discontinuity and skewness, in par- 

 ticular. That the Poisson distribution must be a series of discrete 

 points and not a continuous curve is a direct result of the assumption 

 that c represents a number of occurrences. That the distribution is 

 skew follows from the fact that the possible number of occurrences is 

 much larger, in fact infinitely larger, than the average number of 

 occurrences. This skewness is quite marked even in the Poisson 

 distribution with a = 5, which is shown in Fig. 3, and it becomes more 

 pronounced as a is decreased toward zero. If, for example, the aver- 

 age number of occurrences in a million trials is one, in any particular 

 group of a million trials it is equally likely that there will be no oc- 

 currence of the event or one occurrence, and it is almost 1.4 times as 

 likely that there will be no occurrence as that there will be two or 

 more occurrences, though zero and two are equally removed from the 

 average. A third important characteristic of the Poisson exponential, 

 which is not brought out by this figure, is its extreme simplicity. 

 The distribution is entirely determined by the value given to a single 



