334 Method of estiniatimj Bacterial Densitij 



these data from the Poisson samples were not, as had been expected, 

 systematic, but were due to the sporadic occurrence of exceptional sets; 

 the curvature in the smooth curves being perhaps largely due to the 

 crudity of the criterion employed in excluding the exceptions. This view 

 impressed the authors with the necessity of studying the distribution of 

 small random samples from the Poisson Series, with the double object 

 of devising a valid criterion for the recognition of exceptions, and of 

 testing accurately whether or not the remainder were in reality such 

 random samples. 



5. Small Samples of the Poisson Series 



The study of small samples, essential as it is to the development of 

 adequate statistical methods, has hitherto been practically confined to 

 the normal curve and surface. The following investigation may serve to 

 show, that by taking account of the fundamental properties of those 

 statistics which are derived by the method of Maximum Likelihood, the 

 sampling problems of even discontinuous distributions admit of material 

 simplification. 



In a sample from a Poisson Series, the chance of any observation 

 having the value of x is 



xi 

 where vi is the parameter of the series. 



Hence the chance of observing a given series of values Xj, X2 ... Xn'is 



x^ . X2 . . . . Xf^ . 

 If we estimate ')n from such a sample by the method of maximum 

 likelihood, we have 



g„(logA/) = -n + ~=0, 



SO that u' is the most hkely value of m, and in consequence, as Fisher has 

 recently shown (3), it may satisfy the criterion of sufficiency, in which 

 case the distribution of any other statistic, for a given value of x, must 

 be independent of m. 



That this is so may be proved directly ; for 



Q—nin 



ajj . ^2 . ... X,j . 



may be put into the form 



(nmY'^ inx) ! 



{nx) ! ' n*^^ x^\x^\ . . . x„ ! ' 



