372 Mr. F. Y. Edgeworth's Problems in Probabilities. 



Errors to determine the probability that the demand will not 

 exceed any proposed limit. The assumption just made has 

 been defended elsewhere * ; the mathematical constructions 

 which rest upon that foundation are the subject of this paper. 

 I. The first and main problem is : Given a series of Bank- 

 ing returns {e.g. of Notes in the hands of the Public, or 

 of the Reserve), to find the probability that the next return, 

 or the returns in the proximate future, will not exceed 

 certain limits. The general method is to find the Meant of 

 the given series and the Modulus ; to put T for the ratio of 

 the distance between the Mean and the proposed limit to the 



Modulus ; to find —7= I e~ t2 dt from the usual tables, and put 



the value so found for the probability that the next observa- 

 tion on the same side of the Mean as the proposed limit will 

 not exceed the limit, or put half that value for the probability 

 that the next observation unconditionally will not exceed the 

 limit. The question here arises, What method is to be adopted 

 in discovering the Mean and Modulus ? I have elsewhere J 

 considered generally the relative advantages and disadvantages 

 of the different methods. Here it will be sufficient to take 

 account of what is special to the statistics under consideration. 

 It is a peculiarity of Banking returns (as of some meteorolo- 

 gical records) that they cannot be regarded as so many inde- 

 pendent observations, like the different measurements of the 

 same object, or like the heights of different individuals. Con- 

 sider, for instance, the weekly returns of Bank-of-England 

 Notes in the hands of the Public between the years 1833 and 

 1844. Every single return from January to September 

 1834 is above the Mean of the whole series, which is about 

 £18,400,000. Every single return from August 1839 to 

 June 1841 is below that Mean. Such sequences are incon- 

 sistent with the hypothesis of independent observations. We 

 must liken the series of returns, not to a series of numbers 

 each one of which is the sum of, say, twenty digits taken at 

 random (from pages of mathematical tables), but rather to an 

 entangled series constituted in the following manner. Form 

 a first term in the manner just described. Then, for the next 

 term (instead of taking twenty fresh digits), remove one of 

 the constituent digits from the first number and add a fresh 

 digit taken at random. Repeat the process continually. Then 

 you will have an entangled series like the following : — 



* Ibid. 



t Supposing that there is no secular variation. 



X " Observations and Statistics," Camb. Phil. Trans. 1885. 



