104 Prof. F. T. Edgeworth's Exercises 



p' 12 = ^-^ + -05: 

 s 



where Sfy is the sum, and / the number, of the y*s cor- 

 responding to values of x which are between a/ and o/ + 'l. 

 Determine similarly p l2 ") Pn") & c - f°* other degrees or 

 differences on the axis a ; and take the Arithmetic Mean 

 °f p»j P&') Ac. as the value of p 12 . As each little group 

 of observations has the same weight in this combination, 

 it follows that there is assigned to each observation a 

 weight inversely proportional to s r} the size of the group 

 to which the observation belongs. But as the values of 

 x are distributed in conformity with a probability-curve 



with modulus unity, the number s r is proportional to e~ r ; 



and accordingly the weight used is proportional to e +x l. 



To determine the accuracy of this method uncorrected by 

 the rejection of observations, we have for the modulus of the 

 weighted mean 



" L s/ttJo J Jo \1T 



which is infinite. There is, then, evidently in this case an 

 upper limit. 



To determine the limits at which to begin rejecting obser- 

 vations, we have the datum that, while the weight used for 



each observation of the form - is ef the true weight is xl. 



x ° r 



Applying the general formula above given (recollecting that 

 the values of x are distributed in conformity with a pro- 

 bability-curve whose modulus is unity), we have for the 

 upper limit, say v, 



i r°° i 



-j/*=2 I — ^ 2 dx^-{v— u), where u is the lower limit ; 



and a similar equation for the lower limit, u. 



To obtain an approximate solution of these equations, we 



may reason thus: — Observing that the curve y^-^e* 2 has 



a minimum ordinate at the point x=l, let us, in order to 

 approximate to u, make abstraction of the observations out- 

 side that point, and determine a limit u' from the equation 



=■• =2 J 1 ? 



W(l-w). 



The value of u thus determined is less than the true value 

 of u. For it mav be shown that some of the observation? 



