248 Prof. F. Y. Edgeworth on the Application of 



of. percentiles (I. c. p. 254 and references there given). Both 

 problems involve the following fundamental principle. 

 Suppose a sample numbering N is extracted at random from 

 an indefinitely large medley of things, divided into classes 

 such as (sets of) balls numbered 1, 2, ... m and mixed in 

 proportions v 1 : v 2 ... v m -\ : v TO , where Xv = N. 



Let F(n l5 n 2 ... n m ; vi, v 2 ... v m ) be the probability that 

 from a given set of z/s there will result a specified set of n's 

 (and accordingly F(v l5 v 2 ... v m \ w 1} n 2 ...n m ) — the variables 

 and constants changing places — the probability that a given 

 set of n's will have resulted from a specified set of v's). 

 Then if the extraction is subjected to a condition which the 

 ft's must satisfy (sets not satisfying that condition being 

 rejected), the most probable set of w's, corresponding to a 

 given set of v's, is found by making F(n 1? n 2 , ...; v i9 v 2 ...) 

 a maximum subject to the imposed conditions (with a like 

 proposition relating to v's inferred from w's). For instance 

 let N=30 ; and ^ 1 = ^ 2 = v 3 = 10. And let there be imposed 

 on the n's the condition « 2 = 2?? 1 . We have then to maximize 

 the expression 



(1/3) 30 30 !/ Wl ! 7 h \ n 8 !+^(ns--2n 1 )+/A(n J + « a + nj-:N), 



where \ and //, are indeterminate multipliers. The required 

 set is found with the use of a Table to be n 1 =7,n 2 ^=14z,n s = 9. 

 Let n u n 2 *... be all large. Then by Stirling's theorem, 

 log F(n 1? n 2 ... vi* v 2 ...) becomes approximately 



Const. — n l log n Y — n 2 \ogn 2 ... +n 2 log v 1 + n 2 log v 2 + ... (10) 



The variable part of this expression may be called H as 

 being in eodem genere with the " H" above defined. (In the 

 inverse problem of percentiles where the v's are to be deter- 

 mined subject to the condition that they are given functions 

 of certain sought quantities, it is presumed that the differ- 

 ences (v — n) are small; whereby Taylor's principle becomes 

 available, and H reduces to the Pearsonian % 2 ; I. c. p. 254.) 



Let the given v's be all equal, the latter terms of (10), 

 n i l°g v \ + n 2 l°g v 2- ■ • ? thus becoming absorbed in the constant 

 part (since 2w = N). Let the n's be represented by columns 

 standing on equal bases, numbered 1, 2, 3 ... r, r+1, .. 

 on the right of a central point, and — 1, —2, —3... —r, 

 — (r + 1) ... on the left. Let there be imposed the condition 



%Mr 2 (n r -hn- r )=~R 2 



(R a given large number) ; the summation extending over 

 all possible values of n. We have thus to minimize 



%n r log n r -\-^n- r \ogn_ r 



+ X(Mr 2 (n r + n.,)-R 2 ) + ^(XKi"-r)-N). 



