388 Mr. L. Isserlis : Application of Solid Hypergeometrical 



I£ the values for the moments be examined it will be seen 

 that a frequency distribution is not hypergeometrical in type 

 unless certain conditions are fulfilled. For example 



P2i2^opo3 =PuPo2Pso • • • • (44) 

 P02 P20 P21 P12 = Pn poz Pso .... (45) 



Other identities will appear later. This explains why we 

 have to use more moments than there are unknowns to be 

 determined. 



§7. The solution. Let p denote the correlation so that 



2 = _Pn_ (46) 



P20P02 ; 



and let X 2 = ^^ 2 ' (47) 



P20P21 



From (21), (22), (23) we get 



1 _ P2offo2 (n—r)(n — r f ) ,.„. 



? — ^F-= ^ • • • • («) 



and using (38), (39) 



x2= r n-r/n-Jr^y 



r' n—r'\n—2rj 



Put^-1 = £ - f — 1=77, then 

 r r 



&=p (50) 



g^-* («) 



whence f = —7 — ~r-\ , (&%) 



p(p + \) 



X being taken of the same sign as p. 

 Equations (50) and (52) give £ and 77. 

 Next from (21), (22), (40), and (41) we have 



Pzl _ (I-*?? . *-l ( w - 2 ^) 2 (53) 



p 03 2 _ (1-2?) 2 n-1 Qi-2r') 2 (54) 



^ »— ^T^)(n-2)»r'(n-r')' 



