198 Expectation of Moments of Frequency Distributions 
E (pi'-piY {pj - Pj) ipk - ph) = - -^JH pjPh (1 - 'ipi) 
+ -^zViPjPh{^-^Pi) 
^ iPi -pi) ip/ - Pj) (Ph - ph) ip/ - Pj) = 2^ PiPjphPf 
-^^PiPjPhP/ 
In the general case we find : 
Eip;'-pd'= 'i 
and hencef 
E(p/ -pd- = 1.3.5... (2r - 1) {^.pni -piY 
+ ^^Pf-' (1 - Pd'-' [(2r - 1) - (1 - Pi) (4r + 1)] 
+ pr' (1 - p,r-' 9^20 - ^^""^ + + 
- ^pi (1 - p,) (40r^ - SOr^ - r + 3) 
+ 4p,:=(l-^,;)n80r^+ 120r^ + 7r-21)] + ...| 
(12), 
E {pi - =1.3.5... (2r + 1) (1 - 2p,) p,'' (1 - p,)''^ 
+ i^r' (1 - '^|j^^[(10r^- or - 3)- 2pUl -i^^^ 
,._. r(r-l)(r-2 ) 
204120 
[(28r^ - 84r'' - 7?-= + 147r - 54) 
(13), 
- 4j9; (1 -2^,:) (56?'^ + 42?-» - 77?--^ - 42?- + 9) 
+ 4>pi^{l -pi)- (112r^ + 504?" + 665?-- + 189r - 72)] + ...| 
* The expressions for E{pj^-p^)*, E (p^ -pif'^ (p/ -pj), and so on, show how dangerous it is to 
reject, without due qualification, terms containing ^ to higher powers. Depending on the magnitude 
3 
of Pi , £ (pi -Pi)'^ may be either greater or less than pl^ (1 -pi)'^, according as p^ (1 -pj) § \ • when p^ 
is very small — of order 1/iV — the term rejected and the term retained are of the same order. 
f Cf. my paper, previously cited, " On the Mathematical Expectation of a Positive Integral Power 
of the Difierence between the Frequency and the Probability of an Event." Both formulae may easily 
be obtained directly from formulae (22) and (23) of Chapter I. 
