200 Expectation of Moments of Frequency Distributions 
If we agi-ee to put ^ 0,0 = 1 and G,. ^ = 0, then if 1\ + j-g + . . . + = 2r, the first 
term can be brought to the form (cf Introthiction (28)) : 
r(orri-l) (or J-.^-l) r-/t, -/i,- ... -/ij— , (or r,- 1) 
(18), 
and when + 7-0+ ... + 7i-= 2r + 1, to the form (cf Introduction (29)) : 
j-+l(or )•,-!) (or r.-l) r+l-A, -/i.^- ... -/u-, (orr;;.-!) 
A, = 0 
hl- = 0 
C'fM 4.,,„5,^,_^,0 
k ' 
Noting that, in accordance with (3), 
K19). 
Ep'n P'>: -P'h = ^k Ph P'. ■ ■ ■ P'k = Ph P'. • • • P', £ (- 1 )' h (20), 
we find, on the other hand, easily : 
J"- ip'h - PO (P'h - PO ■ ■ ■ (P'ik -Pik) 
= Ph Ph ■ ■ ■ Pik ^''^k,h= Pi, Pi, ■ ■ ■ pik ^ Bn, ,h-k 
A-=Ent 
Hence : 
(21). 
r 20 24) 
^iP'i, -pi,)(P'i, -Pi2) ••• (P'H-pk)=P'lPi, ••• P'5 {- + ]^4j 
„ , , , , , ( 15 130 120 
^ipii- p>i) ■■■(Pi,- Pi,) = Pii ■ ■ ■ pi. i-'w'^lf*' W 
(22). 
E {p'i-,-Pi,) ■ ■ ■ {p'i, - Pi,) = pit • • ■ Pi, I 
210 _ 924 720 
II 
^p/^j 
(1) Noting that m,.^(^) = EX ^^.^ = E 
we find 
^,,,,,^E\i pp^f+ i S pj.p'jjjj, 
= V ^fEp;^+ i s ^,jj,Ep\p'j^ 
.7 = 1 ji = i is+ii 
nir, 
Pf+wPji^ -Pj) 
PhPk- -^PkPh 
~ S t.'i 
j=i L -''^ J .?■=! J^*J> 
= j .2 Pj ^ + ^ Pj - ^ I 2^ Pj^ = "if + 2^ ['"2 - 
