152 Expectation of Moments of Frequency Distribntions 
III 
(1) lfmi = 0, then 
and 
yti, = E{X- m,y = EX'- = lllr 
H-r, (JV) = IN)- 
Putting vii = 0 in the formulae of § II, we may, consequently, replace in them 
the quantities m by the corresponding quantities /u.. But when m = 0, Rr,r-?i = 0,. 
if h < r/2 ; but if h then Rr,r-h reduces (cf (5)) to 
'j,!j,!::)i,![^,!}H/(,!p^...[V]^ 
•(12), 
where the summation extends to all positive integer values of ji, jc^, ... jf, satis- 
fying the condition ji-f-^'o+... + jf — r — h, and to all positive integer values of 
h-^, ... /'/, satisfying the conditions: 
2 ^/;, < /(2< ... < hf, 
+ + ...+hfjf = r. 
Hence 
I 2)--l 1 i-1 
M-r -'-2);2r—h~ -'■«>■, r-h 
1\- /, = ,. i\ - 7,=.o 
1 2r 1 r-1 
/( = )• + ! 
(13), 
or 
fJ'i; (A') 
1 r .v{-[^"'-(r)-'0}r 
where Ent. ^0 denotes the greatest integer in ^ . 
If we now put 
we find after some transformations : 
r-l -[ i 
f-irjm = 2 ^ry— -. S {-!)'' ^r-i+l,,hT2r,r-i+h 
i = 0 iV ^' 7, =0 
)•-! I i 
H-2,-+i,(N) = 2 -l^^i 2 (- l)'' /3,-!+l,,h Tor+^,r-i+h 
.(14), 
or 
Ent.(;^j-l 
■©- 
1 
.(15), 
