204 Expectation of Moments of Frequeney Distributions 
After some simple transformations we find : 
1 ^ 
+ ^2 1 ~ (^'1 + ^'■i+'-2 + 2 
+ 2 /^J-a-s + 2 /^J-i-a /^r2+2 
(33). 
+ r.j + r„) /i^j+i (U^^_i + (n + r„) ^^241 - f^n+i M'r^-s f^2 
- i )\^-'^ i^r,-z f^r,+i f^. + in^ r, + 2ri r,) fX,-, f^r, 
- (7-1 - 1 ) r, fJ'r,-! l^i + ''^t'^ A^n-i /^'-a-s M^" + \ ''a Mn-3 -^2' 
+ I ri (n - 1 ) '-2 {r. -\)Hr^ -2 /ir2-2 ^24 + • ■ ■ 
Noting that 
- E {v'r^ - flr^) (v'r^ - fir,) - [riT, fir, f^r, " ^^T, (n - 1) f^r, f^r,-2 
- ^7\ (ri - 1 ) r„_ /Xr,-, f^r, fi^ + i r, (n - 1 ) (r, - 1 ) m,.,_o fir,... ^li^-..., 
we find hence : : 
- i^n P'r, + »'i '"2 /*i-2-i 
+ -p j - ('-1 + /-o) /X^j+r^ + 2 Mri+r2-2 
+ (^-2 - 1) Air2-2 + 2n (n - 1) /^ri-2 /^r^+a 
+ ro (j-i + n) /LAr,_i + n + n) fir^-^ fir,+i - ir^t-^J ^^^^^ 
- Irjt-'^J /U,,j_3 flr,+i fl.. + (rj + r., + 1\7\) fir, fir, 
- {1\ + 1) r., (/-a - 1) /x^j fi, - i\ (?-! - 1) (r., + 1) /i,^_o fir, fi. 
- h\r. + r^) - ^nr., {r., - 1) fir^-i fir,-^ Ms 
- Iri (7-1 - 1) ?\,//.r,-2 Mra-i Ms + i^'i^'s^"'^ Mrj-i Mr^-s M-i' + i^'i^"'' ?-2 flr,-z tJ'r,-y f^i 
+ in (n - 1) n (n - 1) Mrx-2 M»-2-2 M2'| + • • • 
(34). 
