154 Expectation of Moments of Frequency Distributions 
We find in this way : 
T„,r = 1.3.5... (2r-l)y^/ 
= 1.3.5... (2r - 1) {ij-t-^i /ji/-"- + ^^[-^1 ix.^} 
T.,r.r-2 =1.3.5... {2r - 1) < fj,/'' fx, + ^^rt-^i /../-^ 
=1.3.5...(2r + l)ij>,'->, >! 
= 1 . 3 . o . . . (2r + 1 ) { /x/-^ + ^^,4-3] ^./->3/., 
= 1 . 3 . 5 . . . (2r + 1) ^./-s + -i^rt-l /.^/^c 
+ ik^^"'' /^4/^.-, + ik'"'""^ f^f fJ--. + ih '''"'^ f^-^"" f^-^^^' 
+ J^-rt-^J /x/-« /x/ + T^TT, r[--l fi.J'} 
Hence : 
..(20), 
.(21). 
= 1.3.5...(2r-l)|^^-./z,/ 
+ 
.(22), 
54 
(r - 1) (r - 2) 
18 
= 1 . 3 . 5 ... (2r + 1) /x,'"' 
1 
+ 
+ 
n.!-- fi, + tV'"-'^ m/"' /^3M4 
+ gVrt-^i ^.,'-4 _ ,.[-2] ^^r-i 
r-1 
60 
(r-l)(r- 2) 
36 
r[-=] ^,A^4 + T^Vo Ma''"' f^^ 
.(23). 
On the other hand, 
•(24), 
.(25). 
