162 Expectation of Moments of Frequencji Distributions 
(■i) When r = 3, in the case of a Gaussian distribution of the values of X, 
Noting that ''.j'|y) = *X if the distribution of the values of X be Gaussian, 
we find : 
= Nfi./ 1.3.5.9 [5iV + 72], 
^l!^^^^ ^ 3 . 5 . 1) . 15 [25iV- + lONO.V + 15912]. 
When 7' = 4, for the case of a Gaussian distribution of the values of X, we 
have : 
.(1) 
l,(A"-l)f' 
«f =iV;u,^3[3i\^ + 32], 
^M.v) = 27 [.V^ + 32iV + 352]. 
II 
Replacing /• by in, and by E { /j! ,■ — fu i.)'"- in foi'niulae (22) and (23) of 
Chapter I, and replacing /x/, by S (—1/' C'/,^" /x/' /u and putting 
:£ (- i)^r'„v/>:/-/;i. = A',, 
find : 
k = 0 
E(ix', - fi,)-'" = 1 . 3 . 5 . . . (2/u - 1 ) 1^^^^,, X.J" 
1 
III — 1 
~T2~ 
(-^'1 Xj"-' X, + ^i,; A,— "X, 
3/M — 1 
. ^ //J-i A'.;"-^' AV + ^ m^-^ X, 
+ 
