80 Choice in the Distribution of Observations 
(105) may be brought into the form 
\ IX, . {ix^ix^ - (a -V + - {ni - p-^^^f I 
^2 ^ ■ 1 + IhL f 
where the denominator and /X2f(.4 — jj.'^ are always positive. Hence the condition 
for a':, , takin"' its minimum vahie ,^ , is 
jjii ~\- a/x2 = 0 and fxi - - /J-ifMs = 0 
or /i3_Ai2__l (jQ8)_ 
We shall examine the possible distributions consisting of three groups of 
observations with the frequencies y^, and y-^ at i\, and v^. The conditions 
(108) require 
yi^'i i- y2'^2 + 7 3 ^ yi^'i + 72^2 + y^vj ^ y^ v:^ {v^ - Vi) + 73^3 (^3 - ^1) ^ _ 1 
7l% + 72^2 + 73 '^3 7l^? + 72l'2 + 73 ''^3 72^'2 (^^2 - "l) + 73^3 («3 " ^l) » 
7i^i(l + « ^']) _ 72 '^'2(1 + a^^2) ^ 73% ( 1 + aVs) .^^^^ 
Vz-Vs ^3 - «i ^1 - -^^2 
Now — ^- , — and can never all have the same sign and (1 + av) 
V2 - V3 «3 - Vi V^-V^ 
is for any v = — I positive, from which it follows that (109) leads to negative 
frequencies. Nor can (109) be satisfied by two groups of observations as 72 = 0 
requires = v.^ = 0, that is one group of observations at a; = 0 which of course 
gives a-„ = ^. 
(9) We may write (106) 
N V/X2 ■ (fi,^ - f4) (/X2 - /Al) - (fJ-Z - l^ll^tf / ' 
where the last ratio is seen to be positive unless 
+ ajj.2 = 0 and — 1x^1^9, = 0 (HO). 
If therefore any distribution of observations can give ~ its minimum value 1 
^2 
and at the same time fulfil those conditions it will make a'^_ a minimum and equal 
Q.2 
to ^ . But = 1 together with (110) lead to 
= = - a> 
which require ---y — N observations at — 1 and — N at 1, whereas the actual 
distribution must consist of ^— iV observations at — 1 and "^^-^ N at 1. 
Thus the. only distribution which makes a;^ a minimum and equal to is that 
consisting of ^ " iV observations at — 1 and "^^--iV at 1. 
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