5 
2 
B 
— S. — 
160 
and it will be demonstrated that it depends on the value of 
which is an extraction from the estimator rv 
2. DISTRIBUTIONAL RESULTS 
j n Let’s consider again the estimator . It is not too diffi- 
= cult to find how it is distributed. Indeed we have 
The (4) means, among other things, 
n _ V 1 À? 
oi+ci £fc 2 A +oi 
(8) 
because the terms under the sum are in reality normal 
standardised rvs, as it can be easily demonstrated by the 
following equalities 
which implies, passing from the true quantities to the es 
timated ones, that it is possible to give an estimation of 
the variance of the instrument B in the following way 
Ê 
A, 2 
I 
'A. V 
V°A J 
C 2 - Ç 2 _ 
°A 
(6) 
The equations (5) and (6) are always true, independently 
of the value of , and this seems to contradict our 
common-sense, which suggests that the estimation of 
the precision of the instrument B can be effectively 
carried out by comparison with the measurements 
given by A only if the latter is better than the former 
or, more formally, if the following inequality holds 
Now, it is well known that the sum of n independent 
normal standardised rvs equals a chi square rv with n de 
grees of freedom, and this brings us to (8). 
From relation (8) it is immediately deduced that 
c 2 
10 B ~ 
> D 2 
-ln~ 
Va< 
The point is that (6) is an estimation; it is always true, but 
this only means that the mean values of the rvs involved 
coincide, that is 
£[s 2 ]=£[s;- 0 ; (7) 
Moreover the latter equality doesn't guarantee anything 
about the concentration of the rv around its mean 
value. In other words, it is necessary to investigate the 
goodness of the single estimation 
which represents the first significant result of our discus 
sion: the distribution of depends on the value of the 
variance of A. An immediate way to investigate this de 
pendence formally and quantitatively is to consider the 
two most important descriptors of a probability distribu 
tion: mean value and variance. Remembering that for the 
chi square rv the following equalities hold 
VAR[xiy 
whose demonstration can be found in every good statis 
tics textbook, we have