328 
MR. R. A. FISHER OH THE MATHEMATICAL 
may write down the chance for a given value of the parameter 9, that 9, 
the range d0 ] in the form 
<h 
<T \/ 
(A,-»)- 
2ff2 <u,. 
should lie in 
The mean value of 0 ! will be the true value 6, and the standard deviation is <x, the 
sample being assumed sufficiently large for us to disregard the dependence of <r upon 0. 
The likelihood of any value. 9, is proportional to 
this quantity having its maximum value, unity, when 
for 
0 = 0 ,; 
$»'*** = 
9,-9 
Differentiating now a second time 
ao 
- log d> 
Now <T> stands for the total frequency of all samples for which the chosen statistic 
has the value 0 l3 consequently <I> = S' (<•/,), the summation being taken over all such 
examples, where </> stands for the probability of occurrence of a certain specified sample. 
For which we know that 
log 0 = C + S (log/), 
the summation being taken over the individual members of the sample. 
If now we expand log/ in the form 
or 
we have 
log 
f(9) = iog/(e 1 )+0-e 1 ^iog/(0 1 ) + 
c9 
9-9, 
^2 
c 
39 
'^°gf(9i) + ... 
log/ = lo gfi + a 9-9,+ ^9-9, +... , 
log 0 = C + 0 —0iS {a)+^9— 0 k S (b) + ... ; 
now for optimum statistics 
S (a) = 0 , 
and for sufficiently large samples S ( b ) differs from nb only by a quantity of order \/n cr b ; 
moreover, 9—9, being of order n~ h , the only terms in log 0 which are not reduced 
without limit, as n is increased, are 
log 0 = C+^n l>9—9, ; 
