A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter.

Isaac Todhunter

Date:: 1865

Credit: A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter. Source: Wellcome Collection.

605/648 (page 585)

$l’abscisse qui reponcl tl la plus grande ordonnee de la courbe; mais le milieu que nous adoptons, est 6videmment indiqu6 par la tbeorie des probabilites. This extract illustrates a remark which we have already made in Art. 1008, namely that strictly speaking Laplace’s method does not profess to give the most probable result hut one which he con- siders the most advantageous. 1014. Laplace gives an investigation in his pages 335—310 which amounts to solving the following problem: if we take the average of the results furnished by observations as the most pro- bable result, and assume that positive and negative errors are equally likely and that the function of the facility of error is the same at every observation, what function of the facility of error is implicitly assumed ? Let the function of the facility of an error z be denoted by e-Wz2), which involves only the assumption that positive and nega- tive errors are equally likely. Hence the value of y in the pre- ceding Article becomes He-*, where o- — ^\r[x — aj2 + yjr (x — a2)2 + y]r (x — a3)2 -f ... To obtain the most probable result we must determine x so that a shall be a minimum ; this gives the equation {x - a,) (x - aj2 +(x- a2) -f (x - a2f + (*-«„) P (x ~ °s)2 + ••• = 0. Now let us assume that the average result is always the most probable result; suppose that out of s observations i coincide in giving the result av and s — i coincide in giving the result a2; the preceding equation becomes i (cc - ax) yjr' (x - <q)2 + (s - i) (x - a2) -v/r' (x - a2)2 = 0. The average value in this case is iax + (s — i) a2 s Substitute this value of x in the equation, and we obtain v ^ ~ = v {s fa -a*)} •$