UNKNOWN MEAN OF SAMPLED UNIVERSE 635 



Therefore, by the Laplacian generaHzation of the Bayes formula, 

 the a posteriori probabiHty that 



m < mean < m + dm 



is (cancelHng factors which do not involve m or //) 



dm WOn, h)h('''-^"e-''-^''-"'mi 

 Jo 



P{m)dm = — 



r dm r W(m, /0A^l/2)ng-ft2(x,-m).^;^ 



Adm. r W(m, hW^^-e-''-^''-'^^'dh, 



Jq 



(4) 



where ^ is a constant such that 



[m)dm = 1. 



P(; 



00 



II. Introduction of Restrictions on General Equation 



We are now confronted by a difficulty inherent to a posteriori 

 probability problems. What do we know as to the form of the a 

 priori existence probability function W{m, h) ? If in a specific practical 

 problem the form of W{m, h) is unknown, no conclusions can be 

 drawn from the set of observations unless some assumptions are made 

 and then the weight assignable to the conclusions drawn is a deUcate 

 question depending on the reasonableness of the assumptions.^ 



The analysis and results given below are based on assumptions 

 which the writers believe will be found justifiable in many problems 

 of practical interest. 



A first assumption which suggests itself is that m and h are inde- 

 pendent a priori so that we may write 



W{m,h) = Wi(m)W2ih). (5) 



On this assumption 



P(m)dm = AWi{m)dm f" W^2(/z)A(i/2)ne-'^^('.-«)^ci//. (6) 



Jo 



As a second step toward tentative solutions assume that 



- See Poincare: "Calcul des Probabilites"; 2d edition; articles 178 and 179. 

 Mn this connection see italicized paragraph, page 266, "Probability and Its 

 Engineering Uses," T. C. Fry, 1928. 



