636 BELL SYSTEM TECHNICAL JOURNAL 



where K, c and a are constants. This, by means of the change of 



variable 



y = h{a + Y.{xi — w)2] 



and throwing the definite integral 



f 



'0 



in with the constant A, reduces (6) to 



P(m)dm = A'Wi{m)[_a + ZC^i - w)2]-(i/2)(.+2+c)^^_ (g) 



We are still confronted with the a priori existence probability 

 function Wi(m). 



A plausible form, suggested by the well known "Student" ■* distri- 

 bution of the ratio (x — fn)/s for a set of observations to be made 

 from a normal universe of known mean and standard deviation, is 



Wi{m) = yli[l + B{M - m)2]-(i/2)iv^ (9) 



where M is the value of m which is a priori most probable, N and B 

 are positive constants while the equation 





Wi(m)dm = 1 



1^ — 00 



gives 



-i/2r[i(7v - 1)] 



With this assumed form and noting that 



^(xi — my = ns"^ -\- n{x — my 

 equation (8) gives 

 P{m)dm = A"l\ + B{M - m)2]-(i/2)iv 



X 



1 + 



a + ns^ 



-(l/2)(n+2+c) 



dm, (10) 



the integral of P{m)dm between plus and minus infinity determining 

 A". 



Recapitulating: formula (10) gives us the a posteriori frequency 

 distribution for m in terms of the observed data and the arbitrary 

 constants a, c, N, B, M which have entered into the problem in 



^ The writers are aware of the fact that the "Student" frequency function has 

 been put forwaid in more than one place as the solution for an a posteriori problem. 

 But it should be noted that the various deductions of this function which have been 

 given by "Student" and others are entirely a priori. 



