42 BELL SYSTEM TECHNICAL JOURNAL 



dom sample u' of n is less than any stated value r will be denoted by 



p{\ti'\ < r). Then 



p{\u'\ <r) = r Pudu = -^ j exp(^-^,jdu. (4) 



Evidently the number of parameters can be reduced from one (which 

 is Su) to none by taking as chance-variable the ratio u/Su, which may 

 be called the "reduced" chance-variable. Thus, with \u'\ denoted 

 by r' and with r'/Su and r/Su denoted by R' and R respectively, 

 equation (4) becomes 



p{\u'\ < r) = p(R' <R) = erf (i?/V2), (5) 



where erf ( ) is the so-called "error function" defined, for any vari- 

 able z, by the equation 



Vtt, 



erf (s) =^ TexpC- \'')dX (6) 



ITT Jo 



and extensively tabulated '' for real values of z. For some purposes 

 it is more convenient to employ the "error function complement," 

 defined by the equation 



erfc (2) = -^ r exp{- \')d\ 



(7) 



and hence related to erf (s) by the equation 



erf (s) + erfc (2) = 1. (8) 



If lis denotes any fixed value of ti, and if f/3 denotes Us/Su, then 



p{u' > Us) = r Pudu = ^ erfc ^ - (9) 



If Wi and 112 denote any two fixed values of u such that Ui < 112, 

 and if Ui and U2 denote Wi/5„ and U2/S,, respectively, then 



p{ui < u' < U2) = -z i erfc-p — erfc-^ j • (10) 



* To avoid possible confusion, it may be well to remind the reader that there has 

 also been extensively tabulated, for real values of z, the closely related function 



-L r exp i-xy2)dX, 

 V27r-'o 



which is more convenient for some purposes, though less convenient in the present 

 paper. 



