stated in terms of the linear approximations to X and Y, this relation 

 becomes 



(1) {hSLi + CJI2) ^ + (b' li + C JI2) 2 £ r2 , 



or equivalently 



a2[b2 + (b')2] (£1/0^) 2 + 2aia2(bc + h'c') (Si^/a^) {I^/oq) 



+ a2[c2 + (c-) 2] (£2/02)2 ^ r2 . 



where a^ is the standard deviation of £^ , and 02 ^^ ^^ standard deviation 

 of I2 • '^^^ problem is to evaluate the probability that this relation is 

 satisfied. 



The first step in this evaluation is to change, by linear transformations, 

 from the random variables l-^ , £2 to new random variables t, , t2 . This 

 is done so that t^ and t2 are standardized normal and independent. Also, 

 so that relation (1) takes the foanti 



v,t2 + V t2 < r2 . 

 11 2 2 — 



In this form, existing methods are directly usable for approximately eval- 

 uating the probability that this relation holds. 



Statement of Transformations . 



The relation (1) can be expressed as 

 2 



where the matrix 



ID 



I A (£ /a.)(£ /a.) < 

 i,j=l ^ . ^ ^ 



I I equals 



R2 , 



a2[b2 + (b')2] 

 0^02(^0 + b'c') 



■,^a^O:>c + b'c") 

 a2[c2 + (c')2] 



Let I |A^J M =1 Ia^jI |~1 , and use Xi >. X2 to denote the characteristic 

 roots of 1 |a^D I I . since a12 = a21 , the values of Xi and X2 are the 

 roots of 



(All - ;^)(a22 - X) - (Ai2)2 = 



Specifically, 



Xi = j{All + a22 + [(All - a22)2 + ^{K^^)^]^^^} , 

 X2 = ^{All + a22 - [(All _ a22)2 + 4(Ai2)2]^/^} , 

 are values for Xi and X2 1 where both values should be positive. 



Let the y-^ be any set of four numbers (not all zero) satisfying the 



453 



