336 On Linearity of Regression 
we get 
We may write 
S {nx^<^n^^' {x, - ^~') { ijx^ - fj)] = Nay- (1 - X x^, 
where Xs is a purely numerical constant, which is obviously equal to unity in the 
case of normal correlation, when o-„^^'- = o-,/ (1 — r") for all values of p and ?; = r. 
Hence finally 
2o-3/2^,^_S^^^^E^„^^^^ = -^{Xi, -p„o-3/ + 2p„o-,/(l -t)x3} (x). 
Problem IV. To find the standard deviation of the values of defined by 
the equation 
We have 
Hence 
St] Br 
7) r 
But 
V 
(7y ' 
Hence 
Bajir Bay 
V 
and 
r 
_ Pn 
a^ay' 
87- 
_ Bpn ^O-x 
r 
Pn O-x 
.(xi). 
.(xii). 
.(xiii). 
SOy 
Squaring equations (xi), (xii), and (xiii), and forming the product of equations 
(xii) and (xiii), then on summing for all random samples and dividing by the 
number of such samples we get in succession 
S^° = ^ + ^- " (xiv), 
7]' r- Tjr 
— f = -\ V (xv), 
Pn <^x o-y Pn<^x 
^\,^<'y^P,,<'y I ^^'x'^'^y^Vy ^^^-^ 
Pn ft/ 
'^'^ i^iiO'il/ 0"a;O-jlf 0"yO"3/ 
_ ^^Ai/P,, ^ ^<'x^''y^''x''y ^ ^"y ■ • 
a ay a y 
