: THEORY OF STATISTICS. 
2. Consider first the case in which the two variables are com 
pletely independent. Let the distributions of frequency for the 
two variables x; and x,, singly, be 
4 
Y1="¢ 22 
== 
(1) 
2 | 
Fy 
Yo=Ys¢ kr 
Then, assuming independence, the frequency-distribution of pairs 
of values must, by the rule of independence, be given by 
2 2 
33) ® 
Yio Yee Eo 
whera 
oy 
EX 2.00, i (3) 
Equation (2) gives a normal correlation surface for one special 
case, the correlation-coefficient being zero. If we put xz,=a con- 
stant, we see that every section of the surface by a vertical plane 
parallel to the z; axis, 7.e. the distribution of any array of a;’s, is 
a normal distribution, with the same mean and standard-deviation 
as the total distribution of z’s, and a similar statement holds for 
the array of a,’s; these properties must hold good, of course, as 
the two variables are assumed independent (cf. Chap. V. § 13). 
The contour lines of the surface, that is to say, lines drawn on 
the surface at a constant height, are a series of similar ellipses 
with major and minor axes parallel to the axes of x; and «, and 
proportional to o; and oy, the equations to the contour lines being 
of the general form 
z |, 
Tr a : od) 
Pairs of values of 2, and x, related by an equation of this form 
are, therefore, equally frequent. 
3. To pass from this special case of independence to the general 
case of two correlated variables, remember (Chap. XII. § 8) 
that if 
yg =1y — by. 
yy = y= by; 
x, and x, ,, as also z, and «,, are uncorrelated. If they are not 
merely uncorrelated but completely independent, and if the dis- 
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