An Introduction to the theory of statistics

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Monograph

Identifikator:: 1751730271

URN:: urn:nbn:de:zbw-retromon-127610

Document type:: Monograph

Author:: Yule, George Udny http://d-nb.info/gnd/12910504X

Title:: An Introduction to the theory of statistics

Edition:: 8. ed. rev

Place of publication:: London

Publisher:: Griffin

Year of publication:: 1927

Scope:: XV, 422 S; Ill., Diagr

Digitisation:: 2021

Collection:: Economics Books

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Document type:: Monograph

Structure type:: Chapter

Title:: Part III. Theory of sampling

Collection:: Economics Books

: 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- 318

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