An Introduction to the theory of statistics

Bibliographic data

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

Usage license:: Get license information via the feedback formular.

Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: Part III. Theory of sampling

Collection:: Economics Books

XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 293 independence, of the Ng failures of the first event (Ng)g will be associated (on an average) with failures of the second event, and (&g)p with successes of the second event (cf. row 2 of the scheme on p. 292). Similarly of the Ap successful first events, (¥p)g will be associated (on an average) with failures of the second event and (Np)p with successes. In trials of two events we would therefore expect approximately Ng? cases of no success, 2Npg cases of one success and one failure, and Np? cases of two successes, as in row 3 of the scheme. The results of a third event may be combined with those of the first two in precisely the same way. Of the Ng® cases in which both the first two events failed, (¥q2)q will be associated (on an average) with failure of the third also, (N¢®)p with success of the third. Of the 2/Npq cases of one success and one failure, (28pg)g will be associated with failure of the third event and (28pg)p with success, and similarly for the Np? cases in which both the first two events succeeded. The result is that in A trials of three events we should expect Ng? cases of no success, 3 Npg? cases of one success, 3 Np? cases of two successes, and Np? cases of three successes, as in row 5 of the scheme. The scheme is continued for the results of a fourth event, and it is evident that all the results are included under a very simple rule: the frequencies of 0, 1, 2 . . . . successes are given for one event by the binomial expansion of N(g +p) for two events » ” Ng +p)? for three events i N(g+p)? for four events . fs N(g+p)t and soon. Quite generally, in fact :—the Jrequenciesof0,1,2 . . .. successes in IN trials of n events are given by the successive terms wn the binomial expansion of N(q +p)", viz.— n(n —1 n(n—1)(n-2 vy "+ n.g" p+ Lon ) pry ( RS Jolson l This is the first theoretical expression that we have obtained for the form of a frequency-distribution, 3. The general form of the distributions given by such binomial series will have been evident from the experimental examples given in Chapter XIII, i.e. they are distributions of greater or less asymmetry, tailing off in either direction from the mode. The distribution is, however, of so much importance that it is worth while considering the form in greater detail. This form evidently depends (1) on the values of ¢ and p, (2) on the value of the exponent n. If p and ¢ are equal, evidently the distribution must be symmetrical, for

Access restriction

Copyright

fullscreen: An Introduction to the theory of statistics

Monograph

Chapter

Table of contents

Download

Monograph

Chapter

This page

Image fragment

Citation links

Citation recommendation