Die hygienischen Verhältnisse der Insel Formosa

An Introduction to the theory of statistics

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Monograph

Identifikator:: 1028803753

URN:: urn:nbn:de:zbw-retromon-47137

Document type:: Monograph

Title:: Die hygienischen Verhältnisse der Insel Formosa

Place of publication:: Dresden

Publisher:: [Verlag nicht ermittelbar]

Year of publication:: 1911

Scope:: 1 Online-Ressource (IV, 232 Seiten)

Digitisation:: 2018

Collection:: Economics Books

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Chapter

Document type:: Monograph

Structure type:: Chapter

Title:: 12. Wasserversorgung

Collection:: Economics Books

XVL.—NORMAL CORRELATION, 221 isotropic. The student should form one or two other condensations of the original table to 3- x 3- or 4- x 4-fold form : he will probably find them either isotropic, or diverging so slightly from isotropy that an alteration of the frequencies, well within the margin of possible fluctuations of sampling, will render the distribution isotropic. 14. Before concluding this chapter we may note briefly some of the principal properties of the normal distribution of frequency for any number of variables, referring the student for proofs to the original memoirs. Denoting the frequency of the combination of deviations x, @, «,, . . . , x, by Yi2 . ... n We must have in the notation of Chapter XII. if the uncorrelated deviations zy, Typ» ¥3.19: ete. be completely independent (cf. § 3 of the present chapter), Viz... n=Yis. .. ne CHE 2 ta) (12) where z. x = “a1 Pitiy nS Tai... ey Ht... @) AT AL ore (13) gad ¥12 eee? (27)"*6,04103 19 w ele Cet FRYE ae inl) (14) The expression (13) for the exponent ¢ may be reduced to a general form corresponding to that given for two variables, viz.— Be vn ua... .n Hk rE (n—1) (15) : Lp —1%y, 2Ti2s...m Cis... A028. ns Lr nti... 2 Tn-1)1... n-9nOnl... (a=1 Several important results may be deduced directly from the form (13) for the exponent. Clearly this might have been written in a great variety of ways, commencing with any deviation of the first order, allotting any primary subscript to the second deviation (except the subscript of the first), and so on, just-as in § 3 we arrived at precisely the same final form for the exponent whether we started with the two deviations xy and x, or with x, and xy oe Our assumption, then, that the deviations xy, Ty, ¥g49 etc. are normally distributed amounts to the assumption that all devia- tions of any order and with any suffixes are normally distributed, i.e. in the general normal distribution Jor n variables every array of every order is a normal distribution. It will also follow, gen- eralising the deduction of § 6, that any linear function of x, X, + + + . %, is normally distributed. Further, if in (13) any fixed af tC

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