Untersuchungen über die Theorie des Preises

An Introduction to the theory of statistics

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Monograph

Identifikator:: 82998786X

URN:: urn:nbn:de:zbw-retromon-41040

Document type:: Monograph

Author:: Auspitz, Rudolf

Title:: Untersuchungen über die Theorie des Preises

Place of publication:: Leipzig

Publisher:: Duncker & Humblot

Year of publication:: 1889

Scope:: 1 Online-Ressource (XXXI, 555 S.)

Collection:: Economics Books

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3. THEORY OF STATISTICS. random sampling. Since, N being given, fixing the contents of any n of the classes determines the = + 1th, there are only = independent variables. Let us now suppose that the distribution of deviations is normal. Then the equation of the frequency “solid ” is of the type set out in equation (15) of p. 331, which we will write for the present in the form ke ~ 5x2 x® =a constant, is then the equation of the “ellipsoid” delimiting the two portions of the ‘“volume” corresponding to combina- tions more or less likely to occur than my, m,, m, . . . . my, Accordingly, to find the chance of a system of deviations as probable as or less probable than that observed, we have to dissect the frequency solid, adding together the elliptic elements from the ellipsoid x? to the ellipsoid «0, and to divide this summation by the total volume, ¢.e. the summation from the ellipsoid 0 to the ellipsoid o. In this book we have been concerned with summations the elements of which were finite. The reader is probably aware that when the element summed is taken indefinitely small the summation is called an integration, the symbol [ replacing 2 or iS, and the infinitesimal element being written dz. In the present case we have to reduce an n-fold integral the summation relating to n elements dz, dx,, etc. To reduce this n-fold integral to a single integral, the following method is adopted. In the first place the ellipsoid, referred to its principal axes, is transformed into a spheroid by stretching or squeezing, and the system of rectangular co-ordinates transformed into polar co-ordinates. The reason for adopting the latter device is that, when twc rectangular elements dz, dy are transformed to polar co-ordinates, we replace them by an angular element df, a vectorial element dr, and a term in 7, the radius vector. When n= such elements are transformed, the integral vectorial factor is raised to the n» — 1th power and there is an infinitesimal vectorial element, dr, and a “solid ” angular element. But as the limits of integration of the angular (not of the vectorial) element will be the same in the numerator and denominator, these cancel out, while x may be treated as the vectorial element or ray. Hence the multiple integral reduces to a single integral and the expression becomes oc ( ei po. dx £ [ gad AT aly 4 79

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