By Harald Cramér
In this vintage of statistical mathematical idea, Harald Cramér joins the 2 significant traces of improvement within the box: whereas British and American statisticians have been constructing the technological know-how of statistical inference, French and Russian probabilitists remodeled the classical calculus of likelihood right into a rigorous and natural mathematical thought. the results of Cramér's paintings is a masterly exposition of the mathematical equipment of recent information that set the normal that others have due to the fact sought to follow.
For someone with a operating wisdom of undergraduate arithmetic the publication is self contained. the 1st half is an advent to the elemental inspiration of a distribution and of integration with appreciate to a distribution. the second one half comprises the final idea of random variables and chance distributions whereas the 3rd is dedicated to the speculation of sampling, statistical estimation, and assessments of significance.
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During this vintage of statistical mathematical conception, Harald Cramér joins the 2 significant traces of improvement within the box: whereas British and American statisticians have been constructing the technological know-how of statistical inference, French and Russian probabilitists remodeled the classical calculus of chance right into a rigorous and natural mathematical thought.
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In cases 2) and 3) the saUle expressions are both infinite. 2) yields a uniquely determined - finite or infinite - value of L (J). 2. It remains to show that condition b) is also satisfied. Let II, 12 , ••• be a sequence of sets in ~, such that ~,l~ = 0 for It ~ '1-', and let be a representation of IlL in the forIll used above. Then I' I' ~ is also a set in S, and no two of the il,v have a common point. If i', in, .. , is an arrangement of the double series ~ t'/tv in a simple se- quence (e. g.
2 that the set of all rational points in R1 is enumerable. 4 it then follows that the set of all points with rational coordinates in a plrme is eDumerable, and further hy induction that the set (>/ all points in Rn with I'lItiollaZ coordinates is enumerable. 3. Borel sets. The class of all intervals in Rn is, like the corresponding class in R I , not an additive class of sets. In order to extend this class so as to form an additive class we proceed in the same way as in the case of intervals in R 1 • 'rhus we consider first the class ~1I of all sets I that are sums of finite 01' enumerable sequences of intervals in Rn.
0) c:::: 8 1 and (0, ' , ,,0, Xk+l, ' . " Xn) c::::: S2 will be called a rectangle set with the sides 81 and 8 2 , - In the case when n = 2, while 8 1 and 8 2 are one-dimensional intervals, this is an ordinary rectangle ill the (:1;1, x~)-plalle. Finally, let Rm aud R" be spaces of m and. n dimensions respectively. Consider the set of all pairs of points (x, y) where x = (Xl' . • Xm} is a point in R m, while y = (YI, , .. n. This set will be called the product space of Rm and B n, It is a space of m + n dimensions, with all points (Xl' , .