Download e-book for iPad: A Course in Probability Theory (3rd Edition) by Kai Lai Chung
By Kai Lai Chung
Because the ebook of the 1st variation of this vintage textbook over thirty years in the past, tens of millions of scholars have used A direction in chance Theory. New during this variation is an creation to degree idea that expands the marketplace, as this remedy is extra in step with present classes.
While there are numerous books on chance, Chung's ebook is taken into account a vintage, unique paintings in chance thought because of its elite point of sophistication.
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Additional info for A Course in Probability Theory (3rd Edition)
It is clear that Fac is increasing and Fac Ä F. From (4) it follows that if x < x0 0 Fs x 0 Fs x D F x 0 x Fx f t dt ½ 0. x Hence Fs is also increasing and Fs Ä F. 3. 2. f. f. Such a decomposition is unique. EXERCISES 1. f. F is singular if and only if F D Fs ; it is absolutely continuous if and only if F Á Fac . 2. 2. 3. f. (see Exercise 6 of Sec. 2) is of measure zero, then F is singular. The converse is false. 3 ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS 13 4. f. and (3) holds with a continuous f.
It is sufﬁcient to consider the case of two dimensions, since there is no essential difference in higher dimensions apart from complication in notation. 38 RANDOM VARIABLE. EXPECTATION. INDEPENDENCE We recall ﬁrst that in the 2-dimensional Euclidean space R2 , or the plane, the Euclidean Borel ﬁeld B2 is generated by rectangles of the form f x, y : a < x Ä b, c < y Ä dg. A fortiori, it is also generated by product sets of the form B1 ð B2 D f x, y : x 2 B1 , y 2 B2 g, where B1 and B2 belong to B1 .
M. satisfying (4) or any of the relations in (5). m. f. of . Instead of R1 , B1 we may consider its restriction to a ﬁxed interval [a, b]. Without loss of generality we may suppose this to be U D [0, 1] so that we are in the situation of Example 2. We can either proceed analogously or reduce it to the case just discussed, as follows. f. such that F D 0 for x Ä 0 and F D 1 for x ½ 1. The probability measure of F will then have support in [0, 1], since 1, 0 D 0 D 1, 1 as a consequence of (4). Thus the trace of R1 , B1 , on U may be denoted simply by U , B, , where B is the trace of B1 on U .
A Course in Probability Theory (3rd Edition) by Kai Lai Chung