By Kai Lai Chung

Because the e-book of the 1st variation of this vintage textbook over thirty years in the past, tens of hundreds of thousands of scholars have used A direction in chance idea. New during this version is an advent to degree idea that expands the industry, as this remedy is extra in line with present classes. whereas there are a number of books on chance, Chung's ebook is taken into account a vintage, unique paintings in likelihood thought as a result of its elite point of class.

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Lim sup An, lim inf An in terms of those of A I, A 2 , or An. [For the definitions of the limits see Sec. ] *11. F. with respect to which X is measurable. Show that A E ~ {X} if and only if A = X I (B) for some B E qjl. Is this B unique? Can there be a set A rt. /23 1 such that A - X-I (A)? 12. 's. CJ)? Tbe reader is supposed to have some acquaintance with this, at least in the particular case (11, 9'3, m) or (Ml, :f51 , m). ] The general theory is not much different and will be briefly reviewed.

Prove that If 0 :s r < rl and &'(IXI") < 00, then cf(IXl r ) < 00. Also that {(IXl r ) < 00 if and only if cS,(IX - air) < 00 for every a. for a :s x :s I. *12. If X> 0 and Y> 0, p > 0, then rS'{(X + y)P} < 2P{t(X P) + e(yP)}. If p > 1, the factor 2P may be replaced by 2 P - 1 . If 0 :::: p :::: 1, it may be replaced by I. * 13. If Xj > 0, then n > or L cff(Xj) j=l according as p :::: 1 or p 2: 1. *14. If p > 1, we have 1 P n -n "'X· } 0 j=l and so 1 n P :s -n "'IX·I 0 } j=l 52 I RANDOM VARIABLE.

13. If Xj > 0, then n > or L cff(Xj) j=l according as p :::: 1 or p 2: 1. *14. If p > 1, we have 1 P n -n "'X· } 0 j=l and so 1 n P :s -n "'IX·I 0 } j=l 52 I RANDOM VARIABLE. EXPECTATION. INDEPENDENCE we have also Compare the inequalities. 15. If p > 0, e'(IXIP) < 00, then xP~{IXI > x} = 0(1) as x --+ 00. Conversely, if x P3l{IXI > x} = 0(1), then (f'(IXIP-E) < 00 for 0 < E < p. * 16. For any dJ. and any a ~ 0, we have I: [F(x + a) - F(x)] dx = 0, 17. If F is a dJ. such that F(O-) roo [1 F(x)} dx 10 = a.

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