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This implies that A n C e 0) for all A e CD, C 6 C. Now let 8 be the class of sets E in D such that A n £ G 0), all ^ e D. Similarly 8 satisfies (i), (ii), (iii), so 8 = D. This yields 2) closed under n, but by (ii), (iii), D is also closed under complementation, proving the assertion. Let S be the class of sets G in & such that Q(G) = Q'(G). Then S satisfies (i), (ii), (iii) => 0) c Q or 5"0(C) c S. 22. Let A P' be defined on $„ by P(X e B), P'(X' E B), respectively. Let G c &m be the class of all sets of the form C = {x; jCj < ylt.

By definition. 7. , and $ is a a-field in R, the class of sets (X 6 B }, B E $, is a a-field. If & is a a-field in Q,, then the class of subsets B in R such that {X E B} E $ is a a-field. Proof. 6. 8. If there are a-fields & and $, in Q, R respectively, X: Q —»• R is called a random vector if{X E B} E 5, for all B E $. ) We will sometimes refer to (R, $) as the range space of X. But the range of X is the direct image under X of Q, that is, the union of all points X(co), co e Q.. Denote by J"(X) the cr-field of all sets of the form {X G B}, B E $.

An n-dimensional cylinder set in Pv (00) is any set of the form Problems 2. Show that the class of all finite-dimensional cylinder sets is a field, but not a (r-field. 3. Let F be a countable set, F = {/-,-}. Denote by f' 00 ' the set of all infinite sequences with coordinates in F. Show that X: Q —> F is a random variable with respect to (£1, J") iff {o>; X(o>) = r,} e 3r, ally. 4 EXTENSION IN SEQUENCE SPACE 23 4. Given two processes X, X' such that both take values in F(00), show that they have the same distribution iff for every ^-sequence 4.

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A closer look at the distribution of number needed to treat (NNT) a Bayesian approach (2003)(en)(6s) by Thabane L.


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