Intro to the Math. and Stat. Foundations of Econometrics by H. Bierens

By H. Bierens

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21. (Application of Bayes’ rule): Suppose that 1 out of 10,000 people suffer from a certain disease, say HIV+. 9, and the same if you do have the disease. If a randomly selected person is subjected to this test, and the test indicates that this person has the disease, what is the probability that this person actually has this disease? In other words, if you were this person, would you be scared or not? 22. Let A and B in ö be pairwise independent. Prove that A˜ and B are independent (and therefore A and B˜ are independent and A˜ and B˜ are independent).

Without reference to such a given collection Œ the smallest σ-algebra of subsets of Ω is {Ω , i} , which is called the trivial σ-algebra. 4 we can define the smallest algebra of subsets of Ω containing a given collection Œ of subsets of Ω, which we will denote by α(Œ) . For example, let Ω = (0,1], and let Œ be the collection of all intervals of the type (a,b] with 0 # a < b # 1 . Then α(Œ) consists of the sets in Œ together with the empty set i, and all finite unions of disjoint sets in Œ . To see this, check first that this collection α(Œ) is an algebra, as follows.

A collection ö of subsets of a set Ω is called a monotone class if the following two conditions hold: An 0 ö , An d An%1, n ' 1,2,3,..... imply ^n'1An 0 ö , 4 An 0 ö , An e An%1, n ' 1,2,3,..... imply _n'1An 0 ö . 4 Show that an algebra is a F-algebra if and only if it is a monotone class. 12. A collection öλ of subsets of a set Ω is called a λ& system if A 0 öλ implies A˜ 0 öλ , and for disjoint sets Aj 0 öλ , ^j'1Aj 0 öλ . A collection öπ of subsets of a set Ω is called a 4 π& system if A,B 0 öπ implies that A_B 0 öπ .

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