正好有人問，我就順便把9,10的答案寫出來，看誰需要...... 9.(a)Prove that ∩[n=1~∞](1/2-1/2n,1/2+1/2n)={1/2}. (b)Using part(a), show that the probability of selecting 1/2 in a random se- lection of a point from (0,1) is 0. a) {1/2} is contained in (1/2-1/2n,1/2+1/2n) for all n => {1/2} is contained in ∩[n=1~∞](1/2-1/2n,1/2+1/2n) For x!=1/2, there exists N s.t. |x-1/2|>1/N (Otherwise, |x-1/2|=0) So x!ε<1/2-1/2N,1/2+1/2N), i.e. {1/2}=∩[n=1~∞](1/2-1/2n,1/2+1/2n) b> P(∩{i=1,2...n}Ai) <= P(An) if Ai is contained in Aj whenever i>=j (Evidently form the fact: P(A)=P(B)-P(B-A) for A is contained in B ,P(B-A)>=0 => P(A)<=P(B)) So 0<=P({1/2})=P(xε∩[n=1~∞](1/2-1/2n,1/2+1/2n)<=1/n for all n => P({1/2})=0 10.A point is selected at random from the interval(0,1). What is the probabil- ity that it is rational? What is the probability that it is irrational? By the similar manner as which we use in 9, P({a})=0 for all a belongs to (0,1) All rational numbers is countable since |Z|<=|Q|<=|ZXZ|=|Z| So P(x is rational)=P(∪{a is rational)a)=Σ{a is rational}P({a})=0 P(x is irrational)=1-P(x is rational)=1 (Every real number is either rational or irrational.) Q.E.D. 實際上假設P((a,b))=b-a if (a,b) is contained in (0,1)不是必然為真，因 為機率空間是由樣本空間、可測集和機率測度三者組成，所以甚至P((a,b))不必 存在，當然在實數上這樣的測度是自然的。 P.S.我是不是太晚發文了？XD -- "There are three kinds of lies: lies, damned lies, and statistics." －－Mark Twain --

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