Consider a sequence A1,A2,... of events each of which have probability zero. a) Find Pr(Pmn=1 An)...

 

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Consider a sequence A1,A2,... of events each of which have probability zero. a) Find Pr(Pmn=1 An) and find limm!1 Pr{Pmn=1 An). What you have done is to show that the sum of a countably infinite set of numbers each equal to 0 is perfectly well defined as 0. b) For a sequence of possible phases, a1,a2.... between 0 and 2?, and a sequence of singleton events, An = (an), find Pr{Sn An) assuming that the phase is uniformly distributed. c) Now let An be the empty event # for all n. Use (1.1) to show that Pr(#) = 0. Let A1 and A2 be arbitrary events and show that Pr(A1 SA2)+Pr(A1A2) = Pr(A1) + Pr{A2). Explain which parts of the sample space are being double counted on both sides of this equation and which parts are being counted once. Consider a sequence A1,A2,... of events each of which have probability zero. a) Find Pr(Pmn=1 An) and find limm!1 Pr{Pmn=1 An). What you have done is to show that the sum of a countably infinite set of numbers each equal to 0 is perfectly well defined as 0. b) For a sequence of possible phases, a1,a2.... between 0 and 2?, and a sequence of singleton events, An = (an), find Pr{Sn An) assuming that the phase is uniformly distributed. c) Now let An be the empty event # for all n. Use (1.1) to show that Pr(#) = 0. Let A1 and A2 be arbitrary events and show that Pr(A1 SA2)+Pr(A1A2) = Pr(A1) + Pr{A2). Explain which parts of the sample space are being double counted on both sides of this equation and which parts are being counted once.

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