a). (Borel-Cantelli Lemma) Suppose, A, A,... is a sequence of events i) If P(An)...

  

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a). (Borel-Cantelli Lemma) Suppose, A₁, A₂,... is a sequence of events i) If ΣP(An) <∞o then, P(An, i.o) = P(Lim Sup An) = 0 ii) If ΣP(An) = ∞o and A₁, A₂,... are independent then P(A,, i.o) = P(Lim Sup An) = 1. Prove. (10 marks) b). If F(X) = P[X ≤ x] is continuous in x. Show that Y = F(X) is measurable and that Y has a Uniform distribution P[Y ≤y] = y,0 ≤ y ≤ 1 a). (Borel-Cantelli Lemma) Suppose, A₁, A₂,... is a sequence of events i) If ΣP(An) <∞o then, P(An, i.o) = P(Lim Sup An) = 0 ii) If ΣP(An) = ∞o and A₁, A₂,... are independent then P(A,, i.o) = P(Lim Sup An) = 1. Prove. (10 marks) b). If F(X) = P[X ≤ x] is continuous in x. Show that Y = F(X) is measurable and that Y has a Uniform distribution P[Y ≤y] = y,0 ≤ y ≤ 1

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a Proof of BorelCantelli Lemma i Suppose PA Then we know that the sum of the probabilities of the ev... View the full answer