Let (Q, T, P) Be A Probability Space And A1, A2, A3 Be 3 Events (Ai E .F) - Statistics Assignment Help

Assignment Task

I. Let (Q, T, P) be a probability space, and A1, A2, A3 be 3 events (Ai E .F). We let o-(Ai) and o-(Ai, A.j) be the o--algebras generated by the collections {Ai} and {Ai, Ai}, respectively. We use "I" and •c to denote independence and complement. 
(a) (10 points) Show that a(Ai) I o-(Aj) if and only if P(A, fl Ai) = P(A)P(A3). Show that a(Ai) I a(A2, A3) if and only if (i) P(Ai fl A2) = P(Ai)P(A2), (ii) Poi fl A3) = P(A1)P(A3), and (iii) P(Ai fl A2 fl A3) = P(A1)P(A2 fl As). Hint: Show that a(A2, A3) =a(A2 fl A3, A2 fl 4 A fl A3, A fl AD. 
(b) (10 points) Let A E 1. with P(A) > 0. Let P(B1A) P(B fl AV P(A) for every 
B E T. Show that P(. A) is a probability measure on (Q,T). Verify that P(B1A) = 0 if B fl A = 0 and P(B A) = 1 of B D A. Interpret. 
(c) (10 points) Suppose that 0 < P(A3) < 1. Let a(Ai) and a(A2) be conditionally independent given o-(A), denoted a(Ai) I 0-(A2) 1 o-(A3), if and only if a(Ai) and a(A2) are independent in the probability spaces [Q, .F, Pe 03)] and [11,,F, Pe 00]. Show that a(Ai) I a(A2, A3) if and only if a(Ai) I a(A2) a(A3) and a(Ai) ± a (A3). 
II. (20 points) Let (Q, F, P) be a probability space, A1, A2, . . . a sequence of events, i.e., An E .F for each n, and X, X1, X2 . . . real random variables on (S , .F, P). 
(a) (10 points) (i) Show that {cA., : Xn —> X} = n7,11 [4:11 {w : supi> X — X 
Show that Xn c19.> X if and only if for any € > 0, limn,,, P[supi>„ X2 — X 
< 11,1. (ii) 

(b) (10 points) (i) Borel-Cantelli Lemma: If Erii_i_ P[Ai] converges as n Do, show that limsupn,00 An E 7. and P[lim supn, An] = 0 where lim sup, An ac,',°_, U A. (ii) Let on be an estimator of 0,, G IRk , k> 1, based on a sample of size n. Using a-(ii) and b-(i) give a sufficient condition for 0„ to be strongly consistent for 0,:. Justify your answer. 

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