Please choose the right answers whether it is a,b,c, d, or e. The problem is clear and all details are there. (2) Show that two events, A, B, are independent if and only if at least one of the following holds: (i) P(A|B) = P(A) or (ii) P(B|A) = P(B), or (iii) P(A) = 0, or (iv) P(B) = 0.The following proofs are proposed.(a) Although the result is true, to prove it, it requires deeper results from Calculus that are beyond the prerequisite of this course.(b) The statement of the problem is, in fact, false.(c) If P(A n B) = P(A)P(B), then either (iii) holds or (iv) holds or else the conditional probabilities are defined and we haveP(A n B)P(B)P(A) · P(B)P(A n B)P(A) · P(B)P(A|B) =P(A),or P(BIΑ)-= P(B).%DP(B)P(A)P(A)This finishes the proof.(d) Assume at least one of the the four conditions holds. Condition (i) implies thatP(A n B)P(B)= P(A);Р(An B) 3D Р(А)Р(B).orThis is exactly what independence requires. Similarly, condition (ii) impliesP(B N A)= P(B);orP(B n A) = P(A)P(B).P(A)This is exactly what independence requires. If condition (iii) or (iv) holds then automatically0 < P(A n B) < P(A) = P(A) · P(B) = 0.0 < P(A n B) < P(B) = P(B) · P(A) = 0.orThis is exactly what independence requires and the proof is finished.(e) Assume at least one of the the four conditions holds. Condition (i) implies thatP(A n B)Р(А);P(A n B) = P(A)P(B).orP(B)This is exactly what independence requires. Similarly, condition (ii) impliesP(B n A)= P(B);P(B N A) = P(A)P(B).orP(A)This is exactly what independence requires. If condition (iii) or (iv) holds then automatically0 < P(A n B) < P(A) = P(A) · P(B) = 0.0< ΡPAn B)

Name: Please choose the right answers whether it is a,b,c, d, or e. The prob
Uploaded: 2024-03-20T07:07:50.000Z
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Description: Please choose the right answers whether it is a,b,c, d, or e. The problem is clear and all details are there. (2) Show that two events, A, B, are independent if and only if at least one of the following holds: (i) P(A|B) = P(A) or (ii) P(B|A) = P(B),