It is given that A1, A2, ... is an expanding sequence of events, i.e. A1 is contained in A2, A2 is contained in A3, and so on.
Therefore, P(A1 U A2 ) = P( A2 )
P( A2 U A3 ) = P( A3 )
Therefore, P( A1 U A2 U A3 ) = P( A3 )
Then for n number of events, P( A1 U A2 U .... ) = lim n --> inf P( An ), provided the limit exists.
6. Show that if A1, A2, ... is an expanding sequence of events, that is, AC...
Show: Let A1, A2, ... be a infinitely countable collection of events, then P lim P (UA m+00 i=1
3. Let (12, F,P) be a probability space, and A1, A2, ... be an increasing sequence of events; that is, A1 CA2 C.... Using only the Kolmogorov axioms, prove that P is continuous from belour: lim P(An) = P(U=1 An). Hint: Work with a new sequence of events By := A and B := An An-1. n+00 [1]
6) (10 points) The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.60. It is also known that P(A1 or A2) = 1. Suppose P(BA1) = 20 and P(B|A2) =0.05. a. Are A1 and A2 mutually exclusive? Explain. (2 point) b. What is the probability that A1 does not occur? (2 point) C. Compute P(A2 and B) if A1 and B are independent (3 points) d. Compute P(A1 and B) (3 points)
LIDO D EDIUL 9. Let A1, A2, ... be a sequence of events. Show that PA A - A) = P(A) - PA A - UAA-2 UAA) for i = 2, 3,.... Hint: You don't need induction to prove this. You can assume, without proof, that A A-2 UA A-2 UAA = A (A-2 UA-2 UA) and A = AB Ü AB
The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.45. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. (a) Are A1 and A2 mutually exclusive? - Select your answer -YesNoItem 1 Explain your answer. The input in the box below will not be graded, but may be reviewed and considered by...
The prior probabilities for events A1 and A2 are P(A1) = .50 and P(A2) = .50. It is also known that P(A1 A2) = 0. Suppose P(B | A1) = .10 and P(B | A2) = .04. Are events A1 and A2 mutually exclusive? Compute P(A1 B) (to 4 decimals). Compute P(A2 B) (to 4 decimals). Compute P(B) (to 4 decimals). Apply Bayes' theorem to compute P(A1 | B) (to 4 decimals). Also apply Bayes' theorem to compute P(A2 |...
The prior probabilities for events A1 and A2 are P(A1) = 0.45 and P(A2) = 0.50. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. a) Are A1 and A2 mutually exclusive? b) Compute P(A1 ∩ B) and P(A2 ∩ B). c) Compute P(B). d) Apply Bayes’ theorem to compute P(A1 | B) and P(A2 | B).
The prior probabilities for events A1 and A2 are P(A1) = 0.30 and P(A2) = 0.40. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. (a) Are A1 and A2 mutually exclusive? Explain your answer. The input in the box below will not be graded, but may be reviewed and considered by your instructor. (b) Compute P(A1 ∩...
3. (6 pts) A sequence A = [a1, a2, . . . , anjis called a valley sequence if there is an index i with 1 < t < n such that al > a2 > . . . > ai and ai < ai+1 < . . . < an. A valley sequence must contain at least three elements. (a) (2 pts) Given a valley sequence A of length n, describe an algorithm that finds the element a, as...
(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1). (b) Let A and B1 be independent; similarly, let A and B2 be independent. Show that in this case, A and B1∪B2 are independent if and only if A and B1∩B2 are independent. (c) Given P(A) = 0.42,P(B) = 0.25, and P(A∩B) = 0.17, find (i)P(A∪B) ; (ii)P(A∩Bc) ; (iii)P(Ac∩Bc) ; (iv)P(Ac|Bc).