1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...

Question

Question

1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...

1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B).

2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold?

3. Provethat, for A1 andA2 disjoint, P(A1∪A2|B)=P(A1|B)+P(A2|B).

4. A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.

math Statistics-And-Probability

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

PC2) PLB ,(B) PUB) PCc) N6f Afgc are independent

Add a comment

Answer 2

Similar Homework Help Questions

5. In each step. explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,"...

5. In each step. explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion," that P(AUB) P(A) +P(B)-PAnB). (b) Prove the "uni that P(A UA2) S P(Ai)+ P(A2). Under what conditions does the on bound." equality hold? (c) Prove that, for A1 and A2 disjoint, PAUA2lB)=P(A1B)+P(A21B) (d) A and B are independent events with nonzero probability. Prove whether or not A and B are independent.
5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,''...

5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,'' that PAU B) = P(A)-P(B)-Pan B). (b) Prove the "union bound" that P(Ai UA2) P(Ai) P(A2). Under what conditions does the equality hold? (c) Prove that, for A1 and A-disjoint, PAUA2B)= PAB)-P(A-B). d) A and B are independent events with nonzero probability. Prove whether or not A and B are independent.
5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,"...

5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion," that P(AUB) P(A) +P(B)-P(AnB). (b) Prove the "union bound," that P(AiUAz) < P(A) + P(A2). Under what conditions does the equality hold? (c) Prove that, for Ai and A2 disjoint, P(Ai UA2|B) P(AiB)P(A2|B) (d) A and B are independent events with nonzero probability. Prove whether or not A and B are independent.

Please, I need clear writing if you choose to write by hand for a,b,c, and d....

Please, I need clear writing if you choose to write by hand for a,b,c, and d. thanks, 5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion," that PAU B)-P(A)+P(B)-Pan B) (b) Prove the "union bound" that P(Ai UA2) P(Ai) P(A2). Under what conditions does the equality hold? (c) Prove that, for A and A2 disjoint, P(A UA2 B) P(A B)P(A2 B) (d) A and B are independent events with nonzero probability. Prove whether...
5. In each step, explain clearly what property or axiom you are using (a) Prove "inclusion-exclusion,"...

5. In each step, explain clearly what property or axiom you are using (a) Prove "inclusion-exclusion," that P(AUB) P(A) P(B) P(AnB). (b) Prove the "union bound," that P(Ai UA2) P(A) +P(A2). Under what conditions does the equality hold?
6) (10 points) The prior probabilities for events A1 and A2 are P(A1) = 0.40 and...

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)

2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)

2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)
Let A1, A2, ...An Prove : P(Un k=1 Ak) = P9A1) + P(A1c ...... Problem 4.Let...

Let A1, A2, ...An Prove : P(Un k=1 Ak) = P9A1) + P(A1c ...... Problem 4.Let A1, A2, . . . , An be events. Prove
10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Usi...

10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Using the results in Q3.(a), and clearly describing the events At, A2 and A3. construct a 100(1-a)% joint confidence intervals for estima- tion of three parameters, denoted by 01,02 and 03, say. 10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Using the results in Q3.(a), and...

3. Let A and B be events. Show that P(ABl(AUB)) P(AB|A). When does equality hold?

3. Let A and B be events. Show that P(ABl(AUB)) P(AB|A). When does equality hold?

1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...

Homework Answers

Add Answer to:
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...

Post as a guest

Earn Coins

5. In each step. explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,"...

5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,''...

5. In each step, explain clearly what property or axiom you are using. (a) Prove "inclusion-exclusion,"...

Please, I need clear writing if you choose to write by hand for a,b,c, and d....

5. In each step, explain clearly what property or axiom you are using (a) Prove "inclusion-exclusion,"...

6) (10 points) The prior probabilities for events A1 and A2 are P(A1) = 0.40 and...

2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)

Let A1, A2, ...An Prove : P(Un k=1 Ak) = P9A1) + P(A1c ...... Problem 4.Let...

10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Usi...

3. Let A and B be events. Show that P(ABl(AUB)) P(AB|A). When does equality hold?

1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...

Homework Answers

Add Answer to: 1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...

Post as a guest

Earn Coins

Add Answer to:
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat,...