as we know that P(X|Y) =P(X n Y)/P(Y)
also for disjoint events P(A1 u A2)=P(A1)+P(A2)
therefore P(A1 u A2|B)=P((A1 u A2) n B)/P(B)=(P(A1)+P(A2)) n P(B))/P(B)
=(P(A1 n B)/P(B)+(P(A2 n B)/P(B)
P(A1 u A2|B)=P(A1|B)+P(A2|B)
(hence proved)
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.
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,'' 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.
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.
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.
Need help with this ASAP
10 Q3. (a) Prove the Bonferroni Inequality on three events Ai, A2 and A: P(AinAnAS) 21- P(A) - P(A2)- P(As) (b) Using the results in Q3.(a), and clearly describing the events Ai, A2 and A3, construct a 100(1-a)% joint confidence intervals for estima- tion of three parameters, denoted by 0,02 and 03, say.
10 Q3. (a) Prove the Bonferroni Inequality on three events Ai, A2 and A: P(AinAnAS) 21- P(A) - P(A2)- P(As) (b) Using...
Problem 4. Let Ai, A2,..., An be events. Prove .+P(Ann
3. [5 pts] Suppose A and A2 are disjoint. Prove that
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai?
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
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?