4. Prove for all ne Zt and all sets A1, A2,..., An, B that (Ax – B) = that ---- () --
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.
L: R3 to R3 defined by L([a1 a2 a3]) = [a1 a2^2+a3^2 a3^2]. Prove that this is a linear transformation or not.
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
Question 2 7 pts Theorem If A1, A2, .., A, are sets for n > 2, then (A, UA, U... A.) = (A) n(A)n... n(A) Upload Choose a File Question 3 6 pts o el DLL
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...
Let {a1,a2,...,ak} be a set of activities, where, a1 has the largest (latest) start time. Prove there is an optimal solution to the activity selection problem that contains a1.
Let A1, A2, ...An Prove : P(Un k=1 Ak) = P9A1) + P(A1c
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Problem 4.Let A1, A2, . . . , An be events. Prove
(a) Prove that if A1, A2, . . . , are mutually exclusive, then P(An) → 0 as n → oo. (Recall that whenever Σοοι pn is finite and all the pn's are nonnegative, then Pn-+0 as n o.) (b) Suppose 1 flip a fair coin forever. Let An be the event that the rnth flip is a head. Since the coin is fair, P(An)-.Notice that P(An) 0 asnoo. How, then, can the previous problem still be true? Each An...
Use mathematical induction to show that P(A1∩A2∩...∩An) = P(A1)P(A2|A1)P(A3|A1∩A2)....P(An|A1∩A2∩...∩ An-1) You can assume that you know P(A|B) = P(A|B)P(B)