Exercise 9. (Submit a) a) Prove (p4) directly using inclusion-exclusion Hint: With n= pi p?... per...
2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)
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?
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.
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," 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.
Exercise 1.49. An urn has n 3 green balls and 3 red balls. Draw&balls with replacement. Let B denote the event that a red ball is seen at least once. Find P(B) using the following methods. (a) Use inclusion-exclusion with the events Ai = {ith draw is red) Hint. Use the general inclusion-exclusion formula from Fact 1.26 and the binomial theorem from Fact D.2 (b) Decompose the event by considering the events of seeing a red ball exactly k times,...
solve number 6 only Problem 6: (a) Using induction, derive inclusion exclusion formula for 3 events: PCAO BUC) = P(A)+F(B)+P(C) - PAB) - PANC) - P(BNC) + P(ANBAC). (b) Then, for n events: P(U4) - EPA) FAMA) "FAN (c) Using the above solve the following matching problem the deck of numbered cards is allocated only int e nden t and in onemlope). Find the probability that it lost here will match itsetvelopem ber Problem 7: total of 36 memborsota cu...
(3) Let m,n E N. Let p(x), i -1, ..., m, be polynomials with real coefficients in the variables -(x,..., rn). Prove that pi(r) p(a) Un (r)」 is a continuously differentiable map from R" to R". (Suggestion: Use Theorem 9.21.) (3) Let m,n E N. Let p(x), i -1, ..., m, be polynomials with real coefficients in the variables -(x,..., rn). Prove that pi(r) p(a) Un (r)」 is a continuously differentiable map from R" to R". (Suggestion: Use Theorem 9.21.)
Problem set 9 (10 marks). Let K be a KC UFENI The aim of this exercise is to prove that there is n finite union of the open intervals) compact set of R and (I,)rEN be open intervals such that N such that K C I U..U (i.e. K is actually contained in a n E N, select a, K such that 1. Assume that the result does not hold, and explain why we can then, for any n UUIn...
(3) Let m, n є N. Let Pi(x), 1, , m, be polynomials with real coefficients in the variables r = (ri, . . . , r"). Prove that Pr(x) p(x) = | Pm (x) is a continuously differentiable map from R" to R. (Suggestion: Use Theorem 9.21.) (3) Let m, n є N. Let Pi(x), 1, , m, be polynomials with real coefficients in the variables r = (ri, . . . , r"). Prove that Pr(x) p(x) =...