ESE520Homework#1ESE 520 Homework#1Hong LeiProblem 1 (17)Solution: as we know that321,, CCC,..are countable sets. We can assume that   nnknnnnnkkaaaaaCaaaaaCaaaaaC.......,,,.......................................................,,,.......,,,43212242322212114131211121And then we can easily know that the union of321,, CCC....can be shown by the form that nnknnnnnkkiiaaaaaaaaaaaaaaaCB .......,,,,.......,,,,.......,,,:432124232221114131211121So we can conclude that the B is a countable set .Problem 2 (30)Solution:21, PP(1) As are two probability measures,   0,021PP,   0,021 APAPand   1,121 PPThen P(ϕ)=λx0+(1-λ)x0=0.(2)        0)1(,0,0,01,02121APAPAPSoAPAP(3) if An are events that are mutually exclusive. )()()1()()()1()()1(1121121112111nnnnnnnnnnnnnnnAPAPAPAPAPAPAPAP(4)1)()1()()(21 PPPProblem 3 (31)Solution:(1) It is obvious that0)(,0)( A.(2) Without loss of generality, we can assume thatNiAwi ,0,So         )(,0,1.....1103211nnnnnnnAotherwiseAwAAAAAThen we can conclude that μ satisfies the axioms of a probability measure.Problem 4 (37)Solution:For anyone sequence of events)(lim)(11NnnNnnFPFPAnd then use Inclusion-exclusion axiom)...()1()()()()()()()()()()()()()(..1211111niimiknkniimAAAAPAPCBAPCBPCAPBAPCPBPAPCBAPBAPBPAPBAPSo NnnNnnFPFP11)(In a conclusion.   1111lim)(lim)(nnNnnNNnnNnnFPFPFPFPProblem 5 (38)Solution:(1)As we know that01knknBP(2)Then0)(lim)(1NkkNnkkknkknknBPBPBPBPIn a conclusion:01knknBP.Problem 6 (46)Solution:(1):NO;assume r,w are both σ-algebrar, andw,then it is obvious thatwr (2)Assume thatwrx , thenwrorxx ,and A,B are both σ-fields.Sowrxwrorxxccc(3)Without loss of generality, we can assume that setsrywywxrx,,IfAorByx , thenByyxAxyxcc)()(That is to say :BxAyit is a contradictory. In a conclusion, the answer to the problemis NO.Problem 7 (47)Solution:(a): yes. The elements in set E can be shown that n2...8,6,4,2,and the subset of Ecan be shown that 12...5,3,1 n,so E and subset of E are finite.(b)as we know that321,, AAA,..are finite unions . We can assume that   nnknnnnnkkaaaaaAaaaaaAaaaaaA.......,,,.......................................................,,,.......,,,43212242322212114131211121And then we can easily know that the union of321,, AAA....can be shown by the form that nnknnnnnkkniiaaaaaaaaaaaaaaaA .......,,,,.......,,,,.......,,,432124232221114131211121is also in the that set.(c) yes, it is a σ-field.Problem 8 (additional)Solution:(1): H=head, T=tailWe can find that:Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}F=25628events ( such as (HHH) ( ) ( HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) )P(H,H,H)=1/8P(H,H,T)=18P(H,T,H)=1/8P(H,T,T)=1/8P(T,H,H)=1/8P(T,H,T)=1/8P(H,T,T)=1/8P(T,T,T)=1/8(2):let: O=orange, B=blueΩ = {OB, BB, BO, OO}F=1624eventsP(B,B)=1/6P(O,O)=1/6One O and one B:P=2/3(3):Ω =