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1、Chapter1ProbabilityConcepts1.1Thegivensetsare:A={1,2,3,4}B={0,1,2,3,4,5,6,7,8}C={x
2、xrealand1≤x<3}D={2,4,7}E={4,7,8,9,10}.Weobservethat:Aisfiniteandcountable.Disfiniteandcountable.Bisfiniteandcountable.Eisfiniteandcountable.Cisinfiniteanduncountable.1.2Byinspection,(a)AIB=A=
3、{1,2,3,4}.(b)AUBUDUE={1,2,3,4,5,6,7,8,9,10}.(c)(BUE)ID=D={2,4,7}.(d)B−E={1,2,3,5,6}.(e)AIBIDIE={4}.1.3TheuniversalsetisU={0,1,2,3,4,5,6,7,8,9,10,11,12}.ThesubsetsareA={0,1,4,6,7,9},B={2,4,6,8,10,12}andC={1,3,5,7,9,11}.Byinspection,(a)AIB={4,6}.(b)(AUB)IC=AIC={1,7,9}.12Signa
4、lDetectionandEstimation(c)BUC={0}.(d)B−A={0,1,7,9}.(e)(AUB)I(AUC)=AU(BIC)=A={0,1,4,6,7,9}.(f)AIC={0,4,6}.(g)B−C=∅.(h)BIC=B={2,4,6,8,10,12}.1.4Applyingthedefinitions,wehave(a)A−B(b)(AUB)ICUAUBABC(c)AIBICID(d)AUUABACD(e)A∩BUABProbabilityConcepts31.5ThisiseasilyverifiedbyVennd
5、iagrams.Hence,wehaveA⊂BandB⊂C,thenA⊂CUBBACBAC1.6Byinspection,BandCaremutuallyexclusive.1.7LetR,WandBdenoteredballdrawn,whiteballdrawnandblueballdrawn,respectively10R,3W,7BNumberofredballs101(a)P(R)====0.5.Totalnumberofballs10+3+7237(b)P(W)==0.15.(c)P(B)==0.35.2020110+313(d)
6、P(R)=1−P(R)==0.5.(e)P(RUW)===0.65.220201.8LetB1≡firstballdrawnisblue.ndW2≡2balldrawniswhite.rdR3≡3balldrawnisred.(a)Theballisreplacedbeforethenextdraw⇒theeventsareindependentandhence,P(B1IW2IR3)=P(B1)P(W2
7、B1)P(R3
8、B1IW2)=P(B1)P(W2)P(R3)7310210===0.0262520202080004SignalDetec
9、tionandEstimation(b)Sincetheballisnotreplaced,thesamplesizechangesandthus,theeventsaredependent.Hence,P(B1IW2IR3)=P(B1)P(W2
10、B1)P(R3
11、B1IW2)7310==0.03072019181.9B1B210R,3W2R,6W7B1BLetR1andR2denotedrawaredballfromboxB1andB2respectively,andletW1andW2denotealsodrawawhiteballfrom
12、B1andB2.(a)P(R1IR2)=P(R1)P(R2
13、R1)=P(R1)P(R2)sincetheeventsareindependent.Hence,1021P(R1IR2)===0.111.209936(b)Similarly,P(W1IW2)=P(W1)P(W2)==0.1209(c)Sincewecanhaveadifferentcolorfromeachboxseparately,then3167P(WIB)=P(W1IB2)+P(W2IB1)=+=0.25.2099201.10LetB1andB2denoteBox1and2
14、respectively.LetBdenotedrawingablackballandWawhiteball.Then,B1B24W,2B3W,5BProbabil
