Chapter 2 Special Probability Distributions

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1、Chapter2SpecialProbabilityDistributions2.1UniformDistribution∼(01)meansisevenlydistributedintheinterval[01],itsdensityfunctionisdefinedas:()=1for∈[01];()=0elsewhere.Thedistributionfunctionisthen()=0for≤0;()=for∈(01);()=1for≥11Themeanisobviouslyequalto.Tocalculate

2、thevariance,notethat21920CHAPTER2.SPECIALPROBABILITYDISTRIBUTIONS¡¢¡¢µ¶2Z1Z122212121()=−()=−=()−=−20404∙3¸11111=−=−=3434120Exercise1:If∼(01),find(i)Pr(0);(ii)Pr(≤1);(iii)Pr(0);(iv)Pr(≤05);(v)Pr(07);(vi)Pr(04≤08);(vii)Pr(=08)Notethattheareaunderth

3、edensityfunctionhastosumupto1,soifwehavearandomvariablewhichisuniformlydistributedbetween1and3,i.e.if∼(13),thenitsdensityfunctionis1()=for∈[13];2()=0elsewhere.Thedistributionfunctionwillbe()=0for≤1;−1()=for∈(13);2()=1for≥3Exercise2:If∼(12),find(i)();(ii)();(

4、iii)();(iv)()2.2.NORMALDISTRIBUTION21Exercise3:If∼(),where,find(i)();(ii)();(iii)();(iv)()2.2NormalDistributionThenormaldistributionisthemostcommonlyuseddistribution,manyvari-ablesintherealworldfollowapproximatelythisdistribution.Arandomvariablewhichfollowsano

5、rmaldistributionwithmeanandvariance2canbeexpressedas∼(2).Itsdensityfunctionisdefinedas:õ¶!211−()=√exp−−∞∞220.4y0.30.20.1-4-3-2-101234xN(0,1)Exercise4:If∼(14),find(i)Pr(0);(ii)Pr(≤1);(iii)Pr(0);(iv)Pr(≤−1);22CHAPTER2.SPECIALPROBABILITYDISTRIBUTIONS(v)Pr(2);

6、(vi)Pr(1≤3);(vii)Pr(=1)2.3StandardizedNormalDistribution−If∼(2),then=follows(01).Itsdensityfunctionisdefinedas:µ¶112()=√exp−−∞∞22−3Example1:If∼(34),then=follows(01).2µ¶1−3−35−3Pr(1≤≤5)=Pr≤≤222=Pr(−1≤≤1)'067Exercise5:If∼(01),find(i)Pr(0);(ii)Pr(≤

7、1);(iii)Pr(0);(iv)Pr(≤−1);(v)Pr(2);(vi)Pr(1≤3);(vii)Pr(=1)2.4TheLognormalDistributionWhenwestudytherelationshipbetweenaperson’sIQscoreandhisincome,wefindthattheyarepositivelycorrelated.ApersonwithahigherIQscore2.4.THELOGNORMALDISTRIBUTION23usuallymakesmoremoneythanapersonwitha

8、lower