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1、Chapter2SpecialProbabilityDistributions2.1UniformDistribution?～?(0?1)means?isevenlydistributedintheinterval[0?1],itsdensityfunctionisde?nedas:?(?)=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),?nd(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),?nd(i)?(?);(ii)?(?);(
4、iii)?(?);(iv)???(?)?2.2.NORMALDISTRIBUTION21Exercise3:If?～?(???),where???,?nd(i)?(?);(ii)?(?);(iii)?(?);(iv)???(?)?2.2NormalDistributionThenormaldistributionisthemostcommonlyuseddistribution,manyvari-ablesintherealworldfollowapproximatelythisdistribution.Arandomvariablewhichfollowsano
5、rmaldistributionwithmean?andvariance?2canbeexpressedas?～?(???2).Itsdensityfunctionisde?nedas:?μ?!211????(?)=√exp???∞???∞??2?2?0.4y0.30.20.1-4-3-2-101234xN(0,1)Exercise4:If?～?(1?4),?nd(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?de?nedas: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),?nd(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,we?ndthattheyarepositivelycorrelated.ApersonwithahigherIQscore2.4.THELOGNORMALDISTRIBUTION23usuallymakesmoremoneythanapersonwitha
8、lower
