1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 November 14th, 2006-Lecture 22-Instructor: Brian HabingDepartment of StatisticsTelephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today• Last Time - Covariance• Moment Generating Functions• Applications!STAT 702/J702 B.Habing Univ. of S.C. 3CovarianceCov(X,Y)=E[(X -μX)(Y-μY)]2STAT 702/J702 B.Habing Univ. of S.C. 4CorrelationCor(X,Y)=)()(),(YVarXVarYXCovXY=ρSTAT 702/J702 B.Habing Univ. of S.C. 54.5 – Moment Generating FunctionsThe moment-generating function (mgf) of X is M(t )=E(etX)∑=xtXxpetM )()(∫=∞∞−dxxfetMtX)()(STAT 702/J702 B.Habing Univ. of S.C. 6Why “moment generating” ?Assume the mgf exists on some interval around 0…∫=∞∞−dxxfetMtX)()(∫=∞∞−dxxfedtdtMtX)()('3STAT 702/J702 B.Habing Univ. of S.C. 7Other properties:a) The m.g.f. uniquely determines the p.d.f.b) If Y=a +bX then MY(t )=eat MX(bt)c) If X and Y are independent and Z=X+Y then MZ(t )=MX(t ) MY(t )STAT 702/J702 B.Habing Univ. of S.C. 8Example 1) X~Uniform[0,1]MX(t)=MaX+b(t)=STAT 702/J702 B.Habing Univ. of S.C. 9Example 2) Sum of Negative Binomials.)1ln(for ])1(1[)()( pteppetMrtrt−−<−−=4STAT 702/J702 B.Habing Univ. of S.C. 10Application 3:Intelligent Searching and Samplinga) Group Testing: A large number nof blood samples are to be tested for a relatively rare disease. Can we find all the infected samples in fewer than ntests?STAT 702/J702 B.Habing Univ. of S.C. 11Consider the case of splitting each of nsamples in half. Combine half of each one is placed into a large combined pool.Should this work better?STAT 702/J702 B.Habing Univ. of S.C. 12Now consider that we divide the msamples into mgroups of size
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