1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 November 2nd, 2006-Lecture 20-Instructor: Brian HabingDepartment of StatisticsTelephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today• Applications• More On Expectations• Moment Generating FunctionsSTAT 702/J702 B.Habing Univ. of S.C. 3Application 2) Each component in the system below has an independent exponentially distributed lifetime with parameter λ. Find the cdf and density of the system’s lifetime.2STAT 702/J702 B.Habing Univ. of S.C. 4Chapter 4 Revisited: More on Expected ValuesRecall that∫∑⇒=+∞∞−dxxxfxxpXE )()()(∫−⇒∑−=∞+∞−dxxfxxpxXVar)()()()()(22μμSTAT 702/J702 B.Habing Univ. of S.C. 5For constants a and b, E(a +b X)= a + b E(x)Var(a+b X)= b 2 Var(X)STAT 702/J702 B.Habing Univ. of S.C. 6Let X1, X2, … Xnbe mutually independent random variables, then:μΣX= E(ΣiXi) =ΣiE(Xi) = ΣμXiσΣX2 = Var(ΣiXi) = ΣiVar(Xi) =Σ σXi23STAT 702/J702 B.Habing Univ. of S.C. 7What if the Xiare not independent? First, if the Xihave joint p.d.f f(x1,…xn) and Y=g(x1,…xn) thenProvided the integral converges with |g |.∫∫=nnndxdxxxfxxgYE LL111),...(),...()(STAT 702/J702 B.Habing Univ. of S.C. 8Now considerand finding E(Y) and Var(Y).∑+==niiXbaY1STAT 702/J702 B.Habing Univ. of S.C. 9CovarianceCov(X,Y)=E[(X -μX)(Y-μY)]CorrelationCor(X,Y)=)()(),(YVarXVarYXCovXY=ρ4STAT 702/J702 B.Habing Univ. of S.C. 104.5 – 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. 11Why “moment generating” ?Assume the mgf exists on some interval around 0…∫=∞∞−dxxfetMtX)()(∫=∞∞−dxxfedtdtMtX)()('STAT 702/J702 B.Habing Univ. of S.C. 12Other 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