Moment Generating Functions in Statistics: Applications and Examples, Study notes of Statistics

Download Moment Generating Functions in Statistics: Applications and Examples and more Study notes Statistics in PDF only on Docsity! 1 STAT 702/J702 B.Habing Univ. of S.C. 1 STAT 702/J702 November 9th, 2004 -Lecture 22- Instructor: Brian Habing Department of Statistics Telephone: 803-777-3578 E-mail: [email protected] STAT 702/J702 B.Habing Univ. of S.C. 2 Today • Moment Generating Functions (cont.) • Application 1: System Failure • Application 2: Search Methods STAT 702/J702 B.Habing Univ. of S.C. 3 4.5 – Moment Generating Functions The moment-generating function (mgf) of X is M(t )=E(etX) ∑= x tX xpetM )()( ∫= ∞ ∞− dxxfetM tX )()( 2 STAT 702/J702 B.Habing Univ. of S.C. 4 Properties of m.g.f.’s a) If the m.g.f. exists on an interval around zero then M(k)(0)=E(Xk) b)The m.g.f. uniquely determines the p.d.f. c) If Y=a +bX then MY(t )=eat MX(bt ) d)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. 5 Example 1) X~Uniform[0,1] MX(t)= MaX+b(t)= STAT 702/J702 B.Habing Univ. of S.C. 6 Example 2) Sum of Negative Binomials. )1ln(for ])1(1[ )()( pt ep petM rt rt −−< −− =