Fundamentals of Probability and Statistics for Engineers

268 Fundamentals of Probability and Statistics for EngineersLet us define a new random variable Y byEquation (9.30) shows thatMoreover, since Y is a sum of n independent random variables, each with meanzero and variance Ef[q ln f X; )/q] 2 g, the variance of Y is the sum of the nvariances and has the formNow, it follows from Equation (9.29) thatRecall thatorAs a consequence of property 2 1, we finally haveor, using Equation (9.32),The proof is now complete.Y ˆ Xnjˆ1q ln f …X j ; †: …9:31†qEfYg ˆ0: ) 2 Y ˆ nE qlnf …X; † 2: …9:32†q1 ˆ Ef ^Yg: …9:33†Ef ^Yg ˆEf ^gEfYg‡ ^Y ^ Y ;1 ˆ …0†‡ ^Y ^ Y : …9:34†1 2^ 2 Y 1; 2^ 1 ) )qlnf …X; † 2 1 2 ˆ nEYq: …9:35†In the above, we have assumed that differentiation with respect to under anintegral or sum sign are permissible. Equation (9.26) gives a lower bound on theTLFeBOOK