Fundamentals of Probability and Statistics for Engineers

82 Fundamentals of Probability and Statistics for EngineersNow,EfY 2 gˆXnk 2 p Y …k† ˆXnk…kkˆ0kˆ01†p Y …k†‡ Xnkˆ0kp Y …k†;andX nkˆ0kp Y …k† ˆnq:Proceeding as in Example (4.1),Thus,X nkˆ0k…k 1†p Y …k† ˆn…n 1†q 2 Xnˆ n…nkˆ21†q 2 Xmˆ n…n 1†q 2 :jˆ0…n 2†!…k 2†!…n k†! qk 2 …1 q† n k!mq j …1 q† m jjEfY 2 gˆn…n1†q 2 ‡ nq;andEx ample 4. 6. We again use Equation (4.8) to determine the variance of Xdefined in Example 4.2. The second moment of X is, on integrating by parts,Hence, 2 Y ˆ n…n 1†q2 ‡ nq …nq† 2 ˆ nq…1 q†:EfX 2 gˆ2Z 10x 2 e 2x dx ˆ 12 : 2 X ˆ EfX2 g m 2 X ˆ 1214 ˆ 14 : TLFeBOOKExample 4.7. Problem: owing to inherent manufacturing and scaling inaccuracies,the tape measures manufactured by a certain company have a standarddeviation of 0.03 feet for a three-foot tape measure. What is a reasonableestimate of the standard deviation associated with three-yard tape measuresmade by the same manufacturer?