Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 273f Θ2(θ 2 )f ∧ Θ2(θ ∧2 )(a)θ∧ ∧ f Θ1(θ 1 )f ∧ Θ1(θ ∧ 1 )∧∧θ 1 ,θ 2(b)θ∧∧∧∧θ 1 ,θ 2where X is the sample mean based on a sample of size n. The choice of ^ 1 isintuitively obvious since EfXg ˆ, and the choice of ^ 2 is based on a priorprobability argument that is not our concern at this point.SinceandFigure 9.2 Probability density functions of ^ 1 and ^ 2EfXg ˆ;we have 2 Xˆ …1 †nEf ^ 1 gˆ;Ef ^ 2 gˆn ‡ 1n ‡ 2 ; 9=;…9:45†and 2^1 ˆ…1 †;n9> = 2^2 ˆ n2 n…1 †…n ‡ 2† 2 2 ˆX…n ‡ 2† 2 : > ;…9:46†We see from the above that, although ^ 2 is a biased estimator, its variance issmaller than that of ^ 1, particularly when n is of a moderate value. This isTLFeBOOK