SOLN_3b

SOLN_3b - Systems 302 Tony E Smith SOLUTIONS TO PROBLEM SET...

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Systems 302 Tony E. Smith SOLUTIONS TO PROBLEM SET 3 1. (a) Let i x = number of passengers in vehicle, . 500 ,..., 2 , 1 = i The BLU estimator for the mean number of occupants, ) ( X E = µ , is then given by: 500 1 1 1.778 500 i i xx = == å (b) Construct the Bernoulli random variable y = î í ì 1, 1 0, x otherwise > Then BLU estimator for ) ( y E P = is given by 500 1 1 .36 500 i i yy = å Partial Printout of JMPIN spreadsheet: Rows X X_bar Y Y_bar 1 1 1.778 0 0.36 2 2 1.778 1 0.36 3 1 1.778 0 0.36 4 5 1.778 1 0.36 5 2 1.778 1 0.36 6 1 1.778 0 0.36 7 3 1.778 1 0.36 (c) If j X = number of occupants in vehicle j = 1,2,&,12000 and 12000 1 , j j TX = = å Then ( ) ( ) 12000 ( ), j j E TE X E X å where X is occupancy of a randomly sampled vehicle. Hence 12000 =⋅ is a linear unbiased estimator of () E T . Moreover, since ± var( ) var( ) X < µ for any other linear unbiased estimator, ± µ , of () E X implies that ± var(12000 ) var(12000 ) X ⋅< µ , it follows by definition that that T is the unique BLU estimate of () E T . In the present case, the desired estimate is 12000 21,366 tx = vehicles.
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2. For any estimator, 11 2 2 L aX =+ with 1 2 1 = + a a , 112 2 1 2 () ( ) ( ) ( ) EL aEX aEX a a = + µ = µ So all three estimators are unbiased . This means that we need only compare their variances to determine which is most efficient . But by the independence of the random variables 1 X and 2 X : 22 1 1 2 2 var( ) var( ) var( ) L L a X a X Þ 12 ( /35) ( /105) aa + σ So we must have: (i) 2 1 var( ) 105 L σ = (ii) 2 2 2 11 9 1 var( ) ( ) 16 35 16 105 140 L σ + = (iii) 2 2 3 11 1 1 var( ) ( ) 4 35 4 105 105 L σ + ⋅ = Þ 2 L is best (most efficient ) 3.
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SOLN_3b - Systems 302 Tony E Smith SOLUTIONS TO PROBLEM SET...

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