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 x x = = = 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 y y = = = 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 T X = = å Then ( ) ( ) 12000 ( ), j j E T E X E X = = å where X is occupancy of a randomly sampled vehicle. Hence 12000 T X = 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 t x = = vehicles.
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2. For any estimator, 1 1 2 2 L a X a X = + with 1 2 1 = + a a , 1 1 2 2 1 2 ( ) ( ) ( ) ( ) E L a E X a E X 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 : 2 2 1 1 2 2 1 1 2 2 var( ) var( ) var( ) L a X a X L a X a X = + Þ = + 2 2 2 2 1 2 ( /35) ( /105) a a = σ + σ So we must have: (i) 2 1 var( ) 105 L σ = (ii) 2 2 2 1 1 9 1 var( ) ( ) 16 35 16 105 140 L σ = σ + = (iii) 2 2 3 1 1 1 1 var( ) ( ) 4 35 4 105 105 L σ = σ + = Þ 2 L is best (most efficient ) 3. To compute the mean of
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