Homework 5 STAT 305B Fall 2018 Due 11/9(R) Name

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1 1 Homewor 5 STAT 305B Fall 018 Due 11/9(R) Name PROBLEM 1(5pts) Throughout the remaider of the course we will regularly ecouter the terms 1/ ad 1/ ( 1). I this problem we will edeavor to illustrate the role that each oe plays. (a)(6pts) For { } 1 ~ iid(, ), cosider the estimator of E(*) to show that Solutio: E( ). 1 give by: ( ). Use the liearity of 1 (b)(1pts) For { } 1 ~ iid( 0, ), use the liearity of E(*) to show that E ( ) ( 1). 1 Solutio: 1 (c)(7pts) Now cosider the estimator ( ). Use (b) to show that 1 Solutio: 1 E( ). Remar 1. What this problem shows is that if you wat a ubiased estimator of ad you ow, use If you do t ow, the use the estimator ( ) ( ) 1 1 ( )

2 PROBLEM (5pts) The mass specificatios for a 3-D prited object are 00 grams ad grams. (a)(5pts) Suppose that you are cofidet that the 00 grams spec. is beig met, but there is doubt as to whether the grams spec. is. To ivestigate this, the followig 10 mass measuremets were made: Compute the ubiased estimate of. Give your code HERE Solutio: (b)(5pts) Assume ow that you are ot cofidet about the spec. 00 grams. (i) Use the std commad to obtai the ubiased estimate of Solutio:. The (ii) verify your aswer by computig the estimate directly. Give your code HERE. (c)(10pts) I Appedix 1 of this homewor we have (C): / ~ whe is used. The quatity to the chi-squared distributio with degrees of freedom. Use (C), alog with your aswer i (a), to arrive at a estimated -sided 95% cofidece iterval (CI) for. Show all steps. Solutio: [See (c).] refers (d)(5pts) I Appedix 1 we also have (D): to arrive at the CI estimate. Solutio: [See (c).] ( 1) ~ 1 whe is used. Use (D), alog with your aswer i (b) Remar. ou should have foud that the lower boud i (d) was oly slightly higher (3.4%) tha that i (c), but that the upper boud i (d) was 9.% higher tha that i (c). Hece, ot owig ca otably icrease the CI estimate for.

3 3 PROBLEM 3(5pts) (a)(5pts) O p.74 the authors defie the ( 1)% -sided CI for [ / / whe is ow as the iterval: x z /, x z / ]. (8-5) Use the authors descriptio of the CI as a radom iterval just prior to (8-5) to explai why (8-5) is icorrect. Explaatio: (b)(5pts) I view of (a), commet o the correctess of (i) equatios (8-7) ad (8-8) o p.77, (ii) equatio (8-11) o p.79, ad (iii) equatio (8-1) o p.80. Commets: (c)(5pts) I Appedix 1 we have: (B): T ~ t 1. Use this to show that the ( 1)% -sided CI for / ( 1) ( 1) ( 1) is uow as the iterval t1 / /, t1 / /, where t1 / tiv(1 /, 1). Solutio: whe (d)(10pts) I relatio to PROBLEM, suppose that is the focus of the study. Compute the 95% -sided CI estimate for i (i) the case where the CI edpoits that results by ot owig. Solutio: [See (d).] is ow ad (ii) the case where it is ot ow. The (iii) compute the percet icrease i

4 4 PROBLEM 4(5pts) Whe you fly a plae the roughess you experiece is related to the spatial roughess, as well as how fast you are goig; which results i temporal roughess. (a)(5pts) Suppose that the spatial turbulece alog a give directio is modeled by: ( x) ( x 1) U( x) ; 1 (1) where { U( x)} x0 ~ iid N(0, U) ad (0) ~ N(0, ). Use iductio (for to show that E[ ( x)] 0 for ay Solutio: x 1ad x x 0 (b)(5pts) Obtai Solutio: Var[ (1)] as a fuctio of, U, ad. [Hit: (1) ad U (1) are idepedet.] (c)(5pts) Suppose that i (b) you require that Var[ (1)]. The, if your aswer i (b) is correct, you should fid that this requires that (1 ). I this case, show that Var[ ()]. [It should the be clear that Var[ ( )] Solutio:.] U (d)(5pts) Now let 1. For each of 0.9 ad 0.1 simulate 99 0 { ( x)} x, ad overlay the two plots. The commet as to which plot exhibits more correlatio. Solutio: [See 4(d).] Commet: (e)(5pts) Recall that Solutio:. Also recall that ( ) Figure 4(d) Simulated turbulece for 0.9 ad 0.1. E. Use these to show that ( x) ( x 1) Remar 4. The Matlab wid turbulece models are essetially of the form of (1) above.

5 5 APPENDI 1. Some hady-dady Test Statistics For ~ N(, ) ad associated iid data collectio variables { } : 1 (A): Z ( ) /( / ) ~ N(0,1 ) ; (B): T ~ t 1 / (C): ( 1) / ~ whe is used ; (D): ~ 1 whe is used. / 1 1 / 1 1 (E): F ~ f 1, whe are used ; (F): F ~ f 1 1, 1 whe / / & & used.

6 6 APPENDI. Matlab Code %PROGRAM NAME: hw5.m rg('shuffle') %PROBLEM mu=00; x=[ ]; =legth(x); %(a): %(b): %(c): %(d): %============================================ %PROBLEM 3 %(d): %(i): %(ii): %==================================== %PROBLEM 4 =100; var=1; figure(40) title('simulatios of () for a=0.9 ad a=0.1') xlabel('x') ylabel('y(x)') leged('0.9','0.1') grid

1the 1it is said to be overdamped. When 1, the roots of

1the 1it is said to be overdamped. When 1, the roots of Homework 3 AERE573 Fall 08 Due 0/8(M) ame PROBLEM (40pts) Cosider a D order uderdamped system trasfer fuctio H( s) s ratio 0 The deomiator is the system characteristic polyomial P( s) s s (a)(5pts) Use