Homework for 2/3. 1. Determine the values of the following quantities: a. t 0.1,15 b. t 0.05,15 c. t 0.1,25 d. t 0.05,40 e. t 0.
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1 Name: ID: Homework for /3. Determie the values of the followig quatities: a. t 0.5 b. t c. t 0.5 d. t e. t f. χ 0.0 g. χ 0.0 h. χ 0.00 i. χ j. χ a. t b. t c. t d. t e. t f. χ g. χ h. χ i. χ j. χ Silicoe implat augmetatio rhioplasty is used to correct cogeital ose deformities. The success of the procedure depeds o various biomechaical properties of the huma asal periosteum ad fascia. The article Biomechaics i Augmetatio Rhioplasty J. of Med. Egr. ad Tech. 005: 4-7 reported that for a sample of 5 ewly deceased adults the mea failure strai % was 5.0 ad the stadard deviatio was 3.5. Assume that the distributio of failure strai is ormal with mea µ. a Fid a 98% cofidece iterval for the parameter µ. b If we kow that the stadard deviatio σ of the distributio of failure strai is 3.5 based o previous kowledge for example what is a 98% cofidece iterval for the parameter µ ow? a The 98% cofidece iterval for µ whe σ is ukow is give by s s x t 0.99 x + t Here /. Sice 5 we have t Therefore the cofidece iterval is b The 98% cofidece iterval for µ whe σ is kow is give by σ s x z 0.99 x + z Here /. Sice z the cofidece iterval is
2 3. The followig observatios were made o fracture toughess of a base plate of 8% ickel maragig steel [ Fracture Testig of Weldmets ASTM Special Publ. No : i ksi i give i icreasig order]: The sample mea ad stadard deviatio are x ad s 3.55 respectively. Assume the fracture toughess is ormally distributed. Calculate a 99% CI for the stadard deviatio of the fracture toughess distributio. Let σ be the variace of the fracture toughess distributio. The the 99% cofidece iterval for σ is give by s s χ χ / ad /. Sice we have χ ad χ Therefore the cofidece iterval is
3 Name: ID: Homework for /5. [ 8-7] Suppose that X follows a geometric distributio PX k p p k ad assume X... X is a i.i.d. sample of size. If X.5 ad 60 fid a approximate cofidece iterval for the parameter p with cofidece level 98%. The mle for p ad the asymptotic variace for the mle are foud i previous homework. The mle for p is ˆp X where X X i ad the asymptotic variace of ˆp is Ip pp. Thus a 98% approximate cofidece iterval for p is give by ˆpˆp ˆpˆp ˆp z 0.99 ˆp + z 0.99 X z X X 0.99 X + z X X
4 . [ 8-6] Cosider a i.i.d. sample of radom variables X... X with desity fuctio fx σ σ exp x σ < x < σ > 0. If X i 9 ad 8 fid a 90% approximate cofidece iterval for σ. The mle for σ is ˆσ X i ˆσ 9 for the give sample ad the asymptotic variace of ˆp is Iσ σ. Thus a 90% approximate cofidece iterval for σ is give by ˆσ ˆσ ˆσ z 0.95 ˆσ + z
5 3. [ 8-47] The Pareto distributio has bee used i ecoomics as a model for a desity fuctio with a slowly decayig tail: fx x 0 θ θx θ 0x θ x x 0 θ >. Assume that x 0 ad that X X... X is a i.i.d. sample. If log X i 0 ad 00 fid a 96% approximate cofidece iterval for θ. The mle for θ is ˆθ ˆθ 0. for the give sample ad x 0 log X i log x 0 ad the asymptotic variace of ˆp is Iθ θ. Thus a 96% approximate cofidece iterval for θ is give by ˆθ ˆθ z0.98 ˆθ ˆθ + z
6 Homework for /7. [ 8-6] Suppose that X... X is a radom sample from a Beroulli distributio with parameter p. a. Fid the mle of p. b. Show that mle of part a attais the Cramér-Rao lower boud. a. The pdf of X is Thus fx p p x p x x 0. likp fx... X p lp log likp l p X p l p X p p Xi p Xi ; [X i log p + X i log p] X i log p + X i log p X log p + X log p X p ; X p < 0. Sice l X 0 the mle for p is ˆp X. b. Let X Berp. Sice log fx p X log p + X log p; p log fx p X p X p ; p log fx p X p X p. Thus the Fisher iformatio for p is [ ] Ip E log fx p p [ E X p X ] p E[X] p p p + p p p p 6 + E[X] p
7 ad the Cramér-Rao lower boud is Ip p p p p. Sice Var[ˆp] Var [ X ] Var[X ] p p p p we see that the mle of part a attais the Cramér-Rao lower boud.. [ 8-68] Let X... X be a i.i.d. sample from a Poisso distributio with mea λ ad let T X i. a. Show that the distributio of X... X give T is idepedet of λ ad coclude that T is sufficiet for λ. b. Show that X is ot sufficiet. 7
8 a. The pmf for Poiλ is λ λx fx λ e x 0... x! Sice the sum of idepedet Poisso radom variables is a Poisso radom variable see Example E 4.5 we have T Poiλ. Thus for t x i we have fx... x T t PX x... X x T t PX x... X x T t PT t PX x... X x PT t λxi λ e x i! λ λt e t! e λ λ t x i! e λ λ t t t! e λ λ xi x i! λ λt e t! t! t x i! which is idepedet of λ. By defiitio T is a sufficiet statistic for λ. b. We have fx... x X x PX x... X x X x PX x... X x X x PX x PX PX i x i x... X x PX x PX x λxi λ PX i x i e x i! i i e λ λ i xi x i! which still depeds o λ. Thus X is ot a sufficiet statistic. i 8
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