Name: ID: Homework for /3. Determie the values of the followig quatities: a. t 0.5 b. t 0.055 c. t 0.5 d. t 0.0540 e. t 0.00540 f. χ 0.0 g. χ 0.0 h. χ 0.00 i. χ 0.0050 j. χ 0.990 a. t 0.5.34 b. t 0.055.753 c. t 0.5.36 d. t 0.0540.684 e. t 0.00540.704 f. χ 0.0 4.87 g. χ 0.0.44 h. χ 0.00 8.6 i. χ 0.0050 7.43 j. χ 0.990 37.57. 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 0.99. Here 0.99 + 0.98/. Sice 5 we have t 0.994.64. Therefore the cofidece iterval is 5.64 3.5 5 5 +.64 3.5 5.69 7.37. b The 98% cofidece iterval for µ whe σ is kow is give by σ s x z 0.99 x + z 0.99. Here 0.99 +0.98/. Sice z 0.99.33 the cofidece iterval is 3.5 3.5 5.33 5 +.33.894 7.06. 5 5
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. 38 965: 38-356 i ksi i give i icreasig order]: 69.5 7.9 7.6 73. 73.3 73.5 75.5 75.7 75.8 76. 76. 76. 77.0 77.9 78. 79.6 79.7 79.9 80. 8. 83.7 The sample mea ad stadard deviatio are x 76.55 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 0.995 χ 0.005 0.995 + 0.99/ ad 0.005 0.99/. Sice we have χ 0.0050 7.43 ad χ 0.9950 40.00. Therefore the cofidece iterval is 0 3.55 0 3.55 6.30 33.9. 40 7.43
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 0.99.5.33 0.3068 0.493..5.5 60.5 +.33.5 60.5
. [ 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 0.95 9 9 9.645 8 9 +.645 8 7.355 0.645. 4
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 0.98 0. 0. 0..05 0. +.05 00 00 0.0795 0.05. 5
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
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
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