Institute of Actuaries of India

Transcription

1 Isiue of cuaries of Idia Subjec CT3-robabiliy ad Mahemaical Saisics May 008 Eamiaio INDICTIVE SOLUTION Iroducio The idicaive soluio has bee wrie by he Eamiers wih he aim of helig cadidaes. The soluios give are oly idicaive. I is realized ha here could be oher ois as valid aswers ad eamier have give credi for ay aleraive aroach or ierreaio which hey cosider o be reasoable.

2 . a. Mea ; Media 8 b. The leas wo ois are 3 ad 6. The larges wo ois are 79 ad 39. So, he rimmed mea is [40.4* ]/ c. Comme: i. Mea gives eual weigh o all he observaios so i is more likely o ge imaced by he ereme observaios. ii. Media does deed o he size of he observaios bu oly o he order. So i is leas likely o ge imaced by he ereme observaios. iii. Imac of ouliers o he rimmed mea deeds o he umber of observaios beig rimmed. I ay case i is less imaced by ereme observaios ha he simle mea. d. S ~ χ, Now 300 S > 300 S > 300 > / 3 / / 3 3 / / 3 3 / Toal [8] Toal [] B 36 4 C B C B C age of 0

3 age 3 of 0 However, 36 6 C C C B C B B B, B ad C are o muually ideede. Toal [4] 4...., ; ± ± E G [ ]... [ ]... r r r r ; < < ; < < Toal [3] 5. a. [ ] 0 ; / 3. log e f π 0 for < 0 b. E 3.7 / 3. / e e e μ σ Var σ σ μ e e e 7.4 e- c. > 8 8. Sice log ~ Nμ3., σ, 8 Ф 3. log8 Ф -., Ф deoes he cdf of sadard ormal.

4 0.34 > Toal [4] 6. a. Mos suiable disribuio for N is Biomial 90, where is he robabiliy of head. Esimae of is 50/90. Mea of N is 90*50/ Variace is 90*50/90 * -50/90 50*40/90.. b. N aroimaely follows Normal wih mea 45 ad variace.5. Usig coiuiy correcio, Bi N > 50 Nor N Nor Z where Z ~ N 0, Nor Z c. Usig he above robabiliy, he -value of he es is 0.3. We do o have sufficie evidece o rejec H o a 5% sigificace level. d. H o should be rejeced for -value < Le be he desired umber of heads. The N 0.05 N * So, H o should be rejeced for he umber of heads 53. Toal [7] 7. i ~U0, E i, Var i ; i,,,0 0 Give ha Y i ad hece Y ~ i 5 N 0, 3 Y a. [ ] Y 9. 5/ 3 5 / 3 [Z ] age 4 of 0

5 Φ Y b. [ ] 8.5 Y.7 5 / 3 5/ 3 [ -.6 Z.37] 5/ 3 Φ.37 Φ Toal [3] 8. a. Wai for e miues 0,, assegers will arrive wihi he e e miues e e e ! b. Solvig his ar deeds o he memoryless roery of oisso rocess. The waiig eriod before he arrival of e asseger is eoeially disribued wih a eeced value of hree miues. havig o wai aoher five miues obody arrives i he e five miues θ k 9. a. d θ 0 θ k θ 0 d 5 k θ [ ] 0 θ k θ * θ k For he above iegral o be, k should be. b. The likelihood fucio is θ i ; 0< i < θ i L, θ 3 e Toal [5] Differeiaig his wr θ ad euaig o 0 will o give a soluio for θ i erms of. L is a decreasig fucio of θ L is osiive oly whe θ Ma i So L is he maimum whe θ Ma i ; which is he MLE of θ. Toal [5] age 5 of 0

6 λ 0. a. For Eoeial λ, f, λ λe ad F - e λ, > 0 The likelihood fucio is 7 λ λ L λ, λ * e * e b. The log-likelihood is l, λ 7 l λ 693λ l, λ λ Euaig above o 0 gives MLE of λ 7/ c. l, λ 7 λ l, λ 7 So, E λ λ CRLB for he MLE is which is same as he l, λ 7 E asymoic variace of he MLE d. The 95% cofidece ierval for λ is ˆ λ ±.96. S. D. ˆ λ. Usig he MLE of λ ad he variace he ierval is: * i.e.0.005,0.077 e. The likelihood is 5λ 3 7 λ[ *5] L λ;, [ e ] * λ * e * e [300 3*5] λ.a. The Scaer lo Toal [] 5 4 College grade High school grade age 6 of 0

7 b. ˆ α ; ˆ β 0.85; rˆ ; c. s he esimae of he sloe is lower ha, he eeced erformace i college is lower ha he erformace i high school. d. yˆ ˆ α ˆ β * cual college grade for a high school grade of. for observaio 0 is much higher ha he eeced. This observaio ca be reaed as a oulier while fiig he regressio. Toal [] * Y. B * YB. C * YC.. Y.. B C 8*00 0*0 9* Y a. ˆ μ Y ˆ τ Y ˆ τ Y B. B. Y.. Y ˆ τ C YC. Y SSR 075 ˆ σ k 4.. b. SS B i Y i. Y.. 8*4.50*63.49* i SS T SS B SS R c. The NOV able is: Source DF SS MS F Bewee Residual Toal F 0.05,, 4 is So here is eough evidece o rejec he ull hyohesis of eualiy of all ferilizer effecs. d. The 95% cofidece ierval for μ τ is give by ˆ σ ˆ σ Y. 0.05,4, Y. 0.05, *,00.064* ,.56 Toal [0] age 7 of 0

8 3. a.σ 378, ΣY 333, Σ 6, ΣY 347, ΣY 4340 ΣY Y r Σ ΣY Y H 0 : ρ 0 ; H : ρ 0 Tes saisic is r r The able value of disribuio for 7 df a 5% level is.895. Coclusio: Rejec H 0. b. H 0 : σ σ ; H : σ σ Usually he larger variace is ake i he umeraor 4 ; s 9. 5 ad Y 37 ; s F cal S S 4.87 ad F ab 8, Y / Y Coclusio: Rejec H 0 σ Y sy s c. 90% cofidece ierval for is σ, F α / 8,8 Fα / 8,8 s s [.46,6.757], F , The CI for σ / [.0596, ] σ Y Y d. Chages i Scores d: d Y- d 5 ; s 9.06 ad 9. d H 0 : μ d 0 Vs H : μ d >0 d 5 The Tes saisic is :. 69 s 9.06 d 3 Table value ; Coclusio: Do o rejec H 0 age 8 of 0

9 e. 95% cofidece ierval for he differece bewee he wo oulaio meas μ y - μ is { μ y μ ± α /, s d } which simlifies o -.077,.077. The cofidece ierval for μ - μ y will herefore be -.077,.077. Toal [4] 4 a. Noe ha value of k i he desiy does o affec he codiioal eecaio. E/Yy: f Y k f Y y, y d 0 y 0 d 0 k y-0 3 ; 0 < y < 30 0 Hece, f / Y / y ; 0 < < y y 0 0; oherwise y 0 E / Y y d 0 y 0 0 y 0 ; 0 < y < b. > 0 / Y 5 d 5/ Toal [7] 5. The mf of ; 0,,,... ad N ;, 3, N / N N ; 0,,,... ;, 3 age 9 of 0

10 Hece,, N / 3 6 6, N, N / 6 3, N / 6 Mea of : 5/6 Variace of : 7/36 The, 0 0, N 0, N Toal [5] *************** age 0 of 0