The Institute of Actuaries of India

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1 The Istitute of Actuaries of Idia ubject CT3 Probabilit & Mathematical tatistics 5 th Ma 7 IDICATIVE OLUTIO Itroductio The idicative solutio has bee writte b the Examiers with the aim of helpig cadidates. The solutios give are ol idicative. It is realized that there could be other poits as valid aswers ad examier have give credit for a alterative approach or iterpretatio which the cosider to be reasoable. Arpa Thaawala Chairperso, Examiatio Committee

2 . a) team ad leaf plot 6,5,5,8,9 7,4,4,5,7,8 8,3,3,5,7,8,9 9,,,4,4,5,7,,7,8,,4,5,4,5 + b) Q : : Q : : Q 3 : : c) Box Plot Mi Max 6 Q Q Q [6].a) Total o. of outcomes 3 8 A {( HTH ),( HTT ),( THH ),( THT )} imilarl B ad C have 4 outcomes each A) C) ow A B A {( HTH ),( THT )} imilarl there are two outcomes i B C ad C A, so A B C)C A) 8 4 A A/ P ( A/ C) A C) C) A) A) b) ice P ( A A). P ( C C). P ( A C) C). A) A,B ad C are pairwise idepedet. [4]

3 3.a) k ( x) dx k b) x ( x) dx x 3 x () 6.67 x c) ( x ) dx x < X < 3) P X < 3 / X > 5 X > 5) d) ( ) dx dx [7] 4. Let A{H} - Head o first toss B {T,H} - Tail o first toss ad Head o secod toss C {T,T} - Tail o both tosses Let X deote his radom earigs EX A) E( X / A) + E( X / + C) E( X / C) E[ X / A], E[ X / B] + E( X / C] A) / / 4 C) / 4 EX (-) [4] 5. i i,,3,4 deotes umber of hospitalizatio i eptember, October, ovember ad December respectivel From the give iformatio

4 ~ Poi(), ~ Poi() 3 ~ Poi(), 4 ~ Poi(3) Let X s ' j are assumed to be mutuall idepedet X ~ Poi(+++3)Poi(7) The desired probabilit is P[X<5] 4 x 7 7 e P [ X < 5] x x! [4] 6. a) α λx λ e x Γα α λ α k e λx x α where k Γα Hece, log f(x) log k-λx+(α-)log x. Differetiatig ' ( α ) λ + x ' λx α k( α ) e x λk e -λx x α- f (x) gives α x λ α Hece, the mode is if α > λ ( or) mode if α < α b) From the data, the average demad a λ () (For Gamma, Mea α/λ) Most likel demad b Mode α i.e. b λ ( ) From () ad () (a-b) /λ For Gamma distributio, Var X α/λ α λ λ a(a-b) from () ad ()

5 Mea Mode c) kewess.d. α α λ λ α λ λ/α o, whe α λ, kewess (The computatio of skewess ca be doe usig a other formula also) [] 7. Let X deote the umber of policies i the block for which there is atleast oe claim i the comig period Here X ~ B(,.) x x We have to evaluate P [ X > 3] (.) (.9) x 3 x To appl ormal approximatio EX (.) 5 Var X (.) (.9).5 5,.5 Whe is large, X ~ ( ) X > 3) P[X 3.5] X P [Z.595] - Φ (.595) a) Likelihood : x L θ / x) θ if < x,..., x i ( θ x θ otherwise if < x otherwise < i ( ) θ < [4] ketch

6 L( θ / x) x () θ ice L(θ/ x) is decreasig fuctio of θ wheever θ > x (), the MLE is X (). b) ice X () < θ for all θ implies EX () < θ for all θ. Hece, X () is ot ubiased for θ. [6] 9. a) From the data 34, X 848 X 848 p.8 (app) 34 H : p.5 (p ) H : p.5 p p.8.5 Z pq (.5)(.5) (o roudig the deomiator as.55) Critical value (5% level).96 Cal Z > critical value Reject H. b) 9% cofidece iterval for p is p( p) p± Z α /.8(.8) i.e..8 ± ±.97 Hece, 9% cofidece iterval for p is (.83,.8397) [5]

7 . a) Choice of Polic Whole Edowmet Edowmet Life with Profit without profit Rural ativit Urba E (-E) (-E) /E Critical value of χ for d.f at 5% level is Do t reject H. b) Cosider Whole life Edowmet Rural 4 6 Urba 35 (33) 75 χ Reject H, sice the critical value of χ for d.f. at 5% level is [6]. a) Before Traiig (X) After Traiig (Y) Diferece (d) x d 5, s d 9.6, 9 H μ d ; H : μ d x d Test statistic t. 63 sd / Critical value of t at 5% level for 8d.f. is.36. Do ot reject H b) 95% cofidece iterval for the mea chage i abilit of traiees ( x d t s /, x d t s / ) α d + ( -.88,.88) α d

8 The above cofidece iterval is for (μ X μ Y ). However, the cofidece iterval for (μ Y - μ X ) is (-.88,.88). c) Computatio of correlatio coefficiet : ΣX 378, ΣY 333, ΣX 6, ΣY 347, ΣXY 434 XY X Y r ΣX ( X ) ΣY ( Y ).6796 Test the sigificace of ρ : H : ρ vs. H ρ r Test statistic t r.6796x Critical value of t at 5% level for 8 d.f.is.36, Reject H [3] Q(c) The Idicative solutios has a mior error i the computatio of test statistic t..6796x 7 The correct value is. 45 (approx) a) Assumptios of Regressio model : The errors ( s) are idepedet ad have ormal distributio with mea zero ad a commo (ukow) variace σ. b) Σx i,, Σxi 5,3, Σi 8.35, Σi 9.97 Σxi i 75.4 () 5,3,,3, (8.35) x (8.35)

9 c) x β b.383 a bx. 69 Hece x s e x /.53 ( x) d) 95% cofidece iterval for α is a t s + ± α / t α / ( ).36 (from tables) Hece 95% cofidece iterval is (.64,.3 ) e) H : β vs. H β b β Test statistic t se 8.75 For the two sided test t value for 8 d.f..36 Reject H e ( x x) f) The formula is ( a + bx ) ± tα / s e + where x 9 (.68,.93) is the 95% cofidece iterval for the mea evaporatio coefficiet α +9β [4] 3. H : μ A μ B μ C H : atleast oe pair is ot equal. where μ A, μ B, ad μ C are mea scores awarded to logos A,B,C. AOVA TABLE ource df M F Betwee logos Error Total Critical value of F.5 (,) 3.47 Do ot Reject H [6]

10 4. f ( x, ) f ( ) Hece, ( x+ ) e e ( x+ ) dx if x < otherwise x e e dx e ; > f ( x, ) e f ( ) e e -x ; x > ( x+ ) f ( x / ) E ( X / Y ) e xe xe x x dx dx e e ( + ) + [4] 5. ( c) P[ C c] F C P [ C c / ]. P[ ] [ W + W W c / ] P[ ] [ W + W W c] P[ ] ( QW j s are idep of ) F W *( c) P ( ) where W * is the fold covolutio of W. Let M C (t) be the mgf of C M C (t) E[e tc ] ct E [ E( e / ) ] ( W + W W ) t E [ E{ e / } ] W [ ( ) ( )... ( )] t W t W t E e E e E e Wt E [ E( e ) ] E [ M ( t) ] E ( QW j s are idep of ) W

11 log MW ( t) E [ e ] M [ log M ( t) ] W Hece, log M ( t ) ψ ( t ) ψ ( ψ ( t )) fuctio of C. C which is the cumulat geeratig C W [7] * * * * * * * * * * * * *