Institute of Actuaries of India

Transcription

1 Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of

2 IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6 = lowr bound = 35-*36.5 = 9.7 c oal no. of daa undr h abov limis = 8 = 9% according o mahmaician, minimum proporion of daa in h inrval = -/ 3 = 87.5% Thus, mahmaician s horm is valid for = for h givn daa [6] ol. X is U[,] r in h inrval o X lis in h inrval o X lis in h inrval o i.. o X 3 lis in h inrval o / which do no covr h full inrval Hnc X and X ar valid simulad obsrvaions of X [] ol. 3 auppos ha prson is born on a paricular day i, whr i can b any on of h 365 days. o probabiliy = /365 If prson dos no shar h sam birhday, hn h should hav born on any on of h rs 36 days. o probabiliy = 36/365 Finally, hr can b such 365 combinaions as i can b any of h 365 days Probabiliy of prsons no having h sam birhday is /365 * 36/365 * 365 = 365*36/365 Thus, probabiliy of prsons having h sam birhday is -365*36/365 =.7 b L all 3 prsons do no shar h sam birhday whos probabiliy basd on abov argumn is /365 *36/365 * 363/365*365 = 365*36*363/365 3 Thus, probabiliy of a las prsons having h sam birhday is -365*36*363/365 3 =.8 For a group of prsons, h probabiliy a las prsons having sam birhday would b - 365*36*363*36/365 =.6356 c L h group siz b 5 Thus, for a group of 5 prsons, h probabiliy ha a las prsons hav h sam birhday would b - 365*36* /365 5 = -.77 = 9 [9] Pag of

3 IAI CT3 Novmbr ol. bias = E E X Var X Var ME = varianc + bias Thus, 5 ME 5 Bu, i is givn ME [ bias ] Thus, quaing h abov quaions as givn > [] ol.5 Expcd claims for h insuranc company = *. + *. =. Insuranc Prmium = % *. =. Rvnu =.7 Ohr Expnss = % *.7 =.3 Profi bfor rpair cos =.7.3. =.9 Raind rpair cos = wih probabiliy. = wih probabiliy.6 Profis afr raind rpair cos =.9 wih probabiliy. = -.8 wih probabiliy.6 Thus, xpcd dividnds =.9*. + *.6 =.368 [] ol.6 y P Y y P Y y F Y = PX > y PX n > y as = PX i > y n X s ar muually sochasic indpndn variabls i 3 y u P X y 3 u du 3 u 3y y i 3 Hnc, PX i > y = 3y + y 3 ; Thrfor, F Y y = 3y + y 3 n d dy 3 n f y F y n 3y y X Y Y y n y 3n y 3y y < y < [] Pag 3 of

4 IAI CT3 Novmbr Pag of ol. 7 L = scor 75 ] [ E E E E ] [ ] [ E Var E Var Var Var E 65 9 / 65 65] 9 [ P P P F F F [5] ol. 8 a dx x dx x E M x x X X dx x dx dx x x x x Now, x dx x x x x imilarly x dx x x x x and dx x x Thus, M X

5 IAI CT3 Novmbr. X X. X. X b i M E E. E Y b ii is h man of wo random sampls from U[,] Basd on h rsuls of a and b i abov, w can say ha X has h riangular disribuion as spcifid in a [8] ol.9 a b = y a x y a x y = imilarly, Pag 5 of

6 IAI CT3 Novmbr imilarly, c Th covarianc bwn X and Y is: To prov: Indpndnc implis null covarianc If boh random variabls ar indpndn, his mans ha: f X,Y x,y = f X x * f Y y = Th only way for h join dnsiy o b qual o is o hav α =. Bu if α =, hn CovX,Y =. Thrfor, h indpndnc implis ha h covarianc is qual o. To prov: Null covarianc implis indpndnc If h covarianc is qual o, w mus hav α =. Bu hn if α =, h join dnsiy is qual o, which mans i is qual o h produc of h marginal dnsiis. Thus, h wo random variabls ar indpndn. Hnc, h samn is ru. [] ol. a inc n has xponnial disribuion wih man 5, hnc, h paramr λ is /5 =. Pr[ n > ] = -. =. = 3.3 b n follows lognormal µ, σ dividing, w hav σ =.6935 or σ =.8356 Hnc, µ =.687 Pag 6 of

7 IAI CT3 Novmbr Pr[ n > + = Pr*log n > log + Z ~ N,.36 Giving =.5 c Pr[ = condiioning on h valu of = -3.3/5 3.3/5-6.6/5 =.9895 [] ol. n = 3 x Fx fx Expcd Claims in ach inrval , , , , infiniy..5.5 Hnc, h firs wo and h las wo inrvals nd o b combind so ha h xpcd numbr of claims in ach inrval is a las 5. x Expcd E Obsrvd O O E O E / E , , , infiniy Toal 3 3 Χ = 9.8 Th Χ disribuion has dgrs of frdom bcaus hr ar fiv cagoris 5 =. Th criical valu for a chi-squar s wih dgrs of frdom and 5% lvl of significanc is 9.9. inc, chi-squar calculad is grar han abulad, h hypohsis is rjcd. [5] Pag 7 of

8 IAI CT3 Novmbr ol. a L L Hnc, is h maximum lilihood simaor of b Pu and, X = => y = and x = => y = - = = Hnc,. [8] Pag 8 of

9 IAI CT3 Novmbr ol.3 Rsaing h abl o b h Ciy Auomobil Club s cos afr % paymn by h auo ownr: Towing Cos x px 7 5% 9 % % Thn EX = *7 +.*9 +.* = 86. EX = *7 +.*9 +.* = VarX = =.6 L N b h numbr of owings ach yar. As N is Poisson disribud, EN = VarN = L b h aggrga owing cos, hn E = EXEN = 86.* = 86, Var = ENVarX + EX VarN = * * = 7,95,6 Pr > 9 = [5] ol. a Hypohss ar H : µ TD = µ LAC = µ VEG h man proin ina is sam on all hr dis H : a las wo µ s ar diffrn W hav, G = 3 groups and n = 6 daa poins. o h dof ar G-=, n-g=3 and n-=5. B = *75 + *57 + 6*7 /6 = W = = 3695 MB = B / G = 63.8 Pag 9 of

10 IAI CT3 Novmbr MW = W / n G = 6.65 F = MB / MW =.3 From abls, F, 3, 95% = 3. inc F obsrvd is grar han abulad, H is rjcd and h dis do no all hav h sam Proin ina rjc vn a % b 95% confidnc inrval is givn by: y.5%,3 n.5%, 3 =.69 TD: LAC: VEG: Th man proin ina on h sandard di TD appars o b highr han ha on ohr wo dis as h confidnc inrvals do no ovrlap. Howvr, h CIs for laco-vgarian and sric vgarian di do ovrlap so w canno b sur hir proin ina diffrs. c 95% confidnc inrval is givn by: TD vs LAC: TD vs VEG: VEG vs LAC: Pag of

11 IAI CT3 Novmbr Th wo inrvals involving h andard di do no conain, maning w can b sur ha h proin ina is highr on h sandard di han on ihr of h wo vgarian dis. Howvr, h CI for h wo vgarian dis conains, so w do no hav sufficin vidnc o say ha h proin ina on hs wo dis is diffrn. To do hypohsis ss, w considr wo dis a on im for any diffrnc in h man proin ina. TD vs LAC: H : µ TD = µ LAC h proin ina on h sandard and laco dis is sam H : h wo µ s ar diffrn h proin ina on h sandard and laco dis is no h sam From abls, = =.69 calc.5%,3 Thrfor, null hypohsis is rjcd. TD vs VEG: H : µ TD = µ VEG - h proin ina on h sandard and vgarian dis is sam H : h wo µ s ar diffrn h proin ina on h sandard and vgarian dis is no h sam From abls, = =.69 calc.5%,3 Thrfor, null hypohsis is rjcd. LAC vs VEG: H : µ LAC = µ VEG - h proin ina on h laco and vgarian dis is sam H : h wo µ s ar diffrn h proin ina on h laco and vgarian dis is no h sam From abls, = =.69 calc.5%,3 Thrfor, w do no sufficin vidnc o rjc H. [5] Pag of

12 IAI CT3 Novmbr ol Toal x y ,737 xy, 6 9,5 6,,,5 96 7,6 7, 37,8 8,56 x,6,,,,5 8, 6, 6,9 9,5 xy = x i y i x i y i /n = * 737/ = 395 xx = x i x i /n = / = 5 = xy / xx =.7 = y -- - bx -- = 737/.7 * 85/ = 7. y = x b Gradin rprsns h amoun of hours pr rup spn [5] ********************************** Pag of