THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA

Transcription

1 THE ROYAL STATISTICAL SOCIETY 3 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER I STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the formato of ay other persos usg the examatos The solutos should NOT be see as "model aswers" Rather, they have bee wrtte out cosderable detal ad are teded as learg ads Users of the solutos should always be aware that may cases there are vald alteratve methods Also, the may cases where dscusso s called for, there may be other vald pots that could be made Whle every care has bee take wth the preparato of these solutos, the Socety wll ot be resposble for ay errors or omssos The Socety wll ot eter to ay correspodece respect of these solutos RSS 3

2 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo x ( P ( X = x = P( X = x, Y = y y= x! x y = x! y! x y! θ θ θ θ y= ( ( x y x! x x y x y ( x! ( x! θ y y θ θ θ = = x ( x x = θ θ + θ θ = θ θ x x { } whch s Bomal(, θ It follows by symmetry that Y s Bomal(, θ x ( For x =,,, y, ( =, = PY ( = y x y θθ ( θ θ P X x Y y P X = xy = y = x y ( θ! y! y! = x! y! x y!! θ x y x y θ θ = x θ θ θ so that, codtoal o Y = y, X s Bomal y, θ y y, ( P(double sx = (/6 = /36 P(o sx = (5/6 = 5/36 The jot dstrbuto of X ad Y as defed s gve by the multomal wth θ = /36, θ = 5/36 Therefore by (, E(X = /36 = 5/8, sce X s Bomal(, /36 By (, the case Y =, E(X Y = = /, sce X wll be Bomal(, / (There are ways out of 36 of havg at least oe sx

3 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo ( (a The law of total probablty s P ( A P( A E P( E Bayes' Theorem states that P( Ej A = = = = ( j P( Ej P A E ( P( E P AE (b Sce {F j } parttos S, E ca be wrtte as the dsjot uo of evets {E ad F j } S s the dsjot uo of {E }, so S s also the dsjot uo of evets {E ad F j } Hece {E ad F j } parttos S m Now P ( A P( A E ad Fj P( E ad Fj = = j= m = j= ( j ( j ( = P AE F P F E P E ( (a P(o so haemophlac = P(o so haemophlac woma carrerp(woma carrer + P(o so haemophlac woma ot carrerp(woma ot carrer 3 9 = + = 6 /6 P(woma carrer o so haemophlac = = 9/6 9 P(daughter carrer = P(daughter carrer woma carrerp(woma carrer = = 9 8 (b P(at least oe grl carrer = P(at least oe grl carrer daughter carrerp(daughter carrer = 8 = 4

4 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo 3 ( The space where X ad Y exst jotly s ot a rectagular rego It s possble to fd pots (x, y, eg (½, ¾, where both f(x ad f(y are > but f(x, y = ; thus f(x, y f(xf(y ( x r s r+ s = 6 E X Y x y dy dx s+ ( x r = 6 x + dx s + 6 ( r+! ( s+! = usg the result quoted the paper s + r + s + 3! ( r+ s ( r+ s+ 6!! = 3! for ay o-egatve tegers r, s Hece 6!! 6 E( X = = =, 4! !! E( X = = =, so Var( X = = 5! 543 6!! E( Y = =, 4! 4 6!! 3 E( Y = =, so Var( Y = = 5! 4 8 6!! E( X, Y = =, so Cov ( X, Y = =, 5! 4 4 ad so ρ XY ( XY ( X Var ( Y Cov, 8 = = = Var w w x ( For w, ( + = 6 ( P X Y w x dy dx 6 w w = x w x dx = 6 wx x = w 3 3 3, 3 so that the cumulatve dstrbuto fucto s FW ( w w ( for w probablty desty fucto s f ( w = 3w ( for w W = ad the

5 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo 4 ( As X ad Y are depedet, f xy X Y x y πσ σ (, = ( pdf of ( pdf of = exp ( + (for < x <, < y < For X = Rcos φ, Y = Rs φ, the Jacoba s X X R φ cosφ Rsφ J = = = R( cos φ + s φ = R Y Y sφ Rcosφ R φ So the jot pdf of R,φ s ( φ r r g r, = exp, for r, φ π πσ σ φ= π ( The pdf of R s ( φ r σ r σ g r, dφ = exp, for r φ = [Ths ca also be see because R,φ exst a "rectagular" space, ad the jot pdf r r /( σ ca be wrtte as e whch factorses, so R,φ are depedet] σ π ( The cumulatve dstrbuto fucto of R s r u u r / σ w u F r = exp du e dw puttg w = = σ σ σ r /( σ = e ( σ k / F k = e, whch s to be 5 Ths gves 5 = k = log 5 = 8 k / e, or

6 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo 5 µ x t e µ µ e µ µ ( M X ( t = E e = e = e = e exp µ e x! x! x= x= x tx tx t We have [ ] = ' ad = ''( E X M X { } t t ( µ µ M ' t e exp e {( e t µ } = exp E X M X Dfferetatg M X ( t gves =, so ' ( M = µ, ad { } { } M '' t µ e t exp µ e t µ e t exp µ e t M '' = µ + µ = +, so Hece ( [ ] Var X = E X E X = µ + µ µ = µ (Note The results for E[ X ] ad expaso of M X ( t E X ca also be obtaed from the power seres X µ ( Z = = X µ, so (usg the "lear trasformato" result for µ µ momet geeratg fuctos we have { } t t µ t µ t/ µ ( = X = exp µ ( MZ t e M e e µ Takg logarthms (base e, 3 t t t log M t = t µ + µ e = t µ + µ / µ µ 6µ t / µ ( Z ( Hece MZ ( t exp ( t / 3 t = t + + t as µ 6 µ as µ, ad ths s the momet geeratg fucto of N(, Hece the lmtg dstrbuto of Z s N(, ( W = Y ad the mgf of Y s ( exp ( t = t { } { } M t = e µ By depedece, t MW ( t = exp ( e µ = exp e µ, e the same form as the orgal = = Posso mgf but wth parameter µ, so the dstrbuto of W s Posso wth parameter µ

7 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo 6 w αt αθ θ θ ( For the Webull dstrbuto, θ αw u θ = = exp( α F w e du w have h( w F w = t e dt ; put u = at θ to gve Thus, from the formula h( w θ exp( α w θ ( α w θ αθ w = = αθ w exp t decreases as w creases f θ < θ f u =, we F u Ths hazard fucto s costat f θ = ; ( G( y = P( Y y = P( X y or X y ( ad = P X y + P X y P X y X y = P X y + P X y P X y P X y by depedece = F y + F y F y F y Hece g( y G' ( y f ( y f ( y f ( y F ( y f ( y F ( y ( for y = = + ( ( g y g y g y h( y = = = G y F y F y + F y F y F y F y { } + { } { F( y }{ F( y } f y F y f y F y = = h y + h y θ If X s Webull(, h y = h y + h y = αθ y + αθ y ( α α θy θ = +, whch s the hazard fucto of Webull( α+ α, θ θ α θ, ths gves ( G(y = P(both compoets fal tme y = F (yf (y by depedece For { } detcal compoets, G( y = F( y, whch gves g ( y F( y f ( y F( y f ( y F( y f ( y F( y h( y = = Now, { F( y } { F ( y }{ + F( y } + F( y f ( y ad therefore h( y, as requred F ( y = ad so (as F(y,

8 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo 7 ( 5 45 (a = 5, so P ( = /5 =, ad smlarly P( = 5/5 5 ad P( = /5 Hece the probablty fucto (f(x ad cdf (F(x are x f(x F(x The verse cdf method produces x = f the radom umber s 486, x = f the radom umber s betwee 487 ad 948, ad x = for 949 upwards Hece we obta,,, (b F(x = x 3 (for x The verse cdf method sets u = F(x = x 3, so x = u /3 So we obta 84, 696, 996, 7894 ( Geeratg a N(9, (/ radom varable requres a N(, z, foud as Φ u, followed by a trasformato x = 9 + z For u = 5398, we get z = ad hece x = 95 For u = 337, we get z = 4 ad hece x = 879 For u = 9887, we get z = 8 ad hece x = 4 For u = 49, we get z = ad hece x = 899 Begg at am ad workg decmals of a mute, the tmes take to reach B, C, D, E wll be 95, 879, 4, 899 mutes Notce that ths meas that the bus wll eed to "wat tme" at B ad C The bus leaves B at ad C at It the leaves D at 34 mutes past, to arrve at E at 393 mutes past It wll have wated 95 mutes at B, mutes at C, ad mutes at D A sample of arrval tmes at E could be geerated ths way usg a larger smulato, ad the sample mea used to estmate the expected arrval tme The umber of tmes the sample, say, that E s ot reached utl after 4 am could be used estmatg the probablty of a late arrval: p ˆ = umber of smulatos

9 Graduate Dploma, Statstcal Theory & Methods, Paper I, 3 Questo 8 ( The states of the Markov Cha are (ot obese ad (obese If X s the p = P X = s X = r for r =, state reached at age ( =,,, years ad rs ( + ad s=,, the trasto matrx s [ ] φ φ P = p rs = θ θ ( The two-step trasto matrx s P φ φ φ φ φ + φ θ φ φ + θ = θ θ θ θ = θ φ + θ θ + φ θ All chldre are o-obese (state at years So the probablty that a chld s obese φ φ + θ (state at years s gve by the "top rght" elemet of P, e t s ( The proporto of chldre who have ever bee obese at ay stage up to ad 3 3 cludg 3 years s ( p ( φ = (v θ φ φ ( θ φ p+ = p + p = + p Isertg = the expresso gve the questo gves ( θ φ ( θ φ whch equals as requred (all chldre are o-obese at age Now supposg the result holds for p (, we have p + ( θ φ θ φ ( θ φ ( θ φ = φ + ( θ φ φ = φ + ( θ φ ( θ φ ( θ φ ( θ φ ( θ φ + = φ + = + ( θ φ ( θ φ φ Hece by ducto the result s true for all (v As creases, p φ sce θ φ <, e p ( θ φ φ ( θ φ For θ = 8 ad φ =, p =, so we expect approxmately oe-thrd of 7 3 ths adult populato to be obese