THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA STATISTICAL THEORY AND METHODS PAPER I

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1 THE ROYAL STATISTICAL SOCIETY 5 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA STATISTICAL THEORY AND METHODS PAPER I The Society provides these solutios to assist cadidates preparig for the examiatios i future years ad for the iformatio of ay other persos usig the examiatios The solutios should NOT be see as "model aswers" Rather, they have bee writte out i cosiderable detail ad are iteded as learig aids Users of the solutios should always be aware that i may cases there are valid alterative methods Also, i the may cases where discussio is called for, there may be other valid poits that could be made While every care has bee take with the preparatio of these solutios, the Society will ot be resposible for ay errors or omissios The Society will ot eter ito ay correspodece i respect of these solutios Note I accordace with the covetio used i the Society's examiatio papers, the otatio log deotes logarithm to base e Logarithms to ay other base are explicitly idetified, eg log RSS 5

2 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio (i If {E, E,, E } partitio S, the P( A P( A E P( This is the law of total probability Sice P( A E P( A E P( E P( E A P( A j j j j i i E i, we have (Bayes' Theorem ( j A P E ( j P( Ej P A E P A ( i ( j P( Ej P A E ( P( E P A E i i (ii parameter 3x Let Y be the amout of time spet i the fittig room; Y is expoetial with PY y y < y x t 3 e dt x e e t /3 /3 /3 (a ( t x t x y x (for y > Sice x takes the values,, 3, 4 each with probability ¼, we therefore have y y y y ( ( F y e + e + e + e (usig (i 4 /3 /6 /9 / 4 e e e e 5/3 5/6 5/9 5/ Whe y 5 this is ( ( ad so P(Y > Solutio cotiued o ext page

3 (b Let X be the umber of garmets take to the room The 5 E( X ( , 4 E( X ( , 4 so Var(X [These results may be quoted, as X has a discrete uiform distributio] Now, E( Y X 3 X Also, because Y has a expoetial distributio, ( ( YX X X Var 3 9 Thus 5 E( Y E{ E( Y X } E{ 3X} 3E( X Also, Var ( Y E { Var ( Y X } + Var{ E ( Y X } { 9 } Var{ 3 } E X + X ( ( X 9E X + 9Var

4 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio (i X + Y ca take the values,,,, + m For these values, PX ( + Y PX ( xad Y x x x x PX ( xpy ( x ( θ θ m x x m+ θ x ( ( x x m + x θ θ θ θ m+ x ( θ m x x m+ Thus X + Y has the biomial distributio with parameters m + ad θ [A alterative method is to use probability geeratig fuctios] (ii P( X x X + Y PX ( x X+ Y PX ( + Y PX ( x Y x PX ( + Y m θ ( θ θ ( θ x x m+ m+ θ ( θ m x x m+ x x x m +x (ie a hypergeometric distributio Solutio cotiued o ext page

5 (iii Let X ad Y be the umbers of failed compoets i the two etworks We have, m 3, θ, 6 i the above otatio 3 P(X 3 X + Y

6 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio 3 ( f x, y x (< x< y< (i ( r s + r s E X Y x y dydx x y x s+ + r y x s + y x s + dx + r s+ x ( x dx r+ 3 r+ s+ 4 x x s+ r+ 3 r+ s+ 4 s+ r+ 3 r+ s+ 4 ( s + ( s+ ( r+ 3( r+ s+ 4 ( r+ 3( r+ s+ 4 Hece 3 E( X E( Y (put r, s ; similarly for the others 3 E( X, so Var( X E( Y, so Var( Y (, so Cov ( X, Y E XY Cov ( XY, 3 ρ 5 XY Var ( X Var ( Y 5 75 Solutio cotiued o ext page

7 (ii ( { x y x+ } regio show: PY X xdy dx >, the evaluatio beig over the shaded y - x This is x [ y] dx x ( x d x+ x x x { x ( x } dx ( 3 4 ( ( ( 4 Therefore F( ( 4 ( for ad f ( F' ( 4( 3 ( for

8 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio 4 (i U X X + Y, V X + Y So X UV ad Y ( UV The Jacobia of the trasformatio is x u y u x v y v v u v( u + uv v u v The joit pdf of X, Y is α+ β α β θ( x+ y θ x y e f ( xy,, for x>, y> Γ( α Γ( β Hece the joit pdf of U, V is ( α+ β α β β θv θ uv ( u v e v guv (, f( xy, J (forv>, < u< Γ( α Γ( β α+ β θ Γ( α Γ( β This is of the form of a product { ( }{ α β α+ β θv u u v e costat fuctio of u aloe [g(u, say] fuctio of v aloe [h(v, say] ad so U, V are idepedet g(u is proportioal to u α ( u β, the pdf of a beta distributio, ad so U has a beta distributio h(v is proportioal to v α+β e θv, the pdf of a gamma distributio, ad so V has a gamma distributio The scale parameter of V is θ, as for X ad Y } X (ii U is the required distributio, where X ad Y are the commo X + Y expoetial radom variables Takig α β, g(u u ( u ad so U has the uiform distributio o (,

9 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio 5 πx π tx tx x / / x( (i ( ( t M X t E e e e dx x e dx t < ½ is used i what follows to esure covergece of the itegral Write ( u x t, so that du ( t dx The / u u M X ( t e π du t t / u u e d π t u The itegral here should be recogised as Γ(½ π ; alteratively, refer back to the origial pdf t M X( t t, so M ' X t t t ( ( ( ( ( ( ( E X M ' X / 3/ 3/ 3 M '' X t t 3 t ( ( ( ( ( M X ( E X '' 3 ( X E( X { E( X } 5/ 5/ Var 3 Solutio cotiued o ext page

10 (ii M X + + X ( t { M X ( t } ( / t Now usig M ( t e bt M ( at ay b Y +, t t M ( t e / To fid the limitig form of M ( t, we take logs: Z log MZ ( t t log t 3 t t t t t + t + t as ( / t So M t e as, which is the mgf of N(, Z Therefore i the limit Z becomes N(,, ie the stadard Normal distributio

11 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio 6 (i I a U( θ, θ distributio, f (x θ ad F(x x +, for θ < x < θ θ F(u (, u ( P(U ( u ( P(U ( > u ( ad U ( u ( P(all data u ( P(all data betwee u ( ad u ( {F(u ( } {F(u ( F(u ( } u( u( u(, for θ < u ( < u ( < θ θ θ f (u (, u ( u u ( ( F(u (, u ( ( ( u u ( ( ( θ [A argumet usig the multiomial distributio with oe observatio at each of u ( ad u ( ad with i betwee is also acceptable] (ii Trasformig to R U ( U ( ad T U ( (so that U ( R + T, we have the Jacobia J ( u( u(,, so J ( rt, ( r Hece f(r, t θ ( (for θ < t < θ r, < r < θ f ( r θ r θ ( r ( θ dt r θ r < < θ ( (, for r ( θ Solutio cotiued o ext page

12 (iii ( ( θ E R r ( θ r dr ( θ ( r r dr ( θ θ ( θ + ( θ r r + ( θ ( ( θ θ + + ( θ ( + θ Hece R is a biased estimator of θ (but asymptotically ubiased as

13 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio 7 (i (a The iverse cumulative distributio fuctio method ca be used with tables of the stadard Normal cdf Φ(x The values of are such that Φ( u, ad for the four values of u the correspodig values of are 7, 4, +46, +4 (b (c These ca be trasformed to N(, 8 by w µ + σ or w + 9, to give 963, 378, 586, 74 The chi-squared distributio with oe degree of freedom is the square of N(,, so take values of from (i: 4, 8,, 96 (ii The probabilities ad cumulative probabilities for a Poisso distributio with mea are: r P(r F(r Taxis: 553 correspods to r (it is betwee 46 ad 6767 etc, givig, 3,, 5, Similarly for customers: 3,,,, Time Taxis Arrivals Customers Arrivals

14 Graduate Diploma, Statistical Theory & Methods, Paper I, 5 Questio 8 (i (a If C α β α β C α + β the CDC α β α α + β β α β ( + ( α α β β α α + β β α β α + β α αβ α + αβ α + β αβ + β α + β αβ β α α β β P (b The -step trasitio matrix is P, which ca be writte CDC CDC CDC CDC ad every pair C C is replaced by I ( ( ( ( to give CD C D is simply, ie ( α β λ i the give otatio Sice < α < ad < β <, we have < λ <, ie λ < ; therefore λ Thus P C C which is α β α β α β α β α β α β α β β α Solutio cotiued o ext page

15 (ii Let state be o rai ad state be rai The trasitio matrix is 8 P 9 This is the matrix i (i with α ad β 9 As there is o rai o the first visit, P gives the probabilities for the two states o the ext visit i days' time As is large, P ca be take as approximately equal to the limitig value i (i(b, ie here This gives 9 9 [ ] 9 P, ie P(rai Replacig with for the first visit gives the same aswer because of the form of P I the log ru there are about 9 days without rai for every days with rai

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber