UNIVERSITY OF DUBLIN TRINITY COLLEGE. Faculty of Engineering, Mathematics and Science. School of Computer Science & Statistics

Transcription

1 UNIVERSI OF DUBLIN RINI COLLEGE Facult of Engineering, Mathematics and Science School of Comuter Science & Statistics BA (Mod) Maths, SM rinit erm 04 SF and JS S35 Probabilit and heoretical Statistics Professor John Haslett Date Venue ime Instructions to Candidates: Answer an questions. All questions carr equal marks Materials ermitted for this examination: Non-rogrammable calculators ma be used Mathematical ables and Cambridge Statistical ables are available from the invigilator. A table of formulae for useful distributions is aended. Page of

2 Question he first sreadsheet on the left below illustrates the Accetance Rejection algorithm for a air of ver simle discrete distributions with mf f (x) and f () as shown. a) Illustrate the algorithm b comleting the emt boxes at the bottom right (transferring our answers to the exam booklet). Exlain the logic of each ste. b) Exlain the theoretical basis for the algorithm in terms of general mf f (x) and f (). c) Extend this theor to continuous multivariate random variables with df f (x) and f (). d) Illustrate the continuous multivariate case using the second sreadsheet, below on the right, where the initial samle airs have been drawn indeendentl from the U(0,) distribution. Comlete the entries in the box (transferring our answers to the exam answer booklet). What is the theoretical accetance rate in this case? e) Derive b calculus the theoretical equivalents of the samle summaries shown. Sreadsheet for Question (a) Sreadsheet for Question(d) f For f Prob VLOOKUP Max Acc =x =x Poss cum.50 =x &Acc Acc vals dist 0 Dist ratio r Pr(Acc) 0.4 arget df 4 / 5 ( + + ) on (0,) Avg SD Corr -0.0 x x ratio r(x) Z for ARAcc RUE FALSE FALSE RUE Z for scaled Z for x ratio AR Acc? b AR RUE FALSE Rej FALSE Rej RUE FALSE Rej FALSE Rej Page of

3 Qb from Lecture Week 3 heor,arget Pr( w) Initial Pr( w) ; Note Poss Poss w " Samle Sace" for f. ;so we 'thin' these randoml. But it Choose Pr(Acc w) r( w) k somek s. t. r( w) doesn't give enough of ' '; so we never 'thin' these. We create robabilities Pr(Acc) r( w) k k rx ( )based on the ratio to selectivel thin. w hus Pr( w Acc) choose as large as oss, st.. rw ( ) k M max w Intuitivel f. rooses too man of ' ' Recall: this involves rescaling! eg ratio ratio().5 For the discrete case there are alwas simler algorithms; but the maths is eas. Note k M valid, but unnecessaril inef Qc arget df Initial df ficient For 0 Pr ( w, w ) & Acc r( w)... he arguments for the cts case are effectivel the same, roviding we use robabilities f (.) Choose Pr(Acc w) r( w) k somek s. t. r( w) Pr(Acc) r( w) ( ) f w k dw k w w Pr (, ) Acc ( ) w w f w For 0 df Acc Q(d) Accetance Rate E r( ) max( ratio) E f ( ) max( ratio) Q(e) Marginal densit of i f ( ) f (, ) d 3 ; also densit of ; ; E ; E 0.43; Var 0.073; SD 0.7; also E Cov Corr 5 Page 3 of

4 Question he continuous random variable follows triangular distribution on (0,). a) Write down and sketch the df and the cdf for. b) Exlain in general the inverse cdf method for samling from a continuous distribution. c) Illustrate our theor b reference to the triangular distribution. Draws from the U(0,) distribution ield values 0.5, 0.9 and What are the corresonding values of? d) Contrast with the inverse cdf method for a discrete distribution. Illustrate using the Binomial distribution with arameters and n =, that is B(,) e) Show theoreticall b convolution that an alternative wa to samle from is to form the sum of two indeendent random draws from U(0,). Page 4 of

5 a df f ( ) for 0 ; ; cdf F ( ) for 0 ; ; b ransform F Z c F Pr Pr F Z Pr Z F F for 0 : z z F z when z F ; z z z F z 0.5 (0.5).656 z 0.9 ( 0.9).580 z ( 0.643).55 d Ke Ste function ~ B(, ) 0 Pr F 0 0 F F F 0 when z e If Z, Z inde U(0,), then the event Z Z has Pr given b shaded area. When, Pr ; else Pr with triangular df b differentiation. Equiv Pr Pr Z Z Pr Z Z Pr Z z dz z (ie convolution) with the limits of integration as in the diagram; see Lectures Week 4,4 Page 5 of

6 Question 3 a) Samles from the Bivariate Normal distribution can be simulated as linear combinations of airs of indeendent standard univariate Normal random variables =μ +A, as in the examle to the left. Exlain the imlementation of the algorithm b comleting the blank cells (transferring our answers to the exam answer booklet).. b) Exlain wh the given matrix A generates samles corresonding to the variance matrix shown. c) Provide the theoretical basis for this algorithm b considering the joint df of the vector and hence that of the vector. he simlest form of the bivariate normal densit is as shown below, as is the corresonding matrix A. f, ex ; A, 0 Sreadsheet for BVN Exam 04 mu Cholesk AA A iid N(0,) d) For the simlest case in (c) above, discuss the conditional distribution of given that has the secific value. ou ma note the identit below. e) Extend the result in (d) to the general Bivariate Normal case discussed in (a). Page 6 of

7 From Multivariate Normal Revision notes and S 35 Of course smmetric = 3 b Var A AA if Var( ) I or Var A AA AA AA 3c Simlest BVN iid N(0,), x df f x, x ex x x ex x x x x For an A, the linear transformation to A, whence A,leads to f, ex A A ex A A ex with A A AA. Here AA ; AA 3d Simlest BVN df f Noting, ex df f, ex ex Cond df f f, ex f, N E Var Page 7 of

8 3e GeneralBVN df f, ex Make substitutions W, W ie the simlest case W W w N, E W W w w ; VarW W w Since w N, E W ie E and Var Page 8 of

9 Question 4 Certain mushrooms are fatal to squirrels if (and onl if) more than half is consumed. One squirrel consumed a roortion P of such a mushroom; a second consumed a roortion of what remained, that is a roortion P =Q(- P ) of the mushroom. We are here concerned with the distributions of P and P and the robabilit of death for either squirrel. In the simlest case P and Q are indeendentl distributed as a) Exlain the stud of the simle case b Monte Carlo methods. Use the sreadsheet framework shown. Illustrate using some of the calculations. Simle: both uniform: = z, = z avg var cov heor E[] Var Cov b) he joint df can be written as the roduct of the conditional distribution P given P and the marginal distribution of. Use the simle uniform distributions above to derive from this: Z Z P P (i) the marginal distributions of P and P (ii) the corresonding exected values and variances; (iii) the correlation between P and P ; and (iv) the conditional distribution of P given the value of P. ou ma find it useful to note that: d dx k k k k x ln( x) x ( k ) x ln( x) c) o simulate from the conditional distribution P given that P takes the secific value it is sufficient: to first simulate from P and hence b Imortance Re-samling to use the conditional distribution of P given P., as in the sreadsheet shown. Exlain, relating to our answer in art a. Samling from cond dist b SIR Samle P; resamle with wts ro to /(-) for <-; else zero wt Given P= 0. sum= 66.5 wt ro rescale cum wts P to E Page 9 of

10 4b from Lectures Week 8 Joint and marginal dist P f 0 P P PP P P P 0 ln 0 f 0 f, 0 ;0 f d of course F f d ln F P Hence ; VarP E P ln of course ln 0 ; E P d E P ln d ln ; Var P 9 6 E PP d d d 0 Cov P, P 4 Alt E P E Q P E Q E P E P E Q P E Q E P 3 3 etc Cond Dist P P, ln f f PP 0 P P f P ln f, ln( ) F 0 PP P P f P Page 0 of

11 Formulae for Discrete Dists in Exam Bernoulli: ~ Ber( ) Poss are 0, Pr( 0) ;Pr( ). E[ ] ; Var[ ] ( ) Discrete Uniform: ~ U ( a, b) Poss are integers a, a,..., b Pr( ) ( ba ) E[ ] ( a b); Var[ ] ( b a ) Binomial: ~ B( n, ) Poss are integers 0,,..., n n n Pr( ) ( ) E[ ] n; Var[ ] n( ) Poisson: ~ P( ) Poss are integers 0,,... Pr( ) e! E[ ] ; Var[ ] Geometric: ~ Geo( ) Poss are integers,... Pr( ) ( ) E [ ] ; Var[ ] ( ) Formulae for Continuous Dists in Exam Continuous Uniform: ~ U ( a, b) Poss are a b df f ( ) a b ba cdf F( ) a b E[ ] ( a b); Var[ ] ( b a) Exonential ~ Ex( mean ara ); Poss 0 ba ( ) 0 df f e cdf F( ) e 0 E[ ] Var[ ] Alt Exonential notation ~ Ex( rate ara ) corresonding to Poisson dist df f ( ) e ; 0 cdf F( ) e ; 0 E[ ] Var[ ] Normal Distribution ~ N, Poss df f ( ) e cdf F( ) no formula but F( ) where is tabulated E[ ] Var[ ] UNIVERSI OF DUBLIN 04 Page of