SOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz

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1 STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems Pove (show) tht. ( Pscl s equtio )..

2 A bo of cdy hets cotis hets, of which 9 e white, e t, e pi, e puple, e yellow, e oge, d 6 e gee. If you select 9 pieces of cdy domly fom the bo, without eplcemet, give the pobbility tht ) Thee of the hets e white.,,86,69,,,, b) Thee e white, e t, is pi, is yellow, d e gee.,6,66,69,69,,

3 . Pete tes ompute Sciece clsses, though ot to le, but to meet smt gils. Thee e othe studets i the clss with Pete, of them e gils. Duig the semeste, studets will be woig o poject i tems of studets. Suppose the studets e divided ito tems t dom. Pete s tem Pete ( domly chose ) studets. Studets e selected t dom without eplcemet. ) Fid the pobbility tht t lest out of studets o Pete s tem e gils b) Fid the pobbility tht thee is t lest gil o Pete s tem OR c) Fid the pobbility tht t most out of studets o Pete s tem e gils OR

4 . A smll gocey stoe hd ctos of mil, of which wee sou. ) If Dvid is goig to buy the sith cto of mil sold tht dy t dom, compute the pobbility tht he selects cto of sou mil. sou fesh, fist : P fesh,sou fist : P sou fesh, fist : P b) If si ctos of mil e sold tht dy t dom, wht is the pobbility tht ectly oe cto of sou mil is sold

5 . Suppose the umbe of boes of Hmmemill ppe used by Aytow ollege Sttistics & Pobbility Deptmet ech moth is dom d hs the followig pobbility distibutio: f () f () f () Suppose t the ed of ech moth the deptmet odes the sme umbe of boes s ws used duig the moth. Suppose ech bo costs $. The deptmet hs to py $ delivey fee (the delivey fee does ot deped o the umbe of boes odeed). The the mothly ppe bill is Y X. Fid Aytow ollege Sttistics & Pobbility Deptmet s vege mothly ppe bill d its stdd devitio. E ( X ) ll f ( ).. [ E( X) ] f ( ) [ E( X ] V ( X) E( X ) ) ll SD ( X ).... (. ).. E ( Y ) E ( X ). $6. SD ( Y ) SD ( X ). $.

6 6. Suppose we oll pi of fi 6-sided dice. Let X deote the mimum (the lgest) of the outcomes o the two dice. ostuct the pobbility distibutio of X d compute its epected vlue. (, ) (, ) (, ) (, ) (, ) (, 6 ) (, ) (, ) (, ) (, ) (, ) (, 6 ) (, ) (, ) (, ) (, ) (, ) (, 6 ) (, ) (, ) (, ) (, ) (, ) (, 6 ) (, ) (, ) (, ) (, ) (, ) (, 6 ) ( 6, ) ( 6, ) ( 6, ) ( 6, ) ( 6, ) ( 6, 6 ) f ( ) f ( ) / 6 / 6 / 6 6 / 6 / 6 / 6 / 6 8 / 6 9 / 6 / 6 6 / 6 66 / 6 E ( X ) 6 /6..

7 . oside f ( ) c ( ),,,,. ) Fid c such tht f ( ) stisfies the coditios of beig p.m.f. fo dom vible X. f ( ) f ( ) f ( ) f ( ) f ( ) c c 9 c 6 c c. ll c. b) Fid the epected vlue of X. E ( X ) ll f ( ) c) Fid the stdd devitio of X. E ( X ) ll f ( ) V ( X ) E ( X ) [ E(X) ] SD ( X ) V ( X ).8.

8 8. ) Let X be discete dom vible with p.m.f. f ( ) c,,,,, 6,, whee c ( ). Recll ( Homewo # Poblem ): this vlid pobbility distibutio. Fid µ X E ( X ). E ( X ) f ll ) (. E ( X )... 6 E ( X )... E ( X ) E ( X ). Theefoe, E ( X ). OR E ( X ) f ll ) (

9 ( ) ( ) E ( Y ), whee Y hs Geometic distibutio with pobbility of success p. E ( X ) E ( Y ). b) Let X be discete dom vible with p.m.f. f ( ) c,,,,, 6,, whee c. e Recll ( Homewo # Poblem 8 ): this vlid pobbility distibutio. Fid µ X E ( X ). E ( X ) f ( ) ll e e e e ( ) ( e ) e.96.

10 9. A oil compy believes tht the pobbility of eistece of oil deposit i ceti dillig e is.. Suppose it would cost $, to dill well. If oil deposit does eist, the compy s pofit will be $, (the dillig costs ot icluded). A seismic test tht would cost $, is beig cosideed to clify the lielihood of the pesece of oil. The poposed seismic test hs the followig elibility: whe oil does eist i the testig e, the test will idicte so 9% of the time; whe oil does ot eist i the test e, % of the time the test will eoeously idicte tht it does eist. Thee e fou possible sttes of tue : θ oil deposit does eist d the test esult is positive, θ oil deposit does eist, but (d) the test esult is egtive, θ oil deposit does ot eist, but (d) the test esult is positive, θ oil deposit does ot eist d the test esult is egtive. The compy c te two possible ctios: dill well without pefomig the test, pefom the test d dill well oly if the test shows pesece of oil. 9. ) Fid the pobbilities of ll fou sttes of tue. Tht is, fid P( θ ), P( θ ), P( θ ), d P( θ ). P( Oil )., P( Oil ).9, P( No Oil ).. P( θ ) P( Oil ) P( Oil ) P( Oil )..9.. P( θ ) P( Oil ) P( Oil ) P( Oil ).... P( θ ) P( No Oil ) P( No Oil ) P( No Oil ).... P( θ ) P( No Oil ) P( No Oil ) P( No Oil ) b) Suppose the test shows pesece of oil. Wht is the pobbility tht oil deposit does eist? P( Oil ) P( Oil ) P( )

11 . c) ostuct the pyoff tble (pofit tble) fo this poblem. Tht is, fid the compy s pofit fo ech possible ctio d ech possible stte of tue. θ θ θ θ Oil Oil No Oil No Oil,,,,,, dill w/o test 6, 6,,,,,,,,,, dill oly if 8,,,, d) Fid the epected pyoff (epected pofit, EP) fo both ctios d detemie the optiml ctio. EP( ) 6,. 6,. (,). (,).6 $,. EP( ) 8,. (,). (,). (,).6 $8,. Optiml ctio (pefom the test d dill well oly if the test shows pesece of oil) is the optiml ctio, it hs highe epected pyoff.

12 Fo fu: Suppose the pobbility of eistece of oil deposit i ceti dillig e is uow, p. P( θ ) P( Oil ) P( Oil ) P( Oil ) p.9. P( θ ) P( Oil ) P( Oil ) P( Oil ) p.. P( θ ) P( No Oil ) P( No Oil ) P( No Oil ) ( p ).. P( θ ) P( No Oil ) P( No Oil ) P( No Oil ) ( p ).8. θ θ θ θ Oil Oil No Oil No Oil,,,,,, dill w/o test 6, 6,,,,,,,,,, dill oly if 8,,,, do othig EP( ) 6, p.9 6, p. (,) ( p ). (,) ( p ).8, p,. EP( ) 8, p.9 (,) p. (,) ( p ). (,) ( p ).8 6, p,. EP( ).

13 EP( ) EP( )., p, 6, p,., p 6,. p /. EP( ) EP( )., p,. p /. EP( ) EP( ). 6, p,. p /. If p < /, is optiml. If / < p < /, is optiml. If p > /, is optiml.