SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
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1 STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random variabls X and Y wih join p.d.f. f (, ) 8 < < K, ohrwis < < K a) Find h valu of K so ha f (, ) is a valid join p.d.f. K K 8 d d K. K. b) Find P ( X > Y ). P ( X > Y ) d 8 d d 79. P ( X > Y ) d 8 d.
2 c) Find P ( X Y > ). d d P ( X Y > ) 8 9 ( ) 8 d 8 d d d P ( X Y > ) 8 ( ) d 8 9 d d) Ar X and Y indpndn? If no, find Cov ( X, Y ). f X ( ) d 8, < <, 9 f Y ( ) d 8 9, < <. f (, ) f X ( ) f ( ). X and Y ar indpndn. Cov ( X, Y ). Y
3 . L X dno h numbr of ims a phoocop machin will malfuncion:,,, or ims, on an givn monh. L Y dno h numbr of ims a chnician is calld on an mrgnc call. Th join p.m.f. p (, ) is prsnd in h abl blow: p Y ( ) p X ( )..... a) Find h probabili P ( Y > X ). P ( Y > X ) p (, ) p (, ).... b) Find p X ( ), h marginal p.m.f. of X. c) Find p Y ( ), h marginal p.m.f. of Y. d) Ar X and Y indpndn? If no, find Cov ( X, Y ). X and Y ar NOT indpndn. E ( X ) E ( Y ) E ( X Y ) Cov ( X, Y ) E ( X Y ) E ( X ) E ( Y )...7..
4 . L h join probabili dnsi funcion for ( X, Y ) b f (, ), >, >, zro ohrwis. <, a) Find h probabili P ( X < Y ). inrscion poin: and and d d P ( X < Y ) 9 7 d. P ( X < Y ) ( ) d d 7 d. 8 9 b) Find h marginal probabili dnsi funcion of X, f X ( ). f X ( ) d 9, < <.
5 c) Find h marginal probabili dnsi funcion of Y, f Y ( ). ( ) f Y ( ) d, < <. d) Ar X and Y indpndn? If no, find Cov ( X, Y ). Th suppor of ( X, Y ) is NOT a rcangl. X and Y ar NOT indpndn. f X, Y (, ) f X ( ) f Y ( ). X and Y ar NOT indpndn. E ( X ) f X ( ) d d 9 d Y d 9 d. 8 E ( Y ) f ( ) d d d d E ( X Y ) ( ) ( ) d. Cov ( X, Y ) E ( X Y ) E ( X ) E ( Y ).. 8
6 . L h join probabili dnsi funcion for ( X, Y ) b f (, ) a) Find h probabili P ( X > Y )., < <, < <, zro ohrwis. P ( X > Y ) d d d d. P ( X > Y ) d d P ( X > Y ) d d d d b) Find h marginal probabili dnsi funcion of X, f X ( ). f X ( ) d, < <. c) Find h marginal probabili dnsi funcion of Y, f Y ( ). f Y ( ) d, < <.
7 d) Ar X and Y indpndn? If no, find Cov ( X, Y ). Sinc f (, ) f X ( ) f ( ), X and Y ar NOT indpndn. Y X E ( X ) f ( ) d d. 9 9 Y E ( Y ) f ( ) d d. 9 9 E ( X Y ) d d 9 d. 8 8 Cov ( X, Y ) E ( X Y ) E ( X ) E ( Y )
8 . Two componns of a lapop compur hav h following join probabili dnsi funcion for hir usful lifims X and Y (in ars): f (, ) ( ), ohrwis a) Find h marginal probabili dnsi funcion of X, f X ( ). f X ( ) ( ) d d,. b) Find h marginal probabili dnsi funcion of Y, f Y ( ). f Y ( ) ( ) d ( ),. c) Wha is h probabili ha h lifim of a las on componn cds ar (whn h manufacurr s warran pirs)? d d d d d ( ) d P( X > Y > ) P( X Y ) ( ).8. P( X > Y > ) P( X > ) P( Y > ) P( X > Y > )
9 . L h join probabili dnsi funcion for ( X, Y ) b f (, ), < <, < <, zro ohrwis. a) Find h marginal probabili dnsi funcion of X, f X ( ). If <, If >, f X ( ) f X ( ) d d, <., >. f X ( ) -, < <. ( doubl ponnial ) b) Find h marginal probabili dnsi funcion of Y, f Y ( ). f Y ( ) d, < <. ( Gamma, α, θ ) c) Ar X and Y indpndn? If no, find Cov ( X, Y ). Th suppor of ( X, Y ) is NOT a rcangl. X and Y ar NOT indpndn. f X, Y (, ) f X ( ) f Y ( ). X and Y ar NOT indpndn.
10 E ( X ), sinc h disribuion of X is smmric abou. E ( Y ), sinc Y has a Gamma disribuion, α, θ. E ( X Y ) d d Cov ( X, Y ) E ( X Y ) E ( X ) E ( Y ). d d. Rcall: Indpndn Cov Cov X Indpndn
11 7. Suppos Jan has a fair -sidd di, and Dic has a fair -sidd di. Each da, h roll hir dic a h sam im (indpndnl) unil somon rolls a. (Thn h prson who did no roll a dos h dishs.) Find h probabili ha p J ( ),,,,, p D ( ),,,,. a) h roll h firs a h sam im (afr qual numbr of amps); ( ) ( ) D J p p n n 9. ( J D ) or ( J' D' ) ( J D ) or ( J' D' ) ( J' D' ) ( J D ) or... 9.
12 b) Dic rolls h firs bfor Jan dos. ( ) ( ) D J m p m p m m 8 n n 8. ( J' D ) or ( J' D' ) ( J' D ) or ( J' D' ) ( J' D' ) ( J' D ) or....
13 8. Dic and Jan hav agrd o m for lunch bwn noon (: p.m.) and : p.m. Dno Jan s arrival im b X, Dic s b Y, and suppos X and Y ar indpndn wih probabili dnsi funcions f ( ) f ( ) X ohrwis Y ohrwis a) Find h probabili ha Jan arrivs bfor Dic. Tha is, find P ( X < Y ). f (, ),,. P ( X < Y ) d d d d d ( ) d. P ( X < Y ) d d ( ) d d d. ( ) d
14 b) Find h pcd amoun of im Jan would hav o wai for Dic o arriv. Hin : If Dic arrivs firs ( ha is, if X > Y ), hn Jan s waiing im is zro. If Jan arrivs firs ( ha is, if X < Y ), hn hr waiing im is Y X. Hin : E ( g( X, Y ) ) g (, ) f (, ) d d f (, ),,. > Jan is waiing for Dic. Jan s waiing im > Dic is waiing for Jan. Jan s waiing im d d ( ) d d d d d d d. d hour minus.
15 9. Suppos ha ( X, Y ) is uniforml disribud ovr h rgion dfind b and. Tha is, f (, ) C,,, zro lswhr. a) Wha is h join probabili dnsi funcion of X and Y? Tha is, find C. d d ( ) d. f X, Y (, ),,. b) Find h marginal probabili dnsi funcion of X, f X ( ). f X ( ) d ( ),. c) Find h marginal probabili dnsi funcion of Y, f Y ( ). ± f Y ( ) d,.
16 . L T, T,, T Suppos E ( T i ), i,,,. i b indpndn Eponnial random variabls. Tha is, f T i( ) i i, >, i,,,. Dno T min min ( T, T,, T ). a) Show ha T min also has an Eponnial disribuion. Wha is h man of T min? Hin: Considr P ( T min > ) P ( T > AND T > AND AND T > ). Sinc T, T,, T ar indpndn, P ( T min > ) P ( T > AND T > AND AND T > ) P ( T > ) P ( T > ) P ( T > ) ( ), >. F T min ( ) P ( T min ) P ( T min > ) ( ), >. f T min ( ) ( ) ( ), >. T min has an Eponnial disribuion wih man....
17 b) Find P ( T T min ) P ( T is h smalls of T, T,, T ) P ( T < T AND AND T < T ). Hin : A good plac o sar is o considr T, T and show ha P ( T < T ). P ( T < T ) d d d. Sinc T, T,, T ar indpndn, hir join probabili dnsi funcion is f (,,, )..., >, >,, >. P ( T T min ) P ( T < T AND AND T < T ) d d d... d d d... d....
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