STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P ( 9 < X < ) = P < Z < = P (.3 < X <. ) è.. =.9. =.3. b) Alx boass ha h onc caugh a fish ha was jus big nough o b in h op.% of of h fish populaion. How much did his fish wigh? Wan x =? such ha P ( X > x ) =.. P ( Z >.9 ) =.. x = 8...9 = 3 lbs. c) If on cachs a fish from h boom % of h populaion, h fish mus b rurnd o h la. Wha is h wigh of h smalls fish ha on can p? Wan x =? such ha P ( X < x ) =.. P ( Z <.8 ) =.».. x = 8.. (.8 ) = lbs.
. Suppos h duraion of roubl-fr opraion of a paricular brand of rfrigraors is normally disribud wih man monhs and sandard dviaion monhs. a) If h company ha mas his paricular brand of rfrigraors wishs o s h warrany priod so ha only % of all rfrigraors would nd rpair srvics whil undr warrany, how long a warrany mus b s? Nd x =? such ha P( X < x ) =.. Find z such ha P( Z < z ) =.. Th ara o h lf is. = F ( z ). Using h sandard normal abl, z =.7. x = µ s z. x = (.7 ) = 8 monhs ( 7 yars ). b) Wha is h probabiliy ha a rfrigraor of his brand will wor for ovr 8 yars wihou roubl? 8 yars = 9 monhs. æ 9 - P ( X > 9 ) = P Z > = P ( Z >.7 ) =.773. è c) Sixn rfrigraors of his brand ar purchasd for a -uni aparmn complx. Wha is h probabiliy ha xacly ou of rfrigraors would wor for ovr 8 yars wihou roubl? (Assum indpndnc.) L W dno h numbr of rfrigraors (ou of ) ha wor for ovr 8 yars wihou roubl. Thn W has Binomial Disribuion, n =, p =.773 ( s par (b) ). ( ) ( ) P ( W = ) =.773 -.773».83.
3. Bob slls hingamabobs. His yarly salary is $7, plus a commission of % of his sals. His yarly sals ar normally disribud wih man $, and sandard dviaion $,. a) Find h probabiliy ha Bob arns ovr $, in a givn yar. L X dno Bob s yarly sals. L Y dno h amoun Bob arns in a givn yar. Thn Y =. X 7,. Sinc X has a Normal disribuion wih man $, and sandard dviaion $,, Y also has a Normal disribuion wih man., 7, = $37, and sandard dviaion., = $,. æ, - 37, P ( Y >, ) = P Z > = P ( Z >. ) =.8. è, b) Find h missing valu: Wih probabiliy 7% Bob arns ovr in a givn yar. Nd y =? such ha P ( Y > y ) =.7. Find z such ha P ( Z > z ) =.7. Th ara o h lf is.33 = F ( z ). Using h sandard normal abl, z =.. x = µ s z. x = 37,, (. ) = $3,.
. 3.3-3.- If h momn-gnraing funcion of X is M ( ) = xp ( ), find a) Th man of X. b) Th varianc of X. c) P ( 7 < X < ). d) P ( 8 < X < 7 ).. Suppos X is a normally disribud random variabl. Suppos also ha P ( X > ) =.83 and P ( X < 77 ) =.999. Find h man and h sandard dviaion of X. P ( X > ) =.83 P ( X < 77 ) =.999. P ( Z >. ) =.83 P ( Z <.7 ) =.999. µ. s = µ.7 s = 77 µ. s = µ.7 s = 77 Þ.7 s = Þ s =. Þ µ = 3.
.* L h random variabl X hav h p.d.f. x f ( x ) = -, < x <, zro lswhr. p Find h man and h varianc of X. Hin: ompu E ( X ) dircly ( afr an obvious subsiuion ). Find E ( X ) by comparing h ingral wih h ingral rprsning Var ( Z ) = E ( Z ) of a random variabl Z ha is N (, ). E ( X ) = ò x p - x dx = x u = du = x dx -u = ò du = =. p p p E ( X - x -x ) = ò x dx = ò x dx = E ( Z ) =. p - p ( Z ~ N (, ) ) OR E ( X - x x ) = ò x dx = u = du = x dx p -u -u = ò u du = ò u du = p p æ = G =. p è æ G p è 3 Var ( X ) = E ( X ) [ E ( X ) ] p - = - =. p p
7. A compur indpndnly gnras svn random numbrs from a Uniform (, ) disribuion. a) Wha is h probabiliy ha xacly hr will b in h inrval from ½ o? L X dno h numbr of random numbrs ( ou of 7 ) ha ar bwn ½ and. Thn X has a Binomial disribuion, n = 7, p =.. P ( X = 3 ) = 7 3. 3. =.7337. b) Wha is h probabiliy ha fwr han hr will b in h inrval from ¾ o? L X dno h numbr of random numbrs ( ou of 7 ) ha ar bwn ¾ and. Thn X has a Binomial disribuion, n = 7, p =.. P ( X < 3 ) = 7..7 7 7..7 7..7».33839.3.3 =.787.
8. Alx ss hr alarm clocs ach nigh o nsur ha h dos no slp hrough his 9: a.m. class. His primary cloc wors proprly on 9% of h mornings, his scond alarm cloc wors proprly on 8% of mornings, and his hird alarm cloc is an aniqu, i wors proprly on only % of mornings. Assum h alarm clocs ar indpndn. a) Find h probabiliy ha Alx s hr-alarm sragy prvns him from ovrslping. Tha is, find h probabiliy ha a las on alarm cloc would wor on a givn morning. P ( ' Ç ' Ç 3' ) =... =.8 =.99. b) Find h probabiliy ha all hr alarm clocs would wor on a givn morning. P ( Ç Ç 3 ) =.9.8. =.3. c) Find h probabiliy ha xacly wo alarm clocs would wor on a givn morning. 3'.9.8. =.88 ' 3.9.. =.8 ' 3..8. =.8.88.8.8 =..
9. Alx owns a dog and a ca. H is a nglcful p ownr. On any givn day, Alx rmmbrs o fd h dog wih probabiliy.7. Th ca is lss lucy, h probabiliy ha Alx rmmbrs o fd h ca is only.. Thr is also a % chanc ha nihr animal gs fd. P( D ) =.7, P( ) =., P( D' Ç ' ) =.. a) Wha is h probabiliy ha Alx rmmbrs o fd boh h dog and h ca? P( D È ) = P( D' Ç ' ) =. =.8. P( D È ) = P( D ) P( ) P( D Ç )..8 =.7. P( D Ç ). P( D Ç ) =.. OR P( D' È ' ) = P( D' ) P( ' ) P( D' Ç ' ) =... =.. P( D Ç ) = P( D' È ' ) =. =.. b) Suppos Alx forgs o fd h dog. Wha is h probabiliy ha Alx rmmbrs o fd h ca? ' D...7 D'........ P ( D' ) = = =..
c) Ar vns { Alx rmmbrs o fd h dog } and { Alx rmmbrs o fd h ca } indpndn? Jusify your answr.. = P( D Ç ) ¹ P( D ) P( ) =.7. =.. D and ar NOT indpndn. OR. = P( D' Ç ' ) ¹ P( D' ) P( ' ) =.. =.. D' and ' ar NOT indpndn. Þ D and ar NOT indpndn. OR. = P ( D' ) ¹ P ( ) =.. D' and ar NOT indpndn. Þ D and ar NOT indpndn.
. Alx is a naughy sudn h dos no always sudy for his xams. Thr is only a 7% chanc ha h would sudy for an xam. If h dos sudy for an xam, h probabiliy ha h would pass i is.9. Howvr, if h dos no sudy, hr is an % chanc h would fail. P ( S ) =.7, P ( P S ) =.9, P ( F S' ) = P ( P' S' ) =.. OR P F = P' Toal S.7.9.3.7.7 S'..3..8.3 Toal.7.. a) Suppos you find ou ha Alx passd an xam. Wha is h probabiliy ha h did no sudy for i? P S P ( S' ' Ç P.. P ) = = = =.. P( P ).3.. 7 Bays Thorm: ( ) OR ( ) ( ) ( ) ( ) ( ) ( ) P S' P P S'.3.. P ( S' P ) = = = =.. P S P P S P S' P P S'.7.9.3.. 7
b) Suppos you find ou ha Alx faild an xam. Wha is h probabiliy ha h did sudy for i? P( SÇ F ).7.7 P ( S F ) = = = =.8. P( F ).7.8. OR Bays Thorm: ( ) ( ) ( ) ( ) ( ' ) ( ' ) P S P F S.7..7 P ( S F ) = = = =.8. P S P F S P S P F S.7..3... Alx is a naughy sudn h dos no always do all h problms on a homwor assignmn. Suppos a crain homwor assignmn has 8 problms, bu Alx only did 7 of hm. Th gradr will randomly slc of h 8 problms o grad. Wha is h probabiliy ha Alx did a las 3 of h problms chosn for grading? 8 or 8 or 8 å æ å æ å æ 7 7 7 no no no don don don don don don â â â â â â 3 7 3 7 7 8 8 8 = 33 37 = =».7. 88 3 3 3 88 88 88
. Alx uss a copy machin o ma 7 copis of h xam. Suppos ha h saplr indpndnly malfuncions wih probabiliy 3% for ach copy of h xam. a) Find h probabiliy ha h saplr would malfuncion xacly ims. L X = numbr of ims h saplr malfuncions. Binomial disribuion, n = 7, p =.3. æ7 7 P ( X = ) =.3.97».3. è b) Us Poisson approximaion o find h probabiliy ha h saplr would malfuncion xacly ims. Poisson approximaion: l = n p = 7.3 =.. -.. P ( X = ) =»..! 3. Suppos ha 7%of all cars mad by a crain manufacurr m h nw EPA sandards. If a random sampl of cars is an, wha is h probabiliy ha a las 8 of hm m h nw EPA sandards? L X = numbr of cars (ou of 7) ha m h nw EPA sandards. Thn X has Binomial disribuion, n = 7, p =.8. P ( X ³ 8 ) = (.7) 8 (.3) (.7) 9 (.3) (.7) (.3) 8 9».33..8 =.388.
. Suppos ha h numbr of FUs (colony forming unis) of E. coli in ground bf has Poisson disribuion, wih an avrag of.8 FUs pr pound of ground bf. Wha is h probabiliy ha hr ar xacly 3 FUs in a four-pound pacag of ground bf? pounds Þ l =.8 = 3.. 3-3. 3. P ( X = 3 ) = =.. 3!. Th pumpin diamrs a Pr Pr Pumpin Ear Pumpin Pach ar normally disribud wih man. inchs and sandard dviaion inchs. a) Wha proporions of pumpins hav diamr ovr inchs? æ -. P ( X > ) = P Z > = P ( Z > -.7 ) =.773. è b) Pumpins wih diamrs in h op 3% ar s asid for a pumpin-carving cons. Find h minimum diamr a pumpin mus hav o b s asid for h cons. Wan P ( X > x ) =.3. Find z such ha P ( Z > z ) =.3. z =.88. x = µ s z =..88 =. inchs. c) If 8 pumpins ar slcd a random, wha is h probabiliy ha xacly of hm hav diamrs ovr inchs? L Y = numbr of pumpins ( ou of 8 ) ha hav diamrs ovr inchs. Thn Y has Binomial disribuion, n = 8, p =.773 ( s par (a) ). P ( Y = ) = 8 (. 773 ) (. ) 3».83.
. Suppos a discr random variabl X has h following probabiliy mass funcion: f ( ) = p, f ( ) =, =,, 3,,,, 7,. a) Find h valu of p ha mas his is a valid probabiliy disribuion. ( ) Mus hav å f x =. all x = p å = p = p. Þ f ( ) = p =. = - b) Find h momn-gnraing funcion of X, M X ( ). For which valus of dos i xis? M X ( ) = E ( X ) = = =. = = = =. - - - Nd <. Þ < ln».797. å = å æ è - M X ( ) = =, < ln. - - c) Find E ( X ). æ - - æ - d æ M ' X ( ) = è è = =. d è - æ æ è - è -
OR M ' X ( ) =. E ( X ) = M ' X ( ) = = =.. OR E ( X ) = E ( X ) = Þ E ( X ) = =. Þ E ( X ) =. Thrfor, E ( X ) = =.. OR E ( X ) = = =, whr Y is a Gomric random variabl wih probabiliy of succss.. Thrfor, E ( X ) = =.. è æ - = è æ - è æ - = - è æ - - d d ( ) -... 3 3... 3 3 è æ -... 3 = - = å = å = ú ú û ù ê ê ë é å è æ = - ( ) Y E ( ) Y E = = p
7. Suppos ha numbr of accidns a a consrucion si follows a Poisson procss wih h avrag ra of.8 accidns pr monh. Assum all monhs ar indpndn of ach ohr. a) Find h probabiliy ha xacly accidns will occur in on monh..8 monh Þ l =.8. P( X = ) = =.38.! -.8 b) Find h probabiliy ha a las accidns will occur in on monh. monh Þ l =.8. P( X ³ ) = [ P( X = ) P( X = ) ] = é.8 - ê êë! -.8.8! -.8 ù ú úû = [.93.39 ] =.9. c) Find h probabiliy ha 7 accidns will occur in on yar ( monhs). 7-9. 9. monhs Þ l =.8 = 9.. P( X = 7 ) = =.98. 7! d) Find h probabiliy ha hr will b accidn-fr monhs in on yar. L Y = h numbr of accidn-fr monhs in on yar. Thn Y has Binomial disribuion, n =, p =.93 ( Poisson, l =.8 ) ( ) ( ) 8 P( Y = ) =.93 -.93 =.7.
8. If E ( X ) = 7 and E ( X ) = 99, us hbyshv s inqualiy o drmin µ = E ( X ) = 7, s = Var ( X ) = 99 7 = 3. s = SD ( X ) = 8. a) A lowr bound for P ( < X < ). σ By hbyshv s Inqualiy, for >, P X - µ < ε ³ -. 3 P ( < X < ) = P ( X 7 < 7 ) ³ - =.9. 7 ( ) ε b) A lowr bound for P ( 3 < X < ). 3 P ( 3 < X < ) = P ( X 7 < ) ³ - =.8. c) An uppr bound for P ( X 7 ³ 3 ). σ By hbyshv s Inqualiy, for >, P X - ³. 3 P ( X 7 ³ 3 ) =.3. 3 ( µ ε ) ε
9. A M h Firms, a rcruir is inrviwing candidas for an inrnship. From pas xprinc, h rcruir blivs ha only abou % of h ponial candidas hav h ncssary qualificaions. Assum indpndnc. a) Wha is h probabiliy ha h firs suiabl job candida will b found during h fourh inrviw? F F F S.8 3. =.. Gomric disribuion, p =.. b) Wha is h probabiliy ha i would a a mos 7 inrviws o find h firs suiabl job candida? For Gomric disribuion, X = numbr of indpndn amps ndd o g h firs succss. P ( X > a ) = P ( h firs a amps ar failurs ) = ( p ) a, a =,,, 3,. P ( X 7 ) = P ( X > 7 ) =.8 7 =.7988. c) Wha is h probabiliy ha h firs suiabl job candida will b found during an vn-numbrd inrviw? P(vn) = P() P() P() = 3.8..8..8. = å..8 =. å. n =. = =».. = n= -. 3 9 OR 3 P(vn) =.8..8..8. P(odd) =.8..8..8.
Þ P(vn) =.8 P(odd). P(odd) = P(vn) =. P(vn). Þ 9 = P(odd) P(vn) =. P(vn) = P(vn). P(vn) =».. 9 d) Wha is h probabiliy ha h hird suiabl job candida will b h ighh prson inrviwd? [ 7 calls: S s & F s ] S [ (.) (.8) ].».. 7 Ngaiv Binomial disribuion, = 3, p =.. ) If h rcruir inrviws individuals, wha is h probabiliy ha hr will b xacly 3 suiabl job candidas? 3 7 3 (. ) (. 8)».33. Binomial disribuion, n =, p =.. f) If h rcruir inrviws individuals, wha is h probabiliy ha hr will b a mos 3 suiabl job candidas? 9 (. ) (. 8) (. ) (. 8) 8 3 7 (. ) (. 8) 3 (. ) (. 8)».737.8.399.33 =.8793. Binomial disribuion, n =, p =..
. L X b a coninuous random variabl wih h probabiliy dnsiy funcion 3 x 3 x f ( x ) = - for x, f ( x ) = 3 ohrwis. a) Find P ( X > ). æ 3 3 3 æ 3 3 P ( X > ) = ò x - x dx = x - x = - = =.87. è è b) Find E ( X ). ò - æ 3 3 æ 3 3 3 E ( X ) = x f ( x) dx = ò x 3 x - x dx = ò x - x è è dx æ 3 3 = x - x = -.8 =.. è 8 c) Find SD ( X ). æ Var ( X ) = ( ) - [ E( X) ] æ 3 3 = ò f x dx ò x x x dx è - è 3 x - - [. ] æ 3 3 = [. ] æ 3 ò - x dx - = è è x - [. ] = ( 9. 8 ). =.. =.. SD ( X ) =. =.. x - x 8