3(8 ) (8 x x ) 3x x (8 )
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- Daisy Reed
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1 Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 = a d b 9 d c d d 6 d u u u u du Thn, - +. =, and =
2 -. a cos d sin / / / / b / cos d sin / / / c / / cos d sin / / / / / d / cos d sin / / cos d sin sin. 9 / / Thn, sin = -., and = -. radians. -. a d. 7 b d. 9 8 c 8 d d or 8 8. From par c, 8 =.69. Thrfor, or 8 =.69 =.9 d. 9 Thn, =, and =.7 -. a d. bcaus f for <. 7. b. d.97 bcaus f for >. 7.. c d d. d d d
3 -6. a d, bcaus f for <. This can also b obaind from h fac ha f is a probabiliy dnsiy funcion for <. b d c. From par b,. Thrfor,. 7 d 8 d.89 8 d.8 Thn, = ln. = a., by symmry. b. d c.. d. 67 d < =. < or >. =. 8.. f d Thn, =.9-8. a d. 67 b d. c d. 6 / d d.. Thn, /. 9, and = ln.9 =.6.
4 a.d a. b.8.d Thn,. = 6.7 and = d b < 7.8 or > 7. = < > 7. bcaus h wo vns ar muually clusiv d 7..d Th rsul is =.. c and d a <. or >.7 = <. + >.7 bcaus h wo vns ar muually clusiv. Thn, <. = and >.7 = d....7 b If h probabiliy dnsiy funcion is cnrd a.6 mrs, hn f for. < <.9 and all rods will m spcificaions. -. a 9 bcaus h pdf is no dfind in h rang,9. b.6. c d.6 d
5 d Find a such ha a. a... Thn, a a... 6 a. d a.. a. 6 a 6 a.. Scion - -. a <.8 =.8 bcaus is a coninuous random variabl. Thn, <.8 = F.8 =..8 =.8. b..... c F d 6 F 6 -. a.7.7 F.7 bcaus is a coninuous random variabl. Thn, F b....8 c < - =. d F F Now, f for < and -6. Now,, for <. Thn, F, 8 f for < < 8 and 6 F d 8 u u u u F du for <. Thn, F 6,, 8 6, 8-7. Now, f cos for -/ < < / and F cos udu sin sin / /
6 Thn, F sin, /, / /, / -8. Now, f for > and F du u u Thn, F,, -9. Now, f / 7 for < < and, 9 for <. Thn, F,, u u 9 F du 7.. Now, / f for < and F / y / / dy for <., Thn, F /, > = - = - F = -/ =. -. Now, f = for. < <.9 and F dy. 6. for. < <.9. Thn,,. F.6,..9, F.7.8. bcaus is a coninuous random variabl.
7 / -. Now, f for < and / / / F / d, Thn, F /, a < = F = - - = -.98 =.9 b / -. - / d =.7 for <. c >+ < and >= =.6 d << = F-F = =.7 -. F..d.7 for < <. Thn,, F.7,, -. f, -. f.,., 9-6. f.,.7,. Scion E.d... V.. d
8 -8. E. d V. d. d E d V.8 d d -. E d V 8.67 d d E d 6 6 V 8 8 d 6 V E d.9... V..9.. d d d E * d
9 -. a E V Thrfor,.d d. V b Clarly, cnring h procss a h cnr of h spcificaions rsuls in h gras proporion of cabls wihin spcificaions d. -. a E d ln 9. 6 V 9.6 d 8.ln b. Avrag cos pr par = $.*9.6 = $ a E f d ln.. d E f d d.. 69 Var = E E b *. =.9 c f d.6-7. a E d. Using ingraion by pars wih u = and dv E d d 7 7. d, w obain. Now,. V d. Using h ingraion by pars wih u. dv, w obain V.. d and.
10 Scion - From h dfiniion of E h ingral abov is rcognizd o qual. Thrfor, V.... b. d a E =. +./ =.. V.7, and b c...d..,. F..7,..,. -9. a E = -+/ =., V.8, and. b d...,..9. Thrfor, should qual.., c F..,,.. a f =. for 9.7 < <.. E = / =.,. 9.7 V.8, and.. b F.dy for 9.7 < <.. Thrfor, 9.7, 9.7 F 99., 9.7.,. c. F
11 -. a Th disribuion of is f = 6.67 for.9 < <.. Now,,.9 F ,.9.,. b.. F.. c If > =.9, hn F =.9 and F =.. Thrfor, =. and = d E =. +.9/ =.97 and V = E.min..7 V.8min b d /.7 /.7 / d c..7 F dy /.7 dy /.7 y for.7 < <.. Thrfor,, F /.7.9,, a Th disribuion of is f = for. < <.. Thrfor,,. F.,..,. b. F. [..]. c If > =., hn F =. and F =.8. Thrfor, -. =.8 and =.. d E =. +./ =. m and.. 6 V = 8. m -. L dno h changd wigh. Var = /
12 Sdv = a L b h im in minus bwn arrival and 8: am. f, for 9 9 So h CDF is F, for 9 9 b E, Var = 9 / = 67 c Th vn is an arrival in h inrvals 8:-9: am or 9:-9: am or 9:-: am so ha h probabiliy = /9 = / d Similarly, h vn is an arrival in h inrvals 8:-8: am or 9:-9: am or 9:-9: am so ha h probabiliy = /9 = / -6. a E = / = V, and.7 b L b h volum of a shampoo millilirs 7 7 d c Th disribuion of is f = /6 for7 8., 76 Now, F 76 / 6, 76 7, 7 > =.9, hn F =.9 and F =.. Thrfor, - 76/6 =. and = 76. d Sinc E = 79, hn h man ra cos = 79-7 $. = -$. pr conainr. -7. a L b h arrival im in minus afr 9: A.M. V and.6 b W wan o drmin h probabiliy h mssag arrivs in any of h following inrvals: 9:-9: A.M. or 9:-9: A.M. or :-: A.M. or :-: A.M.. Th probabiliy of his vn is / = /6. c W wan o drmin h probabiliy h mssag arrivs in any of h following inrvals: 9:-9: A.M. or 9:-: A.M. or :-: A.M. or :-: A.M. Th probabiliy of his vn is 6/ = /. -8. a L dno h masurd volag. So h probabiliy mass funcion is, for 6,..., 8 Scion -6 6 b E=, Var= 6. 67
13 -9. a Z<. =.968 b Z<. =.977 c Z>. =.967 =.7 d Z >. = pz <. =.98. < Z <.76 = Z<.76 Z >. = a < Z < = Z < Z > =.8.8 =.6868 b < Z < = Z < [ Z < ] =.9 c < Z < = Z < [ Z < ] =.997 d Z > = Z < =.7 < Z < = Z < Z < =.8. =. -. a Z <.8 =.9 b Z < =. c If Z > z =., hn Z < z =.9 and z =.8 d If Z > z =.9, hn Z < z =. and z =.8.8 < Z < z = Z < z Z <.8 = Z < z.7. Thrfor, Z < z = =.97 and z =.9 -. a Bcaus of h symmry of h normal disribuion, h ara in ach ail of h disribuion mus qual.. Thrfor h valu in Tabl III ha corrsponds o.9 is.6. Thus, z =.6. b Find h valu in Tabl III corrsponding o.99. z =.8. c Find h valu in Tabl III corrsponding o.8. z =. d Find h valu in Tabl III corrsponding o z =.. -. a < = Z < / = Z < =.8 b > 9 = < 9 = Z < 9/ = Z <. = c 6 < < = Z = < Z < = Z < Z < ] =.9. d < < = Z
14 = < Z < = Z < Z < =. < < 8 = < 8 < 8 = Z Z = Z < Z < 6 = a > = Z =.. Thrfor, = and =. b > = Z = Z =.9. Thrfor, Z =. and =.6. Consqunly, = 6.7. c < < = Z = Z Z =. Z =.. Thrfor, Z =. and =.. Consqunly, = d < < + = / < Z < / =.9. Thrfor, / =.6 and =. < < + = / < Z < / =.99. Thrfor, / =.8 and =.6 -. a < = Z = Z <. =.99 b > = Z = Z >.7 = Z <.7 = = c < < 7 = Z =. < Z <. = Z <. Z <. =.89 9 d < < 9 = Z =.7 < Z < = Z < Z <.7] =.88 8 < < 8 = Z =.7 < Z <.7 = Z <.7 Z <.7 =.67
15 -6. a > = Z =.. Thrfor, =. b > = Z =.9. Thrfor, Z =. Thrfor, =.6, and =.6. 7 c < < 7 = Z =.. Thrfor, Z < 7 Z < =. whr Z <. =.696. Thus Z < =.96. Consqunly, = -. and =.9. d < < = Z =.9. Thrfor, Z Z <. =.9 and Z.8 =.9 Consqunly, Z =.8. Bcaus a probabiliy canno b grar han on, hr is no soluion for. In fac, < =. < Z =.696. Thrfor, vn if is s o infiniy h probabiliy rqusd canno qual.9. < < = < < + = Z = Z =.99 Thrfor, / =.8 and = a < 6 = Z = Z <. = b 8 < < 9 = Z = < Z < = Z < Z < = c > = Z =.9. Thrfor, =.6 and = a L dno h im. ~ N9, b
16 c c a b L dno h im. ~ N9, c Hr 9% of h surgris will b finishd wihin.8 minus. d 99 >>.8 so h volum of such surgris is vry small lss han %. -6. L dno h cholsrol lvl. ~ N9., σ 9. a b c d.9.8 f a >.6 = Z. = Z >.8 = Z <.8 = b.7 < <.6 = Z.. =.6 < Z <. = Z <. Z <.6 = = c < = Z. =.9..6 Thrfor, =.8 and =.67..
17 a < 6 = Z = Z < b < 6 = Z = Z < =.86 and 7 7 > 7 = Z = Z > =.86. Thrfor, h proporion of cans scrappd is =.7, or.7% c 7 < < 7 + =.99. Thrfor, Z =.99 Consqunly, Z =.99 and =.8 =.9. Th limis ar 7., a If > 6 =.999, hn Z =.999. Thrfor, 6 =.9 and = b If > 6 =.999, hn Z =.999. Thrfor, 6 =.9 and = a >. = Z. = Z > =. =..... b. < <. = Z.. = < Z < = Z < Z < = c > =.9, hn Z. =.9.. Thrfor, =.8 and = a > 68 = Z
18 Z b < = Z Z..6 c,, bys*8 bis/by 8,, bis 8,, bis. sconds 6, bis/sc -66. L dno h high. ~ N6, a b c d L dno h high. ~ N.,... a b c L dno h dmand for war daily. ~ N7, 7 7 a b c d N, 7. 7
19 a < 6 = Z = Z <.67 = b > =.9. Thrfor, Z =.9 and 6 6 Consqunly, = 66 7 = c > 7 = Z Z. 6 Thr lasrs opraing afr 7 hours = / =/ a >.6 = Z. = Z >. = -Z <. = b. < <.6 = Z.. =. < Z <. = c. < <.6 = Z.. = Z... Thrfor, Z =.997. Thrfor, =.8 and = a >.7 = Z = Z >. = b If <.7 =.999, hn Z = Thrfor,./ =.9 and c If <.7 =.999, hn Z. =.999.
20 Thrfor,.7 =.9 and = a L dno h masurmn rror, ~ N, b From h shap of h normal curv, h probabiliy is maimizd for an inrval symmric abou h man. Thrfor a =. wih probabiliy =.97. Th sandard dviaion dos no affc h choic of inrval a > 9 = Z = Z >.667 = b 8 8 Z Z.. =.77.9 =.76. c > =., hn d > 9 =., hn < 9 =. = Z.99. Thrfor,. and µ = a > 8 = Z = Z >.6 =..9 b < = Z = Z < -.6 =. c > =., hn a > 6 = Z = Z >.8 =.6.9 b > =., hn Scion -7.6 c < = Z = Z < -.86 =.6.9 Th normal disribuion is dfind for all ral numbrs. In cass whr h disribuion is runcad bcaus wai ims canno b ngaiv, h normal disribuion may no b a good fi o h daa a E =. = 6, V =..7 = and
21 . 6 Thn, Z Z b 7 Z.7 Z c Z.77 Z i a. i i! b is approimaly ~ N6,6 6 Thn, Z Z If a coninuiy corrcion wr usd h following rsul is obaind.. 6 Z Z c 9 Z. Z If a coninuiy corrcion wr usd h following rsul is obaind Z 6 6. Z Z is approimaly N, a Z 8 Z If a coninuiy corrcion wr usd h following rsul is obaind Z 8 Z b. If a coninuiy corrcion wr usd, h following rsul is obaind Z Z c
22 If a coninuiy corrcion wr usd, h following rsul is obaind Z 8 8. Z L dno h numbr of dfciv chips in h lo. Thn, E =. =, V =..98 = a Z Z.79 Z b Z Z. -8. L dno h numbr of popl wih a disabiliy in h sampl. ~ Bin, Z is approimaly N, a b L dno h numbr of accouns in rror in a monh. ~ BIN7,,. a E = 7 Sdv = b Z is approimaly N, c v. 9 v d Thn h probabiliy is.9..
23 -8. L dno h numbr of original componns ha fail during h usful lif of h produc. Thn, is a binomial random variabl wih p =. and n =. Also, E =. = and V =..999 = Z Z. Z L dno h numbr of rrors on a wb si. Thn, is a binomial random variabl wih p =. and n =. Also, E =. =. and V =..97 =.7.. Z Z.8 Z L dno h numbr of paricls in cm of dus. Thn, is a oisson random variabl wih,. Also, E = and V = Z Z. If a coninuiy corrcion wr usd h following rsul is obaind.. Z Z is h numbr of minor rrors on a s parn of pags of. is a oisson random variabl wih a man of. pr pag a Th numbrs of rrors pr pag ar random variabls. Th assumpion ha h occurrnc of an vn in a subinrval in a oisson procss is indpndn of vns in ohr subinrvals implis ha h numbrs of vns in disjoin inrvals ar indpndn. Th pags ar disjoin inrvals and h consqunly h rror couns pr pag ar indpndn... b. 67!.67. Th man numbr of pags wih on or mor rrors is. = pags c L Y b h numbr of pags wih rrors Y Z Z. Z L dno h numbr of his o a wb si. Thn, is a oisson random variabl wih a man of, his pr day. Also, V =,. a,,,, Z, Z If a coninuiy corrcion wr usd, h following rsul is obaind.
24 ,.,,, Z, Z. 9,799, b 9,8 9,799 Z.., Z If a coninuiy corrcion wr usd h following rsul is obaind. 9,799., 9,8 9,799 Z.., Z c If > =., hn, Z =..,, Thrfor,. and,, d L dno h numbr of his o a wb si. Thn, is a oisson random variabl wih a man of, pr day. Thrfor, E =, and V =,,,, Z, Z Z If a coninuiy corrcion wr usd, w obain h following rsul,.,, Z.., Z Z and his approimaly quals h rsul wihou h coninuiy corrcion. Th pcd numbr of days wih mor han, his is.7*6 = 8. days pr yar. L Y dno h numbr of days pr yar wih ovr, his o a wb si. Thn, Y is a binomial random variabl wih n = 6 and p =.7. EY = 8. and VY = = Y Z Z.6 Z L dnos h numbr of random ss ha is mor disprsd han h opron. Assum ha has a ru man =. = ss.
25 7.. 7 Z Z.. Z.8 Z Wih, ashma incidns in childrn in a -monh priod, hn man numbr of incidns pr monh is / =. L dno a oisson random variabl wih a man of pr monh. Also, E = = = V. a Using a coninuiy corrcion, h following rsul is obaind.. Z Z Wihou h coninuiy corrcion, h following rsul is obaind Z Z.6 Z b Using a coninuiy corrcion, h following rsul is obaind Z Z. Z Z Z Z. Z c d Th oisson disribuion would no b appropria bcaus h ra of vns should b consan for a oisson disribuion. Scion a d 6 b d. c d.9 d 6 d.7
26 . d and =.7-9. If E =, hn.... a. d b.. c.9... d. d.9 and = a., b.... c Thy ar h sam. -9. L dno h im unil h firs coun. Thn, is an ponnial random variabl wih couns pr minu. a d. /6 /6 / 6 d b. 8 c a E = / = / =. minus b V = / = / =.6, =. c d. 9, = L dno h im unil h firs call. Thn, is ponnial and E calls/minu. a d. b Th probabiliy of a las on call in a -minu inrval quals on minus h probabiliy of zro calls in a -minu inrval and ha is >.
27 .68. Thrfor, h answr is -.68 =.6. Alrnaivly, h rqusd probabiliy is qual o < =.6. c / /. / d < =.9 and. 9. Thrfor, =. minus L b h lif of rgulaor. Thn, is an ponnial random variabl wih / E / a Bcaus h oisson procss from which h ponnial disribuion is drivd is mmorylss, his probabiliy is < = / / d. 6 b Bcaus h failur ims ar mmorylss, h man im unil h n failur is E = yars L dno h im o failur in hours of fans in a prsonal compur. Thn, is an ponnial random variabl and / E.. a >, = b < 7, =... d.8,, 7, 7,...8. d L dno h im unil a mssag is rcivd. Thn, is an ponnial random variabl and / E /. a > = / / d. 679 b Th sam as par a. c E = hours L dno h im unil h arrival of a ai. Thn, is an ponnial random variabl wih / E. arrivals/ minu... a > 6 =. d b < =. d.6 c > =.... d. and = 8.97 minus. d < =.9 implis ha > =.. Thrfor, his answr is h sam as par c... < =. and = 6.9 minus.
28 -. a /. =. pr yar b...6 = =. cl T dno h im bwn sighings T E d. 7. = =. -. L dno h numbr of insc fragmns pr gram. Thn OI./ a /. = b.69! 7 6 c L dno h disanc bwn major cracks. Thn, is an ponnial random variabl wih / E. cracks/km. a > =... d. b L Y dno h numbr of cracks in km of highway. Bcaus h disanc bwn cracks is ponnial, Y is a oisson random variabl wih. cracks pr km. Y = =. 77! c /. d..... d.9 > =.679. By indpndnc of h inrvals in a oisson procss, h answr is Alrnaivly, h answr is > =.. Th probabiliy dos dpnd on whhr or no h lnghs of highway ar conscuiv. f By h mmorylss propry, his answr is > =. from par. -. L dno h lifim of an assmbly. Thn, is an ponnial random variabl wih / E / failurs pr hour. a < = / /. d. 9 / b > = /. 86 c From h mmorylss propry of h ponnial, his answr is h sam as par a., < =..
29 d L U dno h numbr of assmblis ou of ha fail bfor hours. By h mmorylss propry of a oisson procss, U has a binomial disribuion wih n = and p =. from par a. Thn, U U L V dno h numbr of assmblis ou of ha fail bfor 8 hours. Thn, V is a binomial random variabl wih n = and p = < 8, whr dnos h lifim of an assmbly. 8 Now, < 8 = / / d Thrfor, V = = L Y dno h numbr of arrivals in on hour. If h im bwn arrivals is ponnial, hn h coun of arrivals is a oisson random variabl and arrival pr hour. a Y > =. 8!!! Y b From par a, Y > =.8. L W dno h numbr of on-hour inrvals ou of ha conain mor han arrivals. By h mmorylss propry of a oisson procss, W is a binomial random variabl wih n = and p =.8. W = = c L dno h im bwn arrivals. Thn, is an ponnial random variabl wih arrivals pr hour. > =. and d.. Thrfor, =. hours. -. L dno h numbr of calls in minus. Bcaus h im bwn calls is an ponnial random variabl, is a oisson random variabl wih / E. calls pr minu = calls pr minus. a > =.8!!!! b = =.979! c L Y dno h im bwn calls in minus. Thn, Y. and Y.... d. Thrfor, d Y > =..y dy.y 6... and = 9. minus. Bcaus h calls ar a oisson procss, h numbrs of calls in disjoin inrvals ar indpndn. From Ercis -9 par b, h probabiliy of no calls in on-half hour is Thrfor, h answr is 6.. Alrnaivly, h answr is h probabiliy of no calls in wo hours. From par d of his rcis, his is. f Bcaus a oisson procss is mmorylss, probabiliis do no dpnd on whhr or no inrvals ar conscuiv. Thrfor, pars d and hav h sam answr. 6.
30 -6. is an ponnial random variabl wih. flaws pr mr. a E = / mrs... b > =. d. 68 c No d < =.9. Thn, < =. Thrfor,. 9 and =... > 8 =. d Th disanc bwn succssiv flaws is ihr lss han 8 mrs or no. Th disancs ar indpndn and > 8 =.9. L Y dno h numbr of flaws unil h disanc cds 8 mrs. Thn, Y is a gomric random variabl wih p =.9. Y = = f EY = /.9 = a / / d. 679 / b. / c. 98 d Th rsuls do no dpnd on. -8. E d. Us ingraion by pars wih u = and dv =. Thn, E d / V = d. Us ingraion by pars wih u and dv =. Thn, V d Th las ingral is sn o b zro from h dfiniion of E. Thrfor, V = -9. is an ponnial random variabl wih =. days. d.
31 /. /. a < = d.. /.. b 7 d 7 /.. 7 /. c. 9 and. 9 Thrfor, =.ln.9 =.69 d From h lack of mmory propry < > = < 7 and from par b his quals. = a E. 6, hn. 7.6 /.6.6 b d /. 7.6 u /.6 /.6 c du..6 Thn, = -.6ln. = 6.8 Scion a 7 6! 7 / b.9 c / is a gamma random variabl wih h paramrs. and r. Th man is E r / /.. Th varianc is Var r / / a Th im unil h nh call is an Erlang random variabl wih 6 calls pr minu and r =. b E = /6= minus. V = /6 =. minus. c Bcaus a oisson procss is mmorylss, h man im is /6=.67 minus or sconds L Y dno h numbr of calls in on minu. Thn, Y is a oisson random variabl wih 6 calls pr minu.
32 6 6 d Y = =.! Y > = - Y. 98!!! L W dno h numbr of on minu inrvals ou of ha conain mor han calls. Bcaus h calls ar a oisson procss, W is a binomial random variabl wih n = and p =.98. Thrfor, W = = L dno h kilograms of marial o obain paricls. Thn, has an Erlang disribuion wih r = and.. r a E = 7. b V = 7 and kg. -. L dno h im bwn failurs of a lasr. is ponnial wih a man of,. a. Epcd im unil h scond failur E r / /., hours b. N=no of failurs in hours E N k N.6767 k! k -6. L dno h im unil mssags arriv a a nod. Thn, has an Erlang disribuion wih r = and mssags pr minu. a E = / = / minu = sconds. bv =. 8 minu =. scond and. 89 minu =.7sconds. c L Y dno h numbr of mssags ha arriv in sconds. Thn, Y is a oisson random variabl wih mssags pr minu = mssags pr sconds. Y Y.9!!!!! d L Y dno h numbr of mssags ha arriv in 6 sconds. Thn, Y is a oisson random variabl wih. mssags pr 6 sconds. Y Y L dno h numbr of bis unil hr rrors occur. Thn, has an Erlang disribuion wih r = and rror pr bi. r a E = bis. r b V = and 7. bis. c L Y dno h numbr of rrors in bis. Thn, Y is a oisson random variabl wih
33 / rror pr bi = rror pr bis. Y Y!!! r a E r / / minus b min min = min c L Y b h numbr of calls bfor sconds.* Y.7.87!!! -9. a L dno h numbr of cusomrs ha arriv in minus. Thn, is a oisson random variabl wih. arrivals pr minu = arrivals pr minus..!!! b L Y dno h numbr of cusomrs ha arriv in minus. Thn, Y is a oisson random variabl wih arrivals pr minus. Y Y.87 r -. r d. Us ingraion by pars wih u r r r r!!!!! d r r. r and dv = -. Thn, r r r y y dy -. f ;, r d d. L y =, hn h ingral is r. From h r dfiniion of r, his ingral is rcognizd o qual. -. If is a chi-squar random variabl, hn is a spcial cas of a gamma random variabl. Now, r 7 / r 7 / E = 7 and V / / -. L dno h numbr of pains arriv a h mrgncy dparmn. Thn, has a oisson disribuion wih 6. pains pr hour. a E r / / 6..9 hour. b L Y dno h numbr of pains ha arriv in minus. Thn, Y is a oisson random variabl wih EY = 6./ =.667 arrivals pr minus. Th vn ha h hird arrival cds minus is quivaln o h vn ha hr ar wo or fwr arrivals in minus. Thrfor, Y!!!
34 Th soluion may also b obaind from h rsul ha h im unil h hird arrival follows a gamma disribuion wih r = and = 6. arrivals pr hour. Th probabiliy is obaind by ingraing h probabiliy dnsiy funcion from minus o infiniy. -. a E r / 8, hn r 8 Var r / 8/ 6, hn. Thrfor, h paramrs ar. and r b Th disribuion of ach sp is ponnial wih =. and 9 sps produc his gamma disribuion. Scion - -. =. and =8 hours E 8. V 8 8! 6. 8 [. ] 8, -6. a b F F If is a Wibull random variabl wih = and =, h disribuion of is h ponnial disribuion wih =.. f for.. for Th man of is E = / =. -8. L dno lifim of a baring. = and = hours a F 8.99 E. b = 89. hours c L Y dno h numbr of barings ou of ha las a las 8 hours. Thn, Y is a binomial random variabl wih n = and p =.7. Y a E 9 / 9 6/ hours
35 b V hours c 9 F. -. L dno h lifim. a E. Thn. Now,. > =. 6 b < = a = /, = E! hours b V [ ] 6. [ ]!! c < = F = E.. So Var E..777 Sdv=.68 Var E..9. E. Rquirs a numrical soluion o hs wo quaions. / a F. 7 b c /8.6 F.87 8 F F 8 / 8.6 8/
36 /8.6 d F. 9 Thrfor, /8.6 ln.9., and =.79 / -. a F , 6 b 6 6/ F 6.7. / F.698 b If i is an ponnial disribuion, hn = and / F.9 6/ F / F.9 For h Wibull disribuion wih = hr is no lack of mmory propry so ha h answrs o pars a and b diffr whras hy would b h sam if an ponnial disribuion wr assumd. From par b, h probabiliy of survival byond 6 hours, givn h dvic has alrady survivd hours, is lowr han h probabiliy of survival byond hours from h sar im. / -6. a F. 6 6, b 6-7. a 6/ F / F.6 c If i is an ponnial disribuion, hn = / F / F / F.9 For h Wibull disribuion wih =. hr is no lack of mmory propry so ha h answrs o pars a and b diffr whras hy would b h sam if an ponnial disribuion wr assumd. From par b, h probabiliy of survival byond 6 hours, givn h dvic has alrady survivd hours, is grar han h probabiliy of survival byond hours from h sar im. d Th failur ra can b incrasd or dcrasd rlaiv o h ponnial disribuion wih h shap paramr in h Wibull disribuion. / F.7 b Th man of his Wibull disribuion is..77 = If i is an ponnial disribuion wih his man hn /786.6 F.
37 c Th probabiliy ha h lifim cds hours is grar undr h ponnial disribuion han undr his Wibull disribuion modl. Scion is a lognormal disribuion wih = and = 9 W ln a W ln..9 b Find h valu for which =.9 W ln W ln.9 ln θω / 9/ 9. c E 9.7 θω ω V. -9. a is a lognormal disribuion wih =- and =6 W ln W ln ln ln...7 W ln b W ln. ln / 6 c E 6/. V a is a lognormal disribuion wih = and = W ln W ln.6.96 b
38 ln ln ln c Th produc has dgradd ovr h firs hours, so h probabiliy of i lasing anohr hours is vry low. -. is a lognormal disribuion wih =. and = a W ln. W ln W ln. b W ln. ln...6 sconds /.. c E /. 8 V Find h valus of and givn ha E = and V = 8, / 8 L and y hn y and 8 y y y y Squar h firs quaion o obain y and subsiu ino h scond quaion 8 y y 9. Subsiu y ino h firs quaion and solv for o obain. 9. ln..8 and ln a Find h valus of and givn ha E = and =, L / and y hn y and y y y y
39 Squar h firs quaion y and subsiu ino h scond quaion y y Subsiu y ino h firs quaion and solv for o obain 7. 6 ln and ln. 69 b W ln 8.6 W ln W ln 8.6 c W ln..686 ln hours E p /.87 p.9 So ln and ln.87 / L ~ N,, hn Y = follows a lognormal disribuion wih man and varianc. By dfiniion, F Y y = Y y = < y = < log y = F log y = Bcaus Y = and ~ N,, w can show ha f Y f logy Finally, f Y y = Y y logy log y log Y y F y f log y F y y y y -6. has a lognormal disribuion wih = and = W ln a W ln b W ln W ln
40 c W ln W ln.7 ln. Thrfor, = has a lognormal disribuion wih =. and =. θ ω /..9/. a E.7 θω ω.9.9 V.697 W ln8. b 8 8 W ln c for h lognormal disribuion. If h disribuion is normal, hn.7 Z..697 Bcaus waiing ims canno b ngaiv h normal disribuion gnras som modling rror. Scion Th probabiliy dnsiy is symmric. -9. a....!!! b !!!... c E. 667 V.6 -. a.7.7
41 b c E. 9.. V a Mod =. 8. E.6.. V b Mod = E V c Boh h man and varianc from par a ar grar han for par b. -. a b c E
42 V L dno h complion proporion of h maimum im. Th rcis considrs h proporion /. = Supplmnal Erciss -. f. for < <7 7 a 7.d b 6.d c E 6. sconds 7 V.8 sconds 7 -. a < = Z = Z <. =.9979 b < = Z = Z <. =.6.6% ar scrappd 6-6. a < = Z = Z < - = b > 6 = Z = Z > = - Z < = -.8=.86 6 c < = Z =.99
43 6 Thrfor, =. and = 7-7. a > < = Z Z. = Z > + Z < = Z < + Z < =.8 + =.866. Thrfor, h answr is.866. b Th procss man should b s a h cnr of h spcificaions; ha is, a = c 9. < < 9.6 = Z.. = < Z < =.997. Th yild is *.997 = 99.7% d 9 < < 9.6 = Z.. = < Z < =.997 = =.997 =.97 L Y rprsn h numbr of cass ou of h sampl of ha ar bwn 9. and 9.6 ml. Thn Y follows a binomial disribuion wih n= and p=.997. Thus, EY= 9.97 or a 6 < < 7 = Z = < Z < -. = Z <. Z < =.. b > =.. Thrfor, Z =. and =.8 Thrfor, =.6 hours -9. E =. = and V =..8 = 6 a. 6 Z Z b 7 Z. Z c If > =., hn Z =.. 6
44 Thrfor, =. and = Th im o failur in hours for a lasr in a cyomry machin is modld by an ponnial disribuion wih..... a,. d b,. d. 777 c,,.... d L dno h numbr of calls in hours. Bcaus h im bwn calls is an ponnial random variabl, h numbr of calls in hours is a oisson random variabl. Now, h man im bwn calls is.7 hours and /.7 / calls pr hour = calls in hours..67!!!! -6. L dno h im in days unil h fourh problm. Thn, has an Erlang disribuion wih r = 6 and / problm pr day. 6 a E = 8 days. b L Y dno h numbr of problms in days. Thn, Y is a oisson random variabl wih problms pr days. Y.!!!! -6. L dno h lifim a E 7 7 b V 7 7 [ ] c > 6. = a E p /. p So ln.686
45 And ln. / 7.7 b ln p W.7 W ln a. d b 7. d c 6. d..7 d F. d.. Thn, 8, F, 8 8, E. d V d d d L dno h im bwn calls. Thn, / E. calls pr minu... a. d.6. b. 8.. c < =.9. Thn,. d.9. Now, = 6. minus. d This answr is h sam as par a.... d.6
46 This is h probabiliy ha hr ar no calls ovr a priod of minus. Bcaus a oisson procss is mmorylss, i dos no mar whhr or no h inrvals ar conscuiv.... d.68 f L Y dno h numbr of calls in minus. Thn, Y is a oisson random variabl wih Y.6.!!! g L W dno h im unil h fifh call. Thn, W has an Erlang disribuion wih =. and r =. EW = /. = minus L dno h lifim. Thn / E / 8. a / 8 / 8. d b L W dno h numbr of CUs ha fail wihin h n four yars. Thn, W is a binomial random variabl wih n = and p =.9. Thn, W W is a lognormal disribuion wih = and =9 a W ln W ln ln ln W ln b W ln. ln /. c E 9/ 9. V a Find h valus of and givn ha E = and V = / L and y hn y and y y y y Squar h firs quaion and subsiu ino h scond quaion y
47 y.6 y y y y y Subsiu y back ino h firs quaion and solv for o obain.6 ln. and ln b W ln. W ln L dno h numbr of fibrs visibl in a grid cll. Thn, has a oisson disribuion and fibrs pr cm = 6, fibrs pr sampl =. fibrs pr grid cll... a. 9.! b L W dno h numbr of grid clls amind unil conain fibrs. If h numbr of fibrs hav a oisson disribuion, hn h numbr of fibrs in ach grid cll ar indpndn. Thrfor, W has a ngaiv binomial disribuion wih p =.9. Consqunly, EW = /.9 =. clls..9 c VW =. Thrfor, 6. W clls L dno h high of a plan...6 a >. = Z = Z >.7 = Z.7 = b. < <. = Z =. < Z <.8 = c > =.9 = Z. =.9 and.6 =.8.. Thrfor, = a d b Ys, bcaus h probabiliy of a plan growing o a high of. cnimrs or mor wihou irrigaion is small. -7. L dno h hicknss.. a >. = Z = Z >. =. 7.
48 .7. b.7 < <. = Z = -. < Z <. = Thrfor, h proporion ha do no m spcificaions is.7 < <. =.. c If < =.9, hn Z =.9. Thrfor, =.6 and = L dno h do diamr. If. < <.6 =.997, hn Z Z.997. Thrfor,..7 and = If. < <. +, hn /. < Z < /. =.997. Thrfor, /. =.7 and =.8. Th spcificaions ar from. o L dno h lif. a Z Z b If > =.9, hn Z < 7 = -.8. Consqunly, 6 7 = -.8 and = 6 6 hours.,, c If >, =.99, hn Z > 6 =.99. Thrfor, 6 = -. and,98 d Th probabiliy a produc lass mor han hours is [ ], by indpndnc. If [ ] =.99, hn > = Thn, > = Z Thrfor, 6 = -.7 and, 6 hours is an ponnial disribuion wih E = 7 hours 6 a d b d Thrfor,. 9 and 7 ln hours -78. Find h valus of and givn ha E = 7 and = 6 7 L / 6 and y hn 7 y and 6 y y y y
49 Squar h firs quaion 6 7 y y.7 7 y and subsiu ino h scond quaion 7 Subsiu y ino h firs quaion and solv for o obain ln and ln.7. 7 a W ln W ln b W ln 8.8 W ln.9.8 ln hours a Using h normal approimaion o h binomial wih n = 8 = 8,, and p =. w hav: E = 8. = 6 np.7 6. np p Z b np. 6.8 np p Z Using h normal approimaion o h binomial wih bing h numbr of popl who will b sad. Thn ~Bin, a 9 np Z np p b np Z np p c 9.98, Succssivly rying various valus of n: Th numbr of rsrvaions akn could b rducd o abou 99. n Z o robabiliy Z < Z
50 L dno h survival im of AMI pains. a. scal paramr; 6. shap paramr.7 E. V b 68.. p.6 F c Find a such ha 9. a.9. p.6 a a F a a Thn, 6. a. Mind-Epanding Erciss -8. a > implis ha hr ar r - or lss couns in an inrval of lngh. L Y dno h numbr of couns in an inrval of lngh. Thn, Y is a oisson random variabl wih mna EY =. Thn,! r i i i r Y. b! r i i i c!!! r i i i F f r r i r i i i d d -8. L dno h diamr of h maimum diamr baring. Thn, >.6 = -.6. Also, 6. if and only if all h diamrs ar lss han.6. L Y dno h diamr of a baring. Thn, by indpndnc ] [ Z Y Thn, >.6 = a Qualiy loss = k m ke m Ek, by h dfiniion of h varianc. b. ] [ E m k m k ke m m ke m ke m Ek loss Qualiy Th las rm quals zro by h dfiniion of h man.
51 Thrfor, qualiy loss = k m k. -8. L dno h vn ha an amplifir fails bfor 6, hours. L A dno h vn ha an amplifir man is, hours. Thn A' is h vn ha h man of an amplifir is, hours. Now, E = E AA + E A'A' and.9 6,, / 6,, /, d A E.6988 ' 6 / 6,, / A E. Thrfor, E = = from h dfiniion of condiional probabiliy. Now, d Thrfor, -87. a ppm Z b ppm c ppm Z 7,.7 d ppm 66, If h procss is cnrd si sandard dviaions away from h spcificaion limis and h procss man shifs vn on or wo sandard dviaions hr would b minimal produc producd ousid of spcificaions. If h procss is cnrd only hr sandard dviaions away from h spcificaions and h procss shifs, hr could b a subsanial amoun of produc ousid of h spcificaions.
Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
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