Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016

Size: px
Start display at page:

Download "Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016"

Transcription

1 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 = a., by symmry. b.. d c... d.. d < =. < or >. =... f. d.... Thn, = a d.

2 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 b d. c d. 6 / d d.. Thn, / 9., and = ln.9 = a <. or >.7 = <. + >.7 bcaus h wo vns ar muually clusiv. Thn, <. = and.8 >.7 = d....7 b If h probabiliy dnsiy funcion is cnrd a. mrs, hn f for. < <.8 and all rods will m spcificaions.. -. a..p. d p.. b.p. d p.. 68 c a.p. d p. p.a. a. 99 a -6. Bcaus h ingral f d is no changd whhr or no any of h ndpoins and ar includd in h ingral, all h probabiliis lisd ar qual. a.

3 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion a <.8 =.8 bcaus is a coninuous random variabl. Thn, <.8 =F.8=..8 =.6. b c F d 6 F 6 -. Now, f / 8 for < < and F for <. Thn, F, 9, 6, u u du For, F.6.6 For, F F d , , F 6...,, d

4 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion - -. E.d. V. d. -7. E. d. V...6 d. d -. E d -9. a E d. Using ingraion by pars wih u = and dv E d d, w obain. Now,. V d. Using h ingraion by pars wih u. dv, w obain V.. d From h dfiniion of E h ingral abov is rcognizd o qual. Thrfor, V.... b. d and.

5 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion - -. a E = -+/ = V /, and.77 b d.. Thrfor, should qual.9., c F..,, -. a Th disribuion of is f = for.9 < <.. Now,,.9 F 9.,.9.,. b.. F.. c If > =.9, hn F =.9 and F =.. Thrfor, - 9. =. and = d E =. +.9/ =. and V = 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 / = /. 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/ = /.

6 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion a Z<. =.968 b Z<. =.9986 c Z>. =.967 =.7 d Z >. = pz <. =.98. < Z <.76 = Z<.76 Z >. = 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. < Z < z = Z < z Z <. = Z < z.79 Thrfor, Z < z = =.979 and z = a < = Z < / = Z <. =.99 b > 9 = < 9 = Z < 9/ = Z <. = c 6 < < = Z = < Z < = Z < Z < ]=.9 d < < = Z = < Z < = Z < Z < =. < < 8 = < 8 < = 8 Z Z = Z < Z < 6 = a > = Z =.. Thrfor, =. b > = Z =.9. Thrfor, Z =. Thrfor, =.6, and =.6. c < < 9 = Z =.. Thrfor, Z < Z < =. whr Z < =.8. Thus Z < =.6. Consqunly, =.6 and = 6.. d < < = Z =.9. Thrfor, Z Z <. =.9 and Z.8 =.9 Consqunly, Z =.8.

7 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 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.8 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 % a > 7 = Z Z b < 8 = Z Z.. 88 c,, bys*8 bis/by 8,, bis 8,, bis. sconds 6, bis/sc a > = Z = Z >.96 =.8.9 b < = Z = Z < -.6 =. c > =., hn

8 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion a E =. = 8, V =..6 = 8 and Thn, 7 Z Z b Z.7 Z c Z.77 Z L dno h numbr of popl wih a disabiliy in h sampl. ~ Bin, Z is approimaly N, a b 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.

9 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, Z Z.7 Z Wihou h coninuiy corrcion, h following rsul is obaind Z Z Z.6 Z c d Th oisson disribuion would no b appropria bcaus h ra of vns should b consan for a oisson disribuion. -9. Approima wih a normal disribuion. np.7 np p a Z b 7 7 Z

10 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion If E =, hn.... a. d b. c d. d. 9 and = 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 >. /.. Thrfor, h answr is -. =.866. Alrnaivly, h rqusd probabiliy is qual o < =.866. c / /. / d < =.9 and. 9. Thrfor, =. minus. -. 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 =. minus. d < =.9 implis ha > =.. Thrfor, his answr is h sam as par c... < =. and = 6.9 minus.

11 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6-7. 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. Thrfor, E = calls pr minus. a > =. 8!!!! b = =. 979! c L Y dno h im bwn calls in minus. Thn, Y. and Y. d Y > =..... d. Thrfor,..y dy.y and = 6. minus. Bcaus h calls ar a oisson procss, h numbrs of calls in disjoin inrvals ar indpndn. Th probabiliy of no calls in on-half hour is Thrfor, h answr 6 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. -9. a / / d. 679 / b. / c. 98 d Th rsuls do no dpnd on.. -. 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 = d.

12 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 -. is an ponnial random variabl wih =. days. /. /. 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. = a p p. 9.8 b p c. p a. p. 7 b 7 p /7. 6 c 6.9 p

13 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Scion a 6! b / / 6 c a Th im unil h nh call is an Erlang random variabl wih calls pr minu and r =. b E = / = minus. V = / =. minus. c Bcaus a oisson procss is mmorylss, h man im is / =. minus or sconds L Y dno h numbr of calls in on minu. Thn, Y is a oisson random variabl wih calls pr minu. d Y = =. 7! Y > = - Y. 87!!! 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 =.87. Thrfor, W = = L dno h im bwn failurs of a lasr. Hr 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! -. L dno h numbr of bis unil fiv rrors occur. Thn, has an Erlang disribuion wih r = and rror pr bi. r a E = bis. r b V = and 67 bis. c L Y dno h numbr of rrors in bis. Thn, Y is a oisson random variabl wih / rror pr bi = rror pr bis. Y Y. 8!!! -. a L dno h numbr of cusomrs ha arriv in minus. Thn, is a oisson random variabl wih. arrivals pr minu = arrivals pr minus.. 9!!!! b L Y dno h numbr of cusomrs ha arriv in minus. Thn, Y is a oisson random variabl wih EY = arrivals pr minus.

14 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Y Y.87!!!!! r r r y y dy -7. f ;, r d d. L y =, hn h ingral is r. From h r dfiniion of r, his ingral is rcognizd o qual. -9. 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!!! 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 Man of.7 paricls pr nanosconds. Thrfor, h man of.7/ =.87 paricls pr nanoscond. Hr r = b dnos h im unil fifh paricl arrivs N dnos h numbr of paricls in nanoscond N has a oisson disribuion wih =.87 paricls pr nanoscond n nanoscond N n n! e -.9E - 7.7E - Scion - -.

15 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 =. and = hours E!, V.. [. ].6 -. If is a Wibull random variabl wih =and =, h disribuion of is h ponnial disribuion wih =.. f.. for Th man of is E = / =. for -7. a E 9 / 9 / hours b V hours 9 c F a =, = E.... b V [ [.] ] 6. c < = F = Var E..9. E. Rquirs a numrical soluion o hs wo quaions. / -6. a F , b 6 6

16 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 F F 6 6/ / c If i is an ponnial disribuion, hn = and F F 6 6/ /..7.7 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. 68 b Th man of his Wibull disribuion is. = 77. If i is an ponnial disribuion wih his man hn /77. F.88 c Th probabiliy ha h lifim cds hours is grar undr h ponnial disribuion han undr his Wibull disribuion modl a 9 b p. 9 a c a p. a p. ln.99 ln ln ln p.7 ln.9 ln ln ln.9.9.

17 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Supplmnal Erciss -9. f. for < <7 7 a 7.d b 6.d c E 6. sconds 7 V.8 sconds -9. a < = Z = Z <. =.9979 b < = Z = Z <. =.6 Hr.6% ar scrappd 6-9. a < = Z = Z < - =. 6 6 b > 6 = Z = Z > = - Z < = -.8=.86 6 c < = Z =.99 6 Thrfor, =. and = 7-9. a > 9. + < = Z + Z = Z > + Z <.. = Z < + Z < =.8 + =.866 b Th procss man should b s a h cnr of h spcificaions; ha is, a c 89.7 < < 9. = Z = < Z < = Th yild is.997 = 99.7%

18 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, d 89.7 < < 9. = Z = < Z < = = =.997 =.97 L Y rprsn h numbr of cass ou of h sampl of ha ar bwn 89.7 and 9. ml. Thn Y follows a binomial disribuion wih n= and p=.997. Thus, EY= a < < 8 = Z =. < Z < - = Z < Z <. =.. b > =.. Thrfor, Z =. and =.8 Thrfor, =.6 hours -97. E =. = and V =..8 = 6 a. 6 Z Z b Z. Z c If > =., hn Z =.. 6 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 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

19 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 bwn calls is. hours and /. calls pr hour. Thrfor, E = 6 calls in hours !!!! -. L dno h im in days unil h fourh problm. Thn, has an Erlang disribuion wih r = and / problm pr day. a E = days. b L Y dno h numbr of problms in days. Thn, Y is a oisson random variabl wih EY = problms pr days. Y.!!!! -. L dno h lifim a E 7 6. b V 7 7 [.] 7 7., c > 6. =.9 -. a E p /. p So ln.686 And ln. / 7.7 b. p W. W ln. ln a.. d.. 6 b. d.. 7. c... d d F. d.. Thn,

20 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 F,,, E. d. 8 V... d 9 9 d d -. L dno h im bwn calls. Thn, / E. calls pr minu.... a. d.9... b. 8.. c < =.9. Thn,. d.9. Now, =. minus. d This answr is h sam as par a.... d..9 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.66 f L Y dno h numbr of calls in minus. Thn, Y is a oisson random variabl wih EY =. Y..!!! 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 / 6. a / 6 / 6. d

21 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 b L W dno h numbr of CUs ha fail wihin h n hr yars. Thn, W is a binomial random variabl wih n = and p =.9. Thn, W W is a lognormal disribuion wih = and = a W ln W ln ln ln W ln b W ln. ln / c E / 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 y y y y y y.6 Subsiu y back ino h firs quaion and solv for o obain.6 ln. and ln b W ln. W ln

22 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6-8. L dno h numbr of fibrs visibl in a grid cll. Thn, has a oisson disribuion and fibrs pr cm. Thrfor, E = 8, 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 has a oisson disribuion, hn h numbrs 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... a >. = Z = Z > -. = - Z -. = b. < <. = Z =- < Z < = c. > =.9 = Z =.9 and = Thrfor, = a d b Ys, bcaus h probabiliy of a plan growing o a high of. cnimrs or mor wihou irrigaion is small. -. L dno h hicknss.. a >. = Z = Z >. = b. < <. = Z = -. < Z <. = Thrfor, h proporion ha do no m spcificaions is. < <. =.. c If < =.9, hn Z =.9. Thrfor, =.6 and = L dno h do diamr. If. < <.6 =.997, hn Z Z Thrfor,. 6 = and If.- < <.+, hn -/. < Z < /. =.997. Thrfor, /. = and =.. Th spcificaions ar from.8 o.. -. L dno h lif..

23 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 a Z Z Z. 6 b If > =.9, hn Z < 7 = -.8. Consqunly, 7 = -.8 and = 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 8 a d b d Thrfor,. 9 and 7 ln hours -6. 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 Squar h firs quaion 6 7 y 7 y and subsiu ino h scond quaion y.7 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

24 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 ln hours -7. a Using h normal approimaion o h binomial wih n = 66 = 6,8and p =. w hav E = 68. = np np p Z b. 6.8 np np p Z Using h normal approimaion o h binomial wih bing h numbr of popl who will b sad. Thn ~Bin, a 8 np Z np p b np Z np p c 8.9, Succssivly ry various valus of n. Th numbr of rsrvaions akn could b rducd o 98. n Z o robabiliy Z < Z a

25 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 6 b. d... c a.9 a.9 a. a -. L dno h ampliud. W know E and V.. Drmin and as follows. E p / V p p. p / p p Thn, p and p.696 /, hn. 667 a p W W ln ln b Find a such ha a. ln a.667 a a W.696 ln a.667.6, hn a..696 ln a... a 6 a 6 a.99 Z.99 Z. 8 8 a a 6.6 a b Z c Th subjcs can b disinguishd wll bcaus h mans ar qui diffrn rlaiv o h sandard dviaions. -.

26 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 a. Th funcion is symmric and hr ar wo paks a and. b Th funcion is symmric and hr ar no paks. Acually his probabiliy dnsiy funcion is h sam as h sandard uniform disribuion. c Th funcion is symmric and hr is on pak a..

27 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 -. np np p. 6. a 6 6 Z Z b 6 Z a. a a Z.. c a. a..9.6 a L dno h im inrval bwn filopodium formaion. a 9 p / 6 9. b p / 6 7 p / c Find a such ha a. 9. a p / 6 a.9, hn a 6 ln.9 and a. 6 -.

28 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, E 6 E 6. V 6 V. -6. L dno h survival im of AMI pains. a. scal paramr;. 6 shap paramr E.7 V..6 b p F. 68. c Find a such ha a. 9 a Thn, a. 6. p a a F a..6.9 Mind-Epanding Erciss -7. 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, Y r i b i c f d d F r i! r i r i! i i r i i i i! i i! r! -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.6 [.6] Y Z Thn, >.6 =.... r

29 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6-9. 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. Thrfor, qualiy loss = k m k. -. 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, -., hn E Suppos, hn E E V E

30 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 Suppos, hn E W hav usd h propry of h gamma funcion r r r in his soluion. -.,, p f a p d p d b p p p d =.6 -. 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.

3(8 ) (8 x x ) 3x x (8 )

3(8 ) (8 x x ) 3x x (8 ) Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6

More information

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b 4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs

More information

Microscopic Flow Characteristics Time Headway - Distribution

Microscopic Flow Characteristics Time Headway - Distribution CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,

More information

Institute of Actuaries of India

Institute of Actuaries of India Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6

More information

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -

More information

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o

More information

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =

More information

Discussion 06 Solutions

Discussion 06 Solutions 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

More information

SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz

SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz 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

More information

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h

More information

Elementary Differential Equations and Boundary Value Problems

Elementary Differential Equations and Boundary Value Problems Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ

More information

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn

More information

H is equal to the surface current J S

H is equal to the surface current J S Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion

More information

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an

More information

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r) 98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w +

More information

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar

More information

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 ) AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc

More information

An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT

An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT [Typ x] [Typ x] [Typ x] ISSN : 974-7435 Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [15197-1521] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI

More information

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form:

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form: Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,

More information

Poisson process Markov process

Poisson process Markov process E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion

More information

The transition:transversion rate ratio vs. the T-ratio.

The transition:transversion rate ratio vs. the T-ratio. PhyloMah Lcur 8 by Dan Vandrpool March, 00 opics of Discussion ransiion:ransvrsion ra raio Kappa vs. ransiion:ransvrsion raio raio alculaing h xpcd numbr of subsiuions using marix algbra Why h nral im

More information

On the Speed of Heat Wave. Mihály Makai

On the Speed of Heat Wave. Mihály Makai On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.

More information

Midterm exam 2, April 7, 2009 (solutions)

Midterm exam 2, April 7, 2009 (solutions) Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions

More information

7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS *

7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS * Andri Tokmakoff, MIT Dparmn of Chmisry, 5/19/5 7-11 7.4 QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS * Inroducion and Prviw Now h origin of frquncy flucuaions is inracions of our molcul (or mor approprialy

More information

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin

More information

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields! Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr

More information

Charging of capacitor through inductor and resistor

Charging of capacitor through inductor and resistor cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.

More information

whereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas

whereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically

More information

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm

More information

Control System Engineering (EE301T) Assignment: 2

Control System Engineering (EE301T) Assignment: 2 Conrol Sysm Enginring (EE0T) Assignmn: PART-A (Tim Domain Analysis: Transin Rspons Analysis). Oain h rspons of a uniy fdack sysm whos opn-loop ransfr funcion is (s) s ( s 4) for a uni sp inpu and also

More information

Chapter 6 Differential Equations and Mathematical Modeling

Chapter 6 Differential Equations and Mathematical Modeling 6 Scion 6. hapr 6 Diffrnial Equaions and Mahmaical Modling Scion 6. Slop Filds and Eulr s Mhod (pp. ) Eploraion Sing h Slops. Sinc rprsns a lin wih a slop of, w should d pc o s inrvals wih no chang in.

More information

1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:

1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to: Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding

More information

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs

More information

EXERCISE - 01 CHECK YOUR GRASP

EXERCISE - 01 CHECK YOUR GRASP DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc

More information

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is 39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h

More information

EE 434 Lecture 22. Bipolar Device Models

EE 434 Lecture 22. Bipolar Device Models EE 434 Lcur 22 Bipolar Dvic Modls Quiz 14 Th collcor currn of a BJT was masurd o b 20mA and h bas currn masurd o b 0.1mA. Wha is h fficincy of injcion of lcrons coming from h mir o h collcor? 1 And h numbr

More information

Double Slits in Space and Time

Double Slits in Space and Time Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an

More information

fiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are

fiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor

More information

A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate

A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate A Condiion for abiliy in an I Ag rucurd Disas Modl wih Dcrasing urvival a A.K. upriana, Edy owono Dparmn of Mahmaics, Univrsias Padjadjaran, km Bandung-umng 45363, Indonsia fax: 6--7794696, mail: asupria@yahoo.com.au;

More information

Chapter 6 Test December 9, 2008 Name

Chapter 6 Test December 9, 2008 Name Chapr 6 Ts Dcmbr 9, 8 Nam. Evalua - ÄÄ ÄÄ - + - ÄÄ ÄÄ ÄÄ - + H - L u - and du u - du - - - A - u - D - - - ÄÄÄÄ 6 6. Evalua i j + z i j + z i 7 j ÄÄÄ + z 9 + ÄÄÄ ÄÄÄ 9 ÄÄÄ + C 6 ÄÄÄ + ÄÄÄ 9 ÄÄÄ + C 9.

More information

Decline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.

Decline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline. Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.

More information

Lecture 4: Laplace Transforms

Lecture 4: Laplace Transforms Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions

More information

Chapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu

Chapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im

More information

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of Sampl Final 00 1. Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy

More information

CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy

More information

Wave Equation (2 Week)

Wave Equation (2 Week) Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris

More information

EE 350 Signals and Systems Spring 2005 Sample Exam #2 - Solutions

EE 350 Signals and Systems Spring 2005 Sample Exam #2 - Solutions EE 35 Signals an Sysms Spring 5 Sampl Exam # - Soluions. For h following signal x( cos( sin(3 - cos(5 - T, /T x( j j 3 j 3 j j 5 j 5 j a -, a a -, a a - ½, a 3 /j-j -j/, a -3 -/jj j/, a 5 -½, a -5 -½,

More information

MGM 562 Probability Theory [Teori Kebarangkalian]

MGM 562 Probability Theory [Teori Kebarangkalian] UNIVERSITI SAINS MALAYSIA Firs Smsr Eaminaion 016/017 Acadmic Sssion Dcmbr 016 / January 017 MGM 56 Probabiliy Thory [Tori Kbarangkalian] Duraion : hours [Masa : jam] Plas chck ha his aminaion ar consiss

More information

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th

More information

UNSTEADY FLOW OF A FLUID PARTICLE SUSPENSION BETWEEN TWO PARALLEL PLATES SUDDENLY SET IN MOTION WITH SAME SPEED

UNSTEADY FLOW OF A FLUID PARTICLE SUSPENSION BETWEEN TWO PARALLEL PLATES SUDDENLY SET IN MOTION WITH SAME SPEED 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. USTEADY FLOW OF A FLUID PARTICLE SUSPESIO BETWEE TWO PARALLEL PLATES SUDDELY SET I MOTIO WITH SAME SPEED M. suniha, B. Shankr and G. Shanha 3

More information

Continuous probability distributions

Continuous probability distributions Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd

More information

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15] S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:

More information

Lecture 2: Bayesian inference - Discrete probability models

Lecture 2: Bayesian inference - Discrete probability models cu : Baysian infnc - Disc obabiliy modls Many hings abou Baysian infnc fo disc obabiliy modls a simila o fqunis infnc Disc obabiliy modls: Binomial samling Samling a fix numb of ials fom a Bnoulli ocss

More information

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +

More information

Transfer function and the Laplace transformation

Transfer function and the Laplace transformation Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and

More information

General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract

General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com

More information

Logistic equation of Human population growth (generalization to the case of reactive environment).

Logistic equation of Human population growth (generalization to the case of reactive environment). Logisic quaion of Human populaion growh gnralizaion o h cas of raciv nvironmn. Srg V. Ershkov Insiu for Tim aur Exploraions M.V. Lomonosov's Moscow Sa Univrsi Lninski gor - Moscow 999 ussia -mail: srgj-rshkov@andx.ru

More information

Lagrangian for RLC circuits using analogy with the classical mechanics concepts

Lagrangian for RLC circuits using analogy with the classical mechanics concepts Lagrangian for RLC circuis using analogy wih h classical mchanics concps Albrus Hariwangsa Panuluh and Asan Damanik Dparmn of Physics Educaion, Sanaa Dharma Univrsiy Kampus III USD Paingan, Maguwoharjo,

More information

a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system:

a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system: Undrdamd Sysms Undrdamd Sysms nd Ordr Sysms Ouu modld wih a nd ordr ODE: d y dy a a1 a0 y b f If a 0 0, hn: whr: a d y a1 dy b d y dy y f y f a a a 0 0 0 is h naural riod of oscillaion. is h daming facor.

More information

Week 06 Discussion Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 8

Week 06 Discussion Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 8 STAT W 6 Discussion Fll 7..-.- If h momn-gnring funcion of X is M X ( ), Find h mn, vrinc, nd pmf of X.. Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) 8 7, f ( ),,, 6, 8,. ( possibl

More information

Mundell-Fleming I: Setup

Mundell-Fleming I: Setup Mundll-Flming I: Sup In ISLM, w had: E ( ) T I( i π G T C Y ) To his, w now add n xpors, which is a funcion of h xchang ra: ε E P* P ( T ) I( i π ) G T NX ( ) C Y Whr NX is assumd (Marshall Lrnr condiion)

More information

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if. Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[

More information

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 11

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 11 8 Jun ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER SECTION : INCENTIVE COMPATABILITY Exrcis - Educaional Signaling A yp consulan has a marginal produc of m( ) = whr Θ = {,, 3} Typs ar uniformly disribud

More information

Part I: Short Answer [50 points] For each of the following, give a short answer (2-3 sentences, or a formula). [5 points each]

Part I: Short Answer [50 points] For each of the following, give a short answer (2-3 sentences, or a formula). [5 points each] Soluions o Midrm Exam Nam: Paricl Physics Fall 0 Ocobr 6 0 Par I: Shor Answr [50 poins] For ach of h following giv a shor answr (- snncs or a formula) [5 poins ach] Explain qualiaivly (a) how w acclra

More information

CHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0)

CHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0) CHATER 6 inar Sysms of Diffrnial Equaions 6 Thory of inar DE Sysms! ullclin Skching = y = y y υ -nullclin Equilibrium (unsabl) a (, ) h nullclin y = υ nullclin = h-nullclin (S figur) = + y y = y Equilibrium

More information

Final Exam : Solutions

Final Exam : Solutions Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b

More information

4.3 Design of Sections for Flexure (Part II)

4.3 Design of Sections for Flexure (Part II) Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon 4. Dsign of Scions for Flxur (Par II) This scion covrs h following opics Final Dsign for Typ Mmrs Th sps for Typ 1 mmrs ar xplaind in Scion

More information

Poisson Arrival Process

Poisson Arrival Process 1 Poisso Arrival Procss Arrivals occur i) i a mmorylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = 1 λδ + ( Δ ) P o P j arrivals durig Δ = o Δ for j = 2,3, ( ) o Δ whr lim =

More information

C From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction.

C From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction. Inducors and Inducanc C For inducors, v() is proporional o h ra of chang of i(). Inducanc (con d) C Th proporionaliy consan is h inducanc, L, wih unis of Hnris. 1 Hnry = 1 Wb / A or 1 V sc / A. C L dpnds

More information

A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER

A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail:

More information

Chemistry 988 Part 1

Chemistry 988 Part 1 Chmisry 988 Par 1 Radiaion Dcion & Masurmn Dp. of Chmisry --- Michigan Sa Univ. aional Suprconducing Cycloron Lab DJMorrissy Spring/2oo9 Cours informaion can b found on h wbsi: hp://www.chmisry.msu.du/courss/cm988uclar/indx.hml

More information

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Inroducion and Linar Sysms David Lvrmor Dparmn of Mahmaics Univrsiy of Maryland 9 Dcmbr 0 Bcaus h prsnaion of his marial in lcur will diffr from

More information

2. The Laplace Transform

2. The Laplace Transform Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin

More information

Differentiation of Exponential Functions

Differentiation of Exponential Functions Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of

More information

10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve

10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve 0. If p and q ar h lnghs of h prpndiculars from h origin on h angn and h normal o h curv + Mahmaics y = a, hn 4p + q = a a (C) a (D) 5a 6. Wha is h diffrnial quaion of h family of circls having hir cnrs

More information

First Lecture of Machine Learning. Hung-yi Lee

First Lecture of Machine Learning. Hung-yi Lee Firs Lcur of Machin Larning Hung-yi L Larning o say ys/no Binary Classificaion Larning o say ys/no Sam filring Is an -mail sam or no? Rcommndaion sysms rcommnd h roduc o h cusomr or no? Malwar dcion Is

More information

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7

More information

Ma/CS 6a Class 15: Flows and Bipartite Graphs

Ma/CS 6a Class 15: Flows and Bipartite Graphs //206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d

More information

Chap.3 Laplace Transform

Chap.3 Laplace Transform Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl

More information

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison Economics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 3/28/2012 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 16 1 Consumpion Th Vry Forsighd dconsumr A vry forsighd

More information

3.9 Carbon Contamination & Fractionation

3.9 Carbon Contamination & Fractionation 3.9 arbon onaminaion & Fracionaion Bcaus h raio / in a sampl dcrass wih incrasing ag - du o h coninuous dcay of - a small addd impuriy of modrn naural carbon causs a disproporionaly larg shif in ag. (

More information

A MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA

A MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA MTHEMTICL MODEL FOR NTURL COOLING OF CUP OF TE 1 Mrs.D.Kalpana, 2 Mr.S.Dhvarajan 1 Snior Lcurr, Dparmn of Chmisry, PSB Polychnic Collg, Chnnai, India. 2 ssisan Profssor, Dparmn of Mahmaics, Dr.M.G.R Educaional

More information

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi

More information

Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )

Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( ) Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division

More information

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions 4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway

More information

where: u: input y: output x: state vector A, B, C, D are const matrices

where: u: input y: output x: state vector A, B, C, D are const matrices Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &

More information

Chapter 17 Handout: Autocorrelation (Serial Correlation)

Chapter 17 Handout: Autocorrelation (Serial Correlation) Chapr 7 Handou: Auocorrlaion (Srial Corrlaion Prviw Rviw o Rgrssion Modl o Sandard Ordinary Las Squars Prmiss o Esimaion Procdurs Embddd wihin h Ordinary Las Squars (OLS Esimaion Procdur o Covarianc and

More information

3.4 Repeated Roots; Reduction of Order

3.4 Repeated Roots; Reduction of Order 3.4 Rpd Roos; Rducion of Ordr Rcll our nd ordr linr homognous ODE b c 0 whr, b nd c r consns. Assuming n xponnil soluion lds o chrcrisic quion: r r br c 0 Qudric formul or fcoring ilds wo soluions, r &

More information

ERROR ANALYSIS A.J. Pintar and D. Caspary Department of Chemical Engineering Michigan Technological University Houghton, MI September, 2012

ERROR ANALYSIS A.J. Pintar and D. Caspary Department of Chemical Engineering Michigan Technological University Houghton, MI September, 2012 ERROR AALYSIS AJ Pinar and D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr, 0 OVERVIEW Exprimnaion involvs h masurmn of raw daa in h laboraory or fild I is assumd

More information

Impulsive Differential Equations. by using the Euler Method

Impulsive Differential Equations. by using the Euler Method Applid Mahmaical Scincs Vol. 4 1 no. 65 19 - Impulsiv Diffrnial Equaions by using h Eulr Mhod Nor Shamsidah B Amir Hamzah 1 Musafa bin Mama J. Kaviumar L Siaw Chong 4 and Noor ani B Ahmad 5 1 5 Dparmn

More information

Ministry of Education and Science of Ukraine National Technical University Ukraine "Igor Sikorsky Kiev Polytechnic Institute"

Ministry of Education and Science of Ukraine National Technical University Ukraine Igor Sikorsky Kiev Polytechnic Institute Minisry of Educaion and Scinc of Ukrain Naional Tchnical Univrsiy Ukrain "Igor Sikorsky Kiv Polychnic Insiu" OPERATION CALCULATION Didacic marial for a modal rfrnc work on mahmaical analysis for sudns

More information

Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12

Engineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12 Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th

More information

Chapter 3 Common Families of Distributions

Chapter 3 Common Families of Distributions Chaer 3 Common Families of Disribuions Secion 31 - Inroducion Purose of his Chaer: Caalog many of common saisical disribuions (families of disribuions ha are indeed by one or more arameers) Wha we should

More information

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields! Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr

More information

Section 4.3 Logarithmic Functions

Section 4.3 Logarithmic Functions 48 Chapr 4 Sion 4.3 Logarihmi Funions populaion of 50 flis is pd o doul vry wk, lading o a funion of h form f ( ) 50(), whr rprsns h numr of wks ha hav passd. Whn will his populaion rah 500? Trying o solv

More information

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm

More information

Midterm Examination (100 pts)

Midterm Examination (100 pts) Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion

More information

DE Dr. M. Sakalli

DE Dr. M. Sakalli DE-0 Dr. M. Sakalli DE 55 M. Sakalli a n n 0 a Lh.: an Linar g Equaions Hr if g 0 homognous non-homognous ohrwis driving b a forc. You know h quaions blow alrad. A linar firs ordr ODE has h gnral form

More information

Asymptotic Equipartition Property - Seminar 3, part 1

Asymptotic Equipartition Property - Seminar 3, part 1 Asympoic Equipariion Propery - Seminar 3, par 1 Ocober 22, 2013 Problem 1 (Calculaion of ypical se) To clarify he noion of a ypical se A (n) ε and he smalles se of high probabiliy B (n), we will calculae

More information