The Distribution of the Inverse Square Root Transformed Error Component of the Multiplicative Time Series Model
|
|
- Loraine Wilkins
- 6 years ago
- Views:
Transcription
1 Journal of Modrn Applid Saisical Mhods Volum 4 Issu Aricl Th Disribuion of h Invrs Squar Roo Transformd Error Componn of h Muliplicaiv Tim Sris Modl Brigh F. Ajibad Prolum Training Insiu, Nigria, qualrigh_brigh@yahoo.com Chinw R. Nwosu Nnamdi Azikiw Univrsiy, Nigria J. I. Mbgdu Univrsiy of Bnin, Nigria Follow his and addiional works a: hp://digialcommons.wayn.du/jmasm Par of h Applid Saisics Commons, Social and Bhavioral Scincs Commons, and h Saisical Thory Commons Rcommndd Ciaion Ajibad, Brigh F.; Nwosu, Chinw R.; and Mbgdu, J. I. (05) "Th Disribuion of h Invrs Squar Roo Transformd Error Componn of h Muliplicaiv Tim Sris Modl," Journal of Modrn Applid Saisical Mhods: Vol. 4 : Iss., Aricl 5. DOI: 0.37/jmasm/ Availabl a: hp://digialcommons.wayn.du/jmasm/vol4/iss/5 This Rgular Aricl is brough o you for fr and opn accss by h Opn Accss Journals a DigialCommons@WaynSa. I has bn accpd for inclusion in Journal of Modrn Applid Saisical Mhods by an auhorizd dior of DigialCommons@WaynSa.
2 Journal of Modrn Applid Saisical Mhods Novmbr 05, Vol. 4, No., Copyrigh 05 JMASM, Inc. ISSN Th Disribuion of h Invrs Squar Roo Transformd Error Componn of h Muliplicaiv Tim Sris Modl Brigh F. Ajibad Prolum Training Insiu Efurum, Warri, Nigria Chinw R. Nwosu Nnamdi Azikiw Univrsiy Awka, Nigria J. I. Mbgbu Univrsiy of Bnin Bnin Ciy, Nigria Th probabiliy dnsiy funcion, man and varianc of h invrs squar-roo ransformd lf-runcad N, rror componn of h muliplicaiv im sris modl wr sablishd. A comparison of ky-saisical propris of and confirmd normaliy wih man bu wih Var Var whn 0.4. Hnc 0.4 is h rquird condiion for 4 succssful ransformaion. Kywords: Muliplicaiv im sris modl, Error componn, Lf runcad normal disribuion, Invrs squar roo ransformaion, Succssful ransformaion, Momns Inroducion Th gnral muliplicaiv im sris modl for dscripiv im sris analysis is X T S C,,,... n () whr for im, X dnos h obsrvd valu of h sris, T is h rnd, S, h sasonal componn, C h cyclical rm and is h random or irrgular componn of h sris. Modl () is rgardd as adqua whn h irrgular componn is purly random. For a shor priod of im, h cyclical componn is Mr. Ajibad, B. F. is a Dpuy Chif Officr in h Dparmn of Gnral Sudis. him a: qualrigh_brigh@yahoo.com. C. R. Nwosu is an Associa Profssor in h Dparmn of Saisics. J. I. Mbgbu is a Profssor in h Dparmn of Mahs and Saisics. 7
3 AJIBADE ET AL. suprimposd ino h rnd (Chafild, 004) o yild a rnd-cycl componn dnod by M and hnc X () M S whr ar indpndn idnically disribud normal rrors wih man and varianc 0 N, According o Uch (003), h lf runcad normal disribuion N, for X is f x 0 x 0 x k 0 x (3) Using Equaion 3, Iwuz (007) obaind h lf runcad normal disribuion N, for X as f LTN x x 0 x 0 0 x (4) wih man E LTN X (5) and 73
4 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Var LTN X Pr (6) Iwuz (007) also showd ha LTN f x > 0 providd < Daa ransformaions ar h applicaion of mahmaical modificaions o valus of a variabl. Thr ar a gra variy of possibl daa ransformaions, including log X, X,,, X, and. In pracic many X X X muliplicaiv im sris daa do no m h assumpions of a paramric saisical analysis; hy ar no normally disribud, h variancs ar no homognous or boh. In analyzing such daa, hr ar wo choics: i. Adjusing h daa o fi h assumpions by making a ransformaion, or ii. Dvloping nw mhods of analysis wih assumpions which fi h daa in is original form. If a saisfacory ransformaion can b found, i will almos always b asir and simplr o us i rahr han dvloping nw mhods of analysis (Turky, 957). Hnc h nd for his work which aims a finding condiions for saisfacory invrs squar roo ransformaion wih rspc o h rror componn of h muliplicaiv im sris modl from a sudy of is disribuion. A ransformaion is considrd saisfacory or succssful, if h basic assumpions of h modl ar no violad afr ransformaion. (Iwuz al., 008)Th basic assumpions of a muliplicaiv im sris modl placd on h rror componn ar: (i) uni man (ii) consan varianc (iii) Normaliy. According o Robrs (008), ransforming daa mad i much asir o work wih - I was lik sharpning a knif. For mor informaion on choic of appropria ransformaions s Osborn (00), Osborn (00) and Wahanachwakul (0). 74
5 AJIBADE ET AL. Daa Classificaion For a im sris daa o b classifid appropria for invrs squar roo ransformaion, i. h daa mus b amnabl o h muliplicaiv im sris modl. Th approprianss of h muliplicaiv modl is accssd by (a) displaying h daa in h Buy s-ballo Tabl. (b) Ploing h priodic (yarly) mans (μ i ) and sandard dviaions σ i agains h priod (yar) i. If hr is a dpndncy rlaionship bwn μ i and σ i, hn h muliplicaiv modl is appropria. ii. h varianc mus b unsabl. Th sabiliy of h varianc of h im sris is ascraind by obsrving boh h row and column mans and sandard dviaions. If h varianc is no sabl h appropria ransformaion is drmind using Barl (947) as was applid by Akpana and Iwuz (009); Y log X, X, (7) Th linar rlaionship bwn h naural log of priodic sandard dviaions (log σ i ) and naural log of h priodic mans (log μ i ) is givn as log log (8) i i Th valu of slop β according o Barl (947) should b approximaly.5 for h invrs squar roo ransformaion (s Tabl ). Tabl. Barl s ransformaions for som valus of β 0 Transformaion No ransformaion X log X 3 X 3 - X X X 75
6 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Background of h Sudy Sinc Iwuz (007) invsigad h ffc of h logarihmic ransformaion on h rror componn, ( ~ N (, σ )) of h muliplicaiv im sris modl, a numbr of sudis invsigaing h ffcs of daa ransformaion on h various componns of h muliplicaiv im sris modl hav bn carrid ou. (S Iwuz al., 008; Iwu al., 009; Ouony al., 0; Nwosu al., 03; and Ohakw al., 03). Th ovrall aim of such sudis is o drmin h condiions for succssful ransformaion. Tha is, o sablish h condiions whr: a. h rquird basic assumpions of h modl ar no violad afr ransformaion, wih rspc o (i) h rror rm (ii) h sasonal componn. b. wih rspc o h rnd componn, hr is no alraion in h form of h rnd curv. In ohr words h form of h rnd curv in h original sris is mainaind in h ransformd sris. componn varianc Iwuz (007) found ha h logarihmic ransformaion of h rror, providd 0. assumpion for h rror rm N o log is normal wih man 0 and, in which cas. I was sablishd ha h, for h addiiv modl obaind afr h logarihmic ransformaion, is valid if and only if σ < 0.0. Obsrv from Tabl ha β for a im sris daa o b classifid fi for logarihmic ransformaion. Ouony al. (0) invsigad h disribuion and propris of h rror componn of h muliplicaiv im sris modl undr squar roo ransformaion, and found ha h squar roo ransformd rror componn is normally disribud wih man and varianc ims ha of 4 h unransformd rror componn. Tha is Var Var 4 whn 0 < σ 0.3. Thus 0 < σ 0.3 is h rcommndd condiion for succssful squar roo ransformaion. Only im sris daa wih ar classifid fi for squar roo ransformaion. Similarly, Nwosu al. (03), whil invsigaing h disribuion of h invrs ransformd rror componn of h muliplicaiv im 76
7 AJIBADE ET AL. sris modl, obaind ha h dsirabl saisical propris of and wr found o b approximaly h sam and normally disribud wih uni man for σ 0.0. Hnc, σ 0.0 is h rcommndd condiion for succssful invrs ransformaion of h muliplicaiv im sris modl. Tim sris daa classifid fi for invrs ransformaion mus hav β. Also, Ohakw al. ha N, in (03) found ha for h squar ransformaion h inrval 0 < σ Hnc, 0 < σ 0.07is h condiion for succssful squar ransformaion. Obsrv ha a im sris daa is classifid fi for squar ransformaion whn β -. No ha h ovrall aim of hs works is o sablish condiions for succssful ransformaion, hnc provid br choic of righ ransformaion. According o Robrs (008), choosing a good ransformaion improvd his analyss in hr ways: (i) incras in visual clariy as graphs wr mad mor informaiv (ii). Rducion or liminaion of oulirs (iii). Incras in saisical clariy; his saisical s bcam mor snsiiv, F and valus incrasd making i mor likly o dc diffrncs whn hy xis. Jusificaion for his Sudy Th valu of h slop, cagorizd im sris daa ino muually xclusiv groups, in h sns ha any im sris daa blongs xclusivly o on and only on group hnc can only b approprialy ransformd by only on of h six ransformaions lisd in Tabl. Thus dspi h fac ha Iwuz (007), Ouony al., (0), Nwosu al. (03), and Ohakw al. (03) carrid ou similar sudis wih rspc o h logarihmic, squar roo, invrs and squar ransformaions rspcivly, his work on invrs squar roo ransformaion is sill vry ncssary sinc rsuls sablishd for h abov lisd four ransformaions canno b applid in h analysis of im sris daa rquiring invrs squar roo ransformaion. Invrs Squar Roo Transformaion 3 Whn, adop invrs squar roo ransformaion on h muliplicaiv im sris modl givn in Equaion o obain 77
8 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Y X M S M S (9) whr M, S and, 0 M S Bcaus dos no admi ngaiv or zro valus, h us of h lf runcad normal disribuion as h pdf of shall b xploid. Thus, i will b of inrs o find wha h disribuion of is. Is iid N,. Wha is h rlaionship bwn and? Aim and Objcivs Th aim of his work is o obain h disribuion of h invrs squar roo ransformd rror componn of h muliplicaiv im sris modl and h objcivs ar: i. o xamin h naur of h disribuion. ii. o vrify h saisfacion of h assumpion on h man of h rror rms; μ =. iii. o drmin h rlaionship bwn and. Mhodology To achiv h abov sad objcivs h following wr conducd: L X = and Y = = X. Obain h pdf of, g(y).. Plo h curvs of h wo pdfs, g(y) and f LTN (x) for various valus of. 3. Obain h rgion whr g(i) saisfis h following normaliy condiions (Bll-shapd condiions). i. Mod Man. ii. Mdian Man. 78
9 AJIBADE ET AL. iii. iv. Approvd normaliy s, Andrson Darling s s saisic (AD) was usd o confirm h normaliy of h simulad rror rms and h invrs squar roo ransformd rror rm. Y = = X for som valus of σ Obain and us h funcional xprssions for h man and varianc of o valida som of h rsuls obaind using simulad daa. Th probabiliy dnsiy funcion of Y, g( y) x Givn h pdf of X in Equaion 4 and h ransformaion Y x hn X dx and y dy y 3 using h ransformaion of variabl chniqu dx g y f x LTN dy (s Frund & Walpol, 986). Hnc y,0 y g y 3 y 0 y 0 (0) 79
10 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Plo of h Probabiliy dnsiy curvs f x f x LTN and g(y) Using h pdf of h wo variabls givn in Equaion 4 and Equaion 0, h curvs f x and g(y) wr plod for som valus of (0, 0.4]. For wan of spac only fiv ar shown in Figurs o 5. Figur. Curv Shaps for σ = 0.06 Figur. Curv Shaps for σ =
11 AJIBADE ET AL. Figur 3. Curv Shaps for σ = 0.5 Figur 4. Curv Shaps for σ = 0.3 Figur 5. Curv Shaps for σ = 0.4 8
12 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Obsrvaions: i. Th curv g(y) is posiivly skwd for σ > 0.5 (s Figurs 3-5). ii. f(x) is posiivly skwd for σ > 0.30 (s Figur 5) as rpord in Iwuz (007). Normaliy Rgion for g(y) From Figurs o 5, i is clar ha h curv g(y) has on maximum poin, y max (mod), and on maximum valu, g(y max ), for all valus of σ. To obain h valus of σ ha saisfy h symmric and bll-shapd condiion of mod = man, w invok Roll s Thorm and procd o obain h maximum poin (mod) for a givn valu of σ. Diffrniaing g(y) in Equaion 0 givs g'( y) 4 4 y 3 y 3y y 3 y y () y ( y ) y y Equaing g`(y) = 0, givs ( y ) 3 y 0 y y y 0 () Puing w = y in Equaion, givs 8
13 AJIBADE ET AL. 3 w w 0 (3) Solving Equaion 3, givs Bcaus y max is posiiv hn 6 w 3 6 w 3 hnc y 6 3 and y 6 3 max Th bll-shapd condiion would imply y max, s Tabl for h numrical compuaion of y max
14 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl. Compuaion of y max 6, for [0.0, 0.3] 3 y max y max 6 3 y max 6 3 y max Thus g(y) is symmrical abou wih Mod Man corrc o wo dcimal placs whn 0 < σ < and corrc o on dcimal plac whn 0 < σ <
15 AJIBADE ET AL. Us of simulad rror rms To find h rgion whr h bll-shapd condiions (ii-iii) lisd in mhodology N, for, ar saisfid, w mad us of arificial daa gnrad from subsqunly ransformd o obain for Valus of h rquird saisical characrisics wr obaind for ach variabl and as shown in Tabls 3 o 6. For ach configuraion of (n = 00, 0.05 σ 0.5), 000 rplicaions wr prformd for valus of σ in sps of 0.0. For wan of spac h rsuls of h firs 5 rplicaions ar shown for h configuraions, (n = 00, σ = 0.06), (n = 00, σ = 0.), (n = 00, σ = 0.5), and (n = 00, σ = 0.). Funcional xprssions for h man and varianc of g(y) By dfiniion, h man of Y, E(Y) is givn by: y 0 E( Y) yg( y) dy dy 0 y (4) l u, hn y and du 3 y u dy, for u 0 u 0 u u du k 3 u 0 (5) E( Y) k u u du whr k l u z, hn z u and du dz for z z z k k E( Y) ( z ) dz ( z ) dz (6) 85
16 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl 3. Simulaion Rsuls whn σ = 0.06 X N,, 0.06 Y,, N, 0.06 Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu No. For ach row, Var Var quals 4. 86
17 AJIBADE ET AL. Tabl 4. Simulaion Rsuls whn σ = 0. X N,, 0. Y,, N, 0. Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu No. For ach row, Var Var quals 4. 87
18 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl 5. Simulaion Rsuls whn σ = 0.5 X N,, 0.5 Y,, N, 0.5 Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu No. For ach row, Var Var quals 4 xcp whr indicad by. For hos rows, Var Var quals 3. 88
19 AJIBADE ET AL. Tabl 6. Simulaion Rsuls whn σ = 0. X N,, 0. Y,, N, 0. Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu < < < < < < < < < < < < < < < No. For ach row, Var Var quals 3. 89
20 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Using h binomial xpansion, 3 n nx n( n ) x n( n )( n ) x ( x)... (7)!! 3! (Smih and Minon, 008). 3 z ( z ) ( z ) ( z )...!! 3! 3 z 3( z ) 5( z )... (8) z k z 3( z ) 5( z ) E( Y)... d 8 48 (9) 3 z z z z z 3( z ) 5( z ) dz dz dz dz z z z dz dz EY ( ) z 3 z 3( z) 5( z) dz dz
21 AJIBADE ET AL. z z z dz dz EY ( ) z 3 3 z 3z 5z dz dz Pr z Pr x () Pr () EY ( ) Pr () () EY ( ) Pr (0) 9
22 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL To find h varianc, firs obain h scond momn; ( ) ( ) E Y y g y dy 0 y 0 y dy du l u hn du dy, 3 and dy 3 for u 0 y y u 0 u ( u) du k 3 0 () E( Y ) k u u du u whr k l u z hn uz and du dz for z z k E( Y ). ( z) dz Using h binomial xpansion on (+zσ) -, givn in Equaion 6 w hav 3 ( z ) z ( z ) ( z )... z k 3 E( Y ) [ z ( z ) ( z )...] dz () 9
23 AJIBADE ET AL. EY ( ) z z dz z dz z 3 z 3 z dz z dz Pr z Pr x () Pr x () 3 () E( Y ) Pr (3) 3 Obsrv h following:. Subsqun rms in sris (0) and (3) for E(Y) and E(Y ) rspcivly all hav as a facor. 93
24 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL 0. for σ 0. corrc o 4 dcimal placs. (S Tabl 7, column 3) 3. Condiions () and () imply ha all subsqun rms for E(Y) and E(Y ) ar all zros for σ 0.. Thus, wihou loss of gnraliy 3 EY ( ) Pr () for 0. 6 (4) and EY ( ) Pr () for 0. (5) hus Var( Y) E( Y ) [ E( Y)] Var( Y ) Pr () 3 Pr () 6 94
25 AJIBADE ET AL. Pr () 8 3 Pr () 6 (6) Numrical compuaions of man and varianc of Y Now compu h valus of E(Y) and Var(Y) for σ [0.0,0.] using h funcional xprssions obaind in Equaions 4 and 6, rspcivly. Tabl 7 shows h compuaions of E(Y) and Var(Y). For hs compuaions w wri and EY 3 B 0. 8A Var Y B 3 B 8A 6A whr A and B Pr From Tabl 7, columns 4 and 5, A = and B = for <0. 3 EY 0. (7) 8 and Var Y (8) 95
26 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Equaion 7 is h rlaionship obsrvd wih simulad daa in Tabls 3-6. Rsuls Th following rsuls wr obaind from h invsigaions carrid ou on h pdf of, g( y ) whr N,, lf runcad a 0. i. Th curv shaps ar bll-shapd, wih mod man whn 0 < σ 0.45 corrc o dcimal plac. Using simulad daa, whnvr σ < 0.5 ii. Mdian Man iii. E 3 Var iv. v. Var 8 4, hus var( ) Var( ) 4 is normally disribud whn σ 0.4. I was obsrvd ha h normaliy of a pdf curv a a poin b implid normaliy a poins 0 a b. Using h funcional xprssions for man and varianc of vi. vii. viii. 3 E 0. 8 corrc o dcimal placs (dp) whn σ 0. corrc o dp whn σ 0. 3 Var( ) Var 4 Var corrc o dp whn σ 0.04 corrc o dp whn σ
27 AJIBADE ET AL. Tabl 7. Compuaions of E(Y) & Var(Y) for σ [0.0, 0.3] A B ( ) EY Var( Y ) VarX / Var Y From h probabiliy dnsiy curvs, h rsuls obaind from simulad daa and h funcional xprssions for h man and varianc, σ 0.4 (inrscing rgion) is h rcommndd condiion for succssful invrs squar roo ransformaion. Th rsuls of his invsigaion oghr wih findings from similar N, undr ohr yps of invsigaions wih rspc o h rror rm 97
28 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl 8. Summary of his and similar findings wih rspc o h rror rm log Disribuion of 0,,,, N, undr diffrn ransformaions Condiion for succssful ransformaion N 0. N 0. Rlaionship bwn σ and σ N 0.59 N,, 0.07 N 0.4 Conclusion From h rsuls of h invsigaions of h disribuions of h rror rm of h muliplicaiv im sris modl and is invrs squar roo ransformd rror rm, i is clar ha h condiion for succssful invrs squar roo ransformaion is σ < 0.4. This is bcaus h wo sochasic procsss and ar normally disribud wih man, bu wih h varianc of invrs squar roo ransformd rror rm bing on quarr of h varianc of h unransformd rror componn whnvr σ < 0.4, ousid his rgion ransformaion is no advisabl sinc h basic assumpion on h rror rm ar violad afr h ransformaion. This rlaionship bwn h wo variancs, Var Var, agrs wih findings of Ouony al. (0) undr squar 4 roo ransformaion, howvr h rgion of succssful ransformaion obaind is closr o h rgion obaind for h logarihmic and invrs ransformaions by Iwuz (007) and Nwosu al. (03). 98
29 AJIBADE ET AL. Rfrncs Akpana, A. C. & Iwuz I. S. (009). On applying h Barl ransformaion mhod o im sris daa. Journal of Mahmaical Scincs, 0(3), Barl, M. S. (947). Th us of ransformaions. Biomrica, 3(), doi:0.307/ Chafild, C. (004). Th analysis of im sris: An inroducion (6h d.). London: Chapman & Hall/CRC. Frund, J. E & Walpol, R. E (986). Mahmaical saisics (4h d.). Uppr Saddl Rivr, NJ: Prnic-Hall, Inc. Iwu, H., Iwuz, I. S., & Nwogu, E. C. (009). Trnd analysis of ransformaions of h muliplicaiv im sris modl. Journal of Nigrian Saisical Associaion,, N, Iwuz, I. S. (007). Som implicaions of runcaing h disribuion o h lf a zro. Journal of Applid Scinc, 7(), Iwuz, I. S., Akpana, A. C., & Iwu, H. C. (008). Sasonal analysis of ransformaions of h muliplicaiv im sris modl. Asian Journal of Mahmaics and Saisics (), doi:0.393/ajms Nwosu, C.R., Iwuz, I.S., & Ohakw J. (03). Condiion for succssful invrs ransformaion of h rror componn of h muliplicaiv im sris modl. Asian Journal of Applid Scinc 6(), -5. doi:0.393/ajaps Ohakw, J., Iwuoha, O., & Ouony, E. L. (03). Condiion for succssful squar ransformaion in im sris modling. Applid Mahmaics, 4(4), doi:0.436/am Osborn, J. (00). Nos on h us of daa ransformaions. Pracical Assssmn Rsarch and Evaluaion, 8(6). Availabl onlin: hp://pareonlin.n/gvn.asp?v=8&n=6 Osborn, J. W. (00). Improving your daa ransformaion. Pracical Assssmn Rsarch & Evaluaion, 5(). Availabl onlin: hp://paronlin.n/gvn.asp?v=5&n= Ouony, E. L., Iwuz, I. S., & Ohakw, J. (0). Th ffc of squar roo ransformaion on h rror componn of h muliplicaiv im sris modl. Inrnaional Journal of Saisics and Sysms, 6(4), Robrs, S. (008). Transform your daa. Nuriion, 4, doi:0.06/j.nu
30 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Smih, R. T., & Minon, R. B. (008) Calculus (3rd d.). NY: McGraw Hill. Turky, J. W. (957). On h comparaiv anaomy of ransformaions. Th Annals of Mahmaical Saisics, 8(3), doi:0.4/aoms/ Uch, P. I (003). Probabiliy: Thory and applicaions. Ikga, Lagos: Longman Nigria PLC. Wahanachwakul, L. (0). Transformaion wih righ skw daa. In S. I. Ao, L. Glmn, D. W. L. Hukins, A. Hunr and A. M. Korsunsky, Eds. Procdings of h World Congrss on Enginring Vol.. London, UK: Nwswood Limid. 00
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 informationMicroscopic 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 informationInstitute 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 informationAn 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 informationOn 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 informationLecture 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 informationApplied 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...
More informationUNIT #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 informationMidterm 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 informationCSE 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 informationBoyce/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 informationSpring 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 informationAR(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 informationA 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 informationUNSTEADY 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 informationElementary 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 informationBoyce/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 information7.4 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 informationA 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 informationLet 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 information1. 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 informationDouble 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 informationA 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 informationCPSC 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 informationDiscussion 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 informationControl 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 information3(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 informationCHAPTER 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 informationImpulsive 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 informationCharging 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 informationERROR 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 informationOn 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 informationMEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control
MEM 355 Prformanc Enhancmn of Dynamical Sysms A Firs Conrol Problm - Cruis Conrol Harry G. Kwany Darmn of Mchanical Enginring & Mchanics Drxl Univrsiy Cruis Conrol ( ) mv = F mg sinθ cv v +.2v= u 9.8θ
More informationfiziks 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 informationH 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 informationGeneral 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 information5. 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 informationMathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013
Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 www.iise.org The ffec of Inverse Transformaion on he Uni Mean and Consan Variance Assumpions of a Muliplicaive rror Model
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationCopyright 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 informationI) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning
I) Til: Raional Expcaions and Adapiv Larning II) Conns: Inroducion o Adapiv Larning III) Documnaion: - Basdvan, Olivir. (2003). Larning procss and raional xpcaions: an analysis using a small macroconomic
More informationFinal 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 informationLogistic 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 informationwhereby 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 informationDEPARTMENT 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 informationS.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 informationEconomics 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 informationEXERCISE - 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 informationLecture 2: Current in RC circuit D.K.Pandey
Lcur 2: urrn in circui harging of apacior hrough Rsisr L us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R and a ky K in sris. Whn h ky K is swichd on, h charging
More informationSOLUTIONS. 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 informationChapter 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 information2.1. 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 informationChapter 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 informationTransfer 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 informationThe 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 informationWave 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 informationCHAPTER. 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 informationRatio-Product Type Exponential Estimator For Estimating Finite Population Mean Using Information On Auxiliary Attribute
Raio-Produc T Exonnial Esimaor For Esimaing Fini Poulaion Man Using Informaion On Auxiliar Aribu Rajsh Singh, Pankaj hauhan, and Nirmala Sawan, School of Saisics, DAVV, Indor (M.P., India (rsinghsa@ahoo.com
More informationStochastic Model for Cancer Cell Growth through Single Forward Mutation
Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com
More informationPhys463.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 information4.3 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( ) 2! l p. Nonlinear Dynamics for Gear Fault Level. ( ) f ( x) ( ),! = sgn % " p. Open Access. Su Xunwen *,1, Liu Jinhao 1 and Wang Shaoping 2. !
Nonlinar Dynamics for Gar Faul Lvl Su Xunwn Liu Jinhao and Wang Shaoping Snd Ordrs for Rprins o rprins@bnhamscinc.a Th Opn Mchanical Enginring Journal 04 8 487496 487 Opn Accss School of Tchnology Bijing
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationB) 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 informationAdvanced Queueing Theory. M/G/1 Queueing Systems
Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld
More informationMidterm 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 informationFeedback Control and Synchronization of Chaos for the Coupled Dynamos Dynamical System *
ISSN 746-7659 England UK Jornal of Informaion and Comping Scinc Vol. No. 6 pp. 9- Fdbac Conrol and Snchroniaion of Chaos for h Copld Dnamos Dnamical Ssm * Xdi Wang Liin Tian Shmin Jiang Liqin Y Nonlinar
More informationCircuits and Systems I
Circuis and Sysms I LECTURE #3 Th Spcrum, Priodic Signals, and h Tim-Varying Spcrum lions@pfl Prof. Dr. Volan Cvhr LIONS/Laboraory for Informaion and Infrnc Sysms Licns Info for SPFirs Slids This wor rlasd
More informationDecline 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 informationEE 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 informationReal time estimation of traffic flow and travel time Based on time series analysis
TNK084 Traffic Thory sris Vol.4, numbr.1 May 008 Ral im simaion of raffic flow and ravl im Basd on im sris analysis Wi Bao Absrac In his papr, h auhor sudy h raffic parn and im sris. Afr ha, a im sris
More informationInstructors 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 informationLagrangian 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 information1973 AP Calculus BC: Section I
97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f
More informationt + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that
ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so
More informationVoltage 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 informationDE 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 informationProbability 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 informationMundell-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 informationThe Science of Monetary Policy
Th Scinc of Monary Policy. Inroducion o Topics of Sminar. Rviw: IS-LM, AD-AS wih an applicaion o currn monary policy in Japan 3. Monary Policy Sragy: Inrs Ra Ruls and Inflaion Targing (Svnsson EER) 4.
More informationThe Variance-Covariance Matrix
Th Varanc-Covaranc Marx Our bggs a so-ar has bn ng a lnar uncon o a s o daa by mnmzng h las squars drncs rom h o h daa wh mnsarch. Whn analyzng non-lnar daa you hav o us a program l Malab as many yps o
More informationLecture 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 information2. 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 informationAlmost power law : Tempered power-law models (T-FADE)
Almos powr law : Tmprd powr-law modls T-FADE Yong Zhang Dsr Rsarch Insiu Novmbr 4, 29 Acknowldgmns Boris Baumr Mark Mrschar Donald Rvs Oulin Par Spac T-FADE modl. Inroducion 2. Numrical soluion 3. Momn
More informationChapter 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 informationPhysicsAndMathsTutor.com
C3 Eponnials and logarihms - Eponnial quaions. Rabbis wr inroducd ono an island. Th numbr of rabbis, P, yars afr hy wr inroducd is modlld by h quaion P = 3 0, 0 (a) Wri down h numbr of rabbis ha wr inroducd
More informationTransient Performance Analysis of Serial Production Lines
Univrsiy of Wisconsin Milwauk UWM Digial Commons Thss and Dissraions Augus 25 Transin Prformanc Analysis of Srial Producion Lins Yang Sun Univrsiy of Wisconsin-Milwauk Follow his and addiional works a:
More informationAsymptotic Solutions of Fifth Order Critically Damped Nonlinear Systems with Pair Wise Equal Eigenvalues and another is Distinct
Qus Journals Journal of Rsarch in Applid Mahmaics Volum ~ Issu (5 pp: -5 ISSN(Onlin : 94-74 ISSN (Prin:94-75 www.usjournals.org Rsarch Papr Asympoic Soluions of Fifh Ordr Criically Dampd Nonlinar Sysms
More informationExperimental and Computer Aided Study of Anisotropic Behavior of Material to Reduce the Metal Forming Defects
ISSN 2395-1621 Exprimnal and Compur Aidd Sudy of Anisoropic Bhavior of Marial o Rduc h Mal Forming Dfcs #1 Tausif N. Momin, #2 Vishal B.Bhagwa 1 ausifnmomin@gmail.com 2 bhagwavb@gmail.com #12 Mchanical
More informationContinous system: differential equations
/6/008 Coious sysm: diffrial quaios Drmiisic modls drivaivs isad of (+)-( r( compar ( + ) R( + r ( (0) ( R ( 0 ) ( Dcid wha hav a ffc o h sysm Drmi whhr h paramrs ar posiiv or gaiv, i.. giv growh or rducio
More informationLecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey
cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd
More informationPart 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 informationEstimation of Metal Recovery Using Exponential Distribution
Inrnaional rd Journal o Sinii sarh in Enginring (IJSE).irjsr.om Volum 1 Issu 1 ǁ D. 216 ǁ PP. 7-11 Esimaion o Mal ovry Using Exponnial Disribuion Hüsyin Ankara Dparmn o Mining Enginring, Eskishir Osmangazi
More informationA Simple Procedure to Calculate the Control Limit of Z Chart
Inrnaional Journal of Saisics and Applicaions 214, 4(6): 276-282 DOI: 1.5923/j.saisics.21446.4 A Simpl Procdur o Calcula h Conrol Limi of Z Char R. C. Loni 1, N. A. S. Sampaio 2, J. W. J. Silva 2,3,*,
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More information3.9 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 informationOn Ψ-Conditional Asymptotic Stability of First Order Non-Linear Matrix Lyapunov Systems
In. J. Nonlinar Anal. Appl. 4 (213) No. 1, 7-2 ISSN: 28-6822 (lcronic) hp://www.ijnaa.smnan.ac.ir On Ψ-Condiional Asympoic Sabiliy of Firs Ordr Non-Linar Marix Lyapunov Sysms G. Sursh Kumar a, B. V. Appa
More informationStability and Bifurcation in a Neural Network Model with Two Delays
Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy
More informationa 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 informationChap.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