M.Sc. MATHEMATIS MAL-5 METHODS OF APPLIED MATHEMATIS DIRETORATE OF DISTANE EDUATION GURU JAMBHESHWAR UNIVERSITY OF SIENE AND TEHNOLOGY HISAR-5
MAL-5 METHODS OF APPLIED MATHEMATIS os Lsso No ad Nam Foi Tasfoms Applicaios of Foi Tasfoms vilia o-odias 4 Radom vaiabl ad Mamaical Epcaio 5 Moms ad Mom gaig fcios 6 Toical Disc Disibios 7 Toical oios Disibios 8 Mlipl ad paial olaio Wi by: Pof.Kldip Basal Dpam of Mamaics G.J.Uivsiy of Scic & Tcology Hisa.
LESSON- FOURIER TRANSFORMS Igal Tasfom: T igal asfom of a fcio f() is dfid by qaio b g(s) f () k(s, ) d, w k(s, ) is a kow fcio of s ad, calld kal of a asfom; s is calld paam of asfom ad f() is calld ivs asfom of g(s). Som of wll kow asfoms a giv as d: () Laplac asfom:- W k(s, ) s, w av Laplac asfom of f() as: g(s) f () s d w ca also wi () Foi asfom:- w k(s, ) L[f()] g(s) o F(s) o f (s) is, w av Foi asfom of f() as F[f()] g(s) () Mlli asfom:- f() is d W k(s, ) s,, w av Mlli asfom of f() as: M[f()] g(s) f () s d (4) Hakl asfom (Foi-Bssl):- W k(s, ) J (s),, w av Hakl asfom of f() as: g(s) f () J (s) d Foi Tasfom:- If f() b a fcio dfid o (, ) ad f b picwis coios i ac fii paial ival ad absolly igabl i (, ), i.., f() b a fcio s.. fcio f () d <
f (s) F[f ()] F(s) f () is d is calld Foi asfom (F.T.) of f(). T F Rmaks: [F(s)] F(s) is ds f() is calld ivs Foi Tasfom of F(s). (i) If F(s) f () is d, F [F(s)] F(s) is ds (ii) If F(s) f () is d, F [F(s)] F(s) is ds (iii) If F(s) f () is d, f() F(s) is ds Foi cosi ad si asfom F c (s) f () cos s d is calld Foi cosi asfom (FT) ad F s (s) f () si s d is calld Foi si asfom (FST) of f(). T fcios f c () F c (s) cos s ds f s () F s (s) si s ds a calld Ivs Foi cosi & ivs Foi si asfom of F c (s) ad F s (s), spcivly. 4
Popis of Foi Tasfom () Liaiy Popy If f () ad f () a fcios wi Foi Tasfom F (s) ad F (s), spcivly ad & a cosa, Foi Tasfom of f () f () is F[c f () c f ()] c F[f ()] c F[f ()] c F (s) c F (s) Poof: By dfiiio L.H.S. is [c f () c f ()] d c is f()d c is f() d F[f ()] F[f ()] F (s) F (s) () ag of scal popy o similaiy om If a is a al cosa ad F(s) F[f()] Poof : Fo a >, F[f(a)] P a d a d a F s a is F[f(a)] f (a) d F[f(a)] s i a f () d a f () a s F[f(a)] F a a s i a d Fo a < :- F[f(a)] is f (a) d 5
P a d d a s i a F[f(a)]. f () a d. a s F[f(a)] F a a Hc, F[f(a)] a F s a Paicla cas:- If a, F[f()] F(s) f () s i a d () Fis sifig Popy If F[f()] F(s), F[f( )] -is F[f()] is F(s) Poof :- F[f( )] f ( ) is d P v d dv Θ F[f( )] f (v) is(v ) dv is f (v) isv dv is f () is d is F[f()]. (4). Scod sifig popy If F(s) F[f()], F[f() ia ] F(s a) 6
Poof :- F[f() ia ] f () ia is d (5) Symmy popy f () i(sa) d F(s a) If F(s) F[f()], F[F()] f(s) Poof : W kow a cagig o, w av Icagig & s, w g f () F(s) is ds is f ( ) F(s) ds is f ( s) F() d f(s) F() f(s) F[F()] is d Eampl.. Fid Foi asfom of f() α,, α > (), < Solio :- By dfiiio, F(s) f () is d is F(s) f () d α is d (α is) d [sig ()] F(s) (α is) (α is) 7
α is α is (α s ) Eampl.. f(),, a > a Solio F(s) a f () is d a a f () is d a f () is d I I I a L I f () P d d is d I f ( ) is (d) a f ( ) a is d Similaly I f () is d a I f () a a is d a i. a is d [Θ f() ] a is ias ias is [ ] is a I ias is ias si as si as., s is s w s, 8
F(s) a cosas () as s [By L-Hospial l] a. Eampl. :- f(),, a > a () Solio :- F(s) is f () d a is is f () d f () d f () a a a a is d a a is d [sig ()] is is a a a a is () d is a ias a ias ias ias ( ) is is is( is) F(s) s ias ias ias ias a ( is a cosas si si as is s [i as cos as i si as] ) ( s ) F(s) i s [as cos as si as] Eampl. 4. If F(s) F[f()]. Fid F.T. of f() cos a Solio :- W kow a ia ia cos a ( ) T F[f() cos a] F[f() ia ] F[f() ia ] 9
o pov by dfiiio [By liaiy popy] F[f() cos a] F(s a) F(s a) [By sig sifig popy] Eampl. 5. If F s (s) ad F c (s) a FST ad FT of f() spcivly, F s [f() cos a] [Fs (s a) F s (s a)] F c [f() cos a] [Fc (s a) F c (s a)] F s [f() si a] [Fc (s a) F c (s a)] F c [f() si a] [Fs (s a) F s (s a)] Solio :- (i) By dfiiio of FST, F s [f() cos a] f () cos a si s d. f () cos a si s d Usig si A cos B si (A B) si (AB), F s [f() cos a] f () [si (s a) si (s a)] d f () si(s a) d f ()si(s a) d (ii) By dfiiio FT, [Fs (s a) F s (s a)] F c [f() cos a] f () cos a cos s d f () [cos(s a) cos (sa) ]d
f ()cos(s a) d f ()cos(s a) d (iii) By dfiiio of FST, [Fc (s a) F c (s a)] F s [f() si a] f () si a si s d f () [cos(s a) cos (s a)] d [Θ si A si B cos (A B) cos (A B)] f ()cos(s a)d f ()cos(s a)d (iv) By dfiiio of FT, [Fc (sa) F c (s a)] F c [f() si a] f () si a cos s d. f () [si (s a) si (sa)] d [sig cosa sib si (A B) si (AB)] f ()si(s a) f ()si(s a) d [Fs (s a) F s (s a) F s (s a)]. Eampl. 6. Fid Foi si & cosi asfom of a, a >. L a cos s d I ()
a Igaig () by pas si s d I () I a cos s a a si s d s a a a si s d s I () a a Igaig () by pas, w av s I I (4) a a s a Θ I si s cos s d a a solvig () ad (4) fo I & I, s s I. I a a a s I a I ad I s. a s s a a a a a s s a I. a s a a Hc F c [ a ] I a (5) s a ad F s [ a ] I s (6) s a Esios :- Diffiaig bo sids of (5) w... a, w fid F c [ a ] (s a ) a.a (s a )
F c [ a ] a s (s a ) F c [ a ] a s (s a ) Diffiaig bo sids of (6) w... a, w g F s [ a ] (s a ) s. a (s a ) F s [ a ] (s as a ) F s [ a ] (s as a ). P a i (5), (6), w g F s [ ] s s s s Rsls: F c [ ]. s α α cosβ d α β α α siβ d α β (α cos β β si β) (α si β β cos β) Eampl. 7 :- Fid F. T. of f() / Solio :- By dfiiio, F is d s ( is) / F[ ]. d s / ( is) d
P ( is) y d dy F[ / ] s / y. dy s / y s / dy. F[ / ] s /, F [ c / ] s / Eampl. 8. Fid Foi cosi asfom of f() Solio :- Fc [ ] cos s d I () Diffiaig w... s, w g di ds si s d Igaig by pas,. ( ) si s d di. si s s cos s. d ds s cos s d di ds s I [sig ()] di s ds I s Igaig, log I log A 4 I A s / 4 () 4
w s, fom (), I d I. () Also w s, fom () I A fom () & (4), w g A (4) () givs I s / 4 Esio :- F c [ a ]. a s 4 a If a F [ c a, w av s 4a s ] [sig cag of scal pobabiliy F[f(a)] F a a a F c [ / ] s / Slf-cipocal fcio: A fcio f() wi popy a F[f()] f(s) is said o b slf-cipocal d Foi asfom,.g. fcio / is slf-cipocal fcio d F.T. T fcio / is also slf cipocal d F..T. / To pov :- F c [ ] s / L F [ c / ] s / / cos s d s / Diffiaig w... s o bo sids, w g F [ s / Hc fcio ] s s / / is slf-cipocal fcio d Foi Si Tasfom. 5
Eampl. :- W kow a if f(),, a > a T F(s) si sa s, s () By dfiiio of F. T., F(s) f() is d f() s F(s) P F(s) fom (), w g ds f() si sa s is ds f() si sa is s, ds, < a > a L.H.S. si sa s si sa s [cos s i si s] ds cos s ds i si sa s si s ds Sic igad i scod igal is a odd fcio, so igal is zo L.H.S. sisa s cos s ds sisa s cos s ds /,,, < a > a a o / sisa cos s ds s / 4,,, < a > a a () si s Evala ds s Solio : P, a i (), w g 6
si s ds w < a. s Rlaio bw Laplac Tasfom ad Foi Tasfom osid fcio f() Takig F.T. of f(), φ,, > < F[f()] F(s) is f() d is φ() d F[f()] F.T. of Divaivs L[φ()] ( is) p φ() d φ() w p i If F[f()] F(s) & f() as ±, Poof :- By dfiiio, F[f ()] i s F[f()] i s F(s) F[f ()] f '() is d is is [ ()] ( is) f () d Now is is. f () F[f ()] is F[f()] is F(s) d [Θ f() as ± ] 7
F[f ()] is F[f ()] (is) F[f()] I gal, w av F[f () ()] is F[f () ()] (i s) F[f()] (is) F(s) Fid Foi si & cosi asfom of f (), f (). Divaio :- By dfiiio F c [f ()] f '() cos s d Assmig f() as, ) [cos s f ()] s f ( si s d F c [f ()] F s [f ()] f() s Fs [f()] () f '() si s d Assmig f() as, f () [ si sf ()] s F s [f ()] s f () cos s d cos s d F s [f ()] s F c [f()] () Now F c [f ()] f () s Fs [f ()] (sig () f () s[s Fc [f()]] [By sig ()] F c [f ()] f () s F c [f()] () ad F s [f ()] s F c [f ()] [sig ()] 8
F s [f ()] s f () s F s [f ()] [sig ()] Tom:- If F[f()] F(s), d ds Poof:- By dfiiio, s f() s F s [f()] (4) F[f()] (i) F[ f()],,, F[f()] F(s) f () is d Diffia w... s d igal sig, w g d F(s) ds f () s ( is ) d d F(s) ds f () (i ) is d is ( i) f () d d F(s) ( i) F[ f()] ds Now d ds F(s) ( i ) f() is d d ds (i) F(s) (i) F[ f()] Galisig sl, w av d ds f() is d F[f ()] ( i) F[ f()],,, Tom :- If F[f()] F(s) 9
ad f ()d F() F f ()d F(s) is Poof : osid φ() f ()d T φ () f() Hc if F[φ()] Φ(s), F[φ ()] F[f()] is Φ(s) Φ(s) F[f()] is Φ(s) F[φ()] F(s) is F(s) is F f ()d is F(s) ovolio :- L f () ad f () b wo giv fcios, covolio of f () & f () is dfid by fcio f() f () f ( )d f () f () Spcial cas :- f () fo < f () fo < T Poof :- f() f() f () f () f () f ( ) d f ()f( )d f()f( )d f()f( ) d
() Now I [Θf () fo < ] () ad I f()f ( ) d Now w < f () w < f ( ) f ( ) fo < o > So I [Θ f ( ) fo > ] () Usig () & () i (), w g f() ommaiv Popy: f () f () f () f () Poof :- By dfiiio, f () f ( )d f f f p y d dy w, y ad w, y () f ( )d f f f ( y) f (y) (dy) f f f (y) f ( y) dy f f Associaiviy popy (f f ) f f (f f ) () Tak g() f f () f f T () bcoms, g() f f ()
ovolio Tom (o Falig Tom) o Foi Tasfoms If w F[f ()] F (f) F[f ()] F (s) i.. f () f () F[f () f ()] F (s) F (s) f () f ( )d Poof :- W av by dfiiio of F.T., F[f () f ()] [f () f ()] is d f()f ( ) d is d [By sig dfiiio of covolio] cagig od of igaio, w g F[f () f ()] is f() f ( ) dd is() f() f ( ) d d is isy is f() f (y) dy d [By pig y d dy] F[f () f ()] f () is F (s)d F (s) f() is d F[f () f ()] F (s) F (s) T covolio ca b sd o obai Foi asfom of podc of wo fcios.
By dfiiio of F.T., w av F[f () f ()] is f ()f() By sig ivs F.T. of f (), w g F[f () f ()] f () d ds' is d is' F (s') i(ss') f() ds'f (s') F[f () f ()] F (s')f (s s')ds' d F (s) F (s) [By sig dfiiio of covolio] F[f () f ()] F F F[f () f ()] F F If F(z) f () iz d T ivs fcio f() F(z) iz dz Eampl :- Fid F.T. of f() ad vify ivs asfom. Sol. Now fo > So fo < f(),, < > F(z) f () iz d (iz) d ( iz) d
i z i z ( z W ca iv asfom sig coo igaio (Rsid om). Fis cosid >, iz f() dz z iz lim R z dz iz z ) dz w c is coo sow i fig. y z i R O R T is a simpl pol a z i ad Rs (z i) Tfo by Rsid om, iz (z i) Lim z i (z i)(z i) iz dz i z i osid <, w coos coo wi a smi-cicla ac lyig blow -ais w c is coo sow i fig. i R O R z i Tfo is a simpl pol a z i. Rs (z ) Lim z i iz (z i) (z i)(z i) i 4
B i is cas coo is i clockwis dicio, c dsid sl is iz dz i z i A, w ca vala dicly ( a z) dz z si w, < < Eampl : Fid F.T. of f(), owis o f() si w H() w, > H(), < is a i sp fcio o Havisid s i sp fcio. Sol. Now F(z) si w iz d iw. [ i i iw i(wz) iz d i(w z) ] d. i iw z) (w z) i( w z) i(w z). i i(z w) i(w z) i z w z w w z w z w z z w w 5
H F(z) (z w w ) is aalyic i z-pla cp a z ± w iz w If > :- f() Lim R z w dz w c is coo. T a wo simpl pol z w & z w isid c. y R z w z w R w f() F(z) iz dz (z w w ) iz dz f() iz w z w dz dz Lim iz w z w R iz w z w dz Rs (z w) iz w Lim(z w) z w (z w)(z w) iw w w iw Rs (z w) iz w Lim (z w) z w (z w)(z w) iw w w iw Tfo by Rsid om, 6
Lim R iz w z w dz (i) (Sm of sid iw iw i f() i(i si w) si w If <, w coos coo wi a smi-cicla ac blow -ais & sic a o pols isid coo, sl is zo. 7
LESSON- APPLIATIONS OF FOURIER TRANSFORMS Solio of Odiay Diffial Eqaio osid od diffial qaio d y d y dy A a... a ay f() d d d Takig F.T. of bo sids, L [a (i s) a (i s) a (i s) a] F[y()] F[f()] F[f()] G(s) F[y()] Y(s) P(i s) a (i s) a (i s). a (i s) a is a polyomial i (i s). T w av Y(s) G(s) P(is) Takig ivs F.T., w av solio is giv by y() Y(s) is ds Eampl:- Solv sig F.T. ciq d y d dy d y () Solio :- Takig F.T. o bo sids, [(i z) (i z) ] F[y()] F[ ] P(i z) F[y()] z Y(z) y() ( z ). P(iz) w Y(z) F[y()] iz dz ( z )[(iz) iz ] iz dz ) (z )( z iz 8
(z )(z iz dz iz ) iz dz (z )(z )(z i)(z i) iz dz (z i)(z i) (z i) Fo > T siglaiy wii coo a a simpl pol a z i ad a dobl pol a z i. Rs. (z i) iz (z i) Lim z i (z i)(z i) (z i) Rs. (z i) L iz z i ) (z i)(z i i.i i iz d (z i) Lim z i dz (z i)(z i) (z i) z i z i z i R Lim z i d dz iz (z i)(z i) iz iz (z i)(z i) (i ) (z i i) Rs. (z i) Lim z i )] [(z i)(z i (z i)(z i)(i ) L iz z i ) (z i) (z i i.( i)(i ) 4i.( i) i iz (z i) i i 4( )( ) i i 4 i i i y().i () 4 9
4i 6i i i i [6 i i 4 i ] 6 6 (6 4 ) () Vificaio :- P () i L.H.S. of DE. () w g Fo < L.H.S. L.H.S. d y d dy d y d d ( ) 4 4 ( ) 9 4 4 4 Rs. (z i) iz (z i) Lim z i (z i)(z i) (z i) ( i) ( i) 4( )( i) i So y() ( i) i 6 Vificaio :- P () i L.H.S. of (), () L.H.S. d y d dy d y d d 6 6 6 6 6 6 6 6. Hc vifid.
Eampl:- Solv sig F.T. ciqs d y dy y H() si w () d d Solio :- T w av o akig F.T. o bo sids, [(i z) w i z ] F[y()] w z T y() iz w dz w z ( z iz ) w y() (z iz d w )(z iz ) Fo > pla. w (z ιz d w )(z i)(z i) I is cas siglaiy a a z ± w, z i, z i ad all s lis isid pp alf iz (z i) Rs. (z i) Lim z i (z w)(z w)(z i)(z i) (i w)(i w)i (4 w )i i w 4 iz (z i) Rs. (z i) Lim z i (zi)(z i)(z w ) i( w ) i ( w ) i(w ) Rs. (z w) iz (z w) Lim z w (z w)(z w)(z i)(z i) iw w(w i)(w i) iw w(w iw ) Rs. (z w) iz (z w) Lim z w (z w)(z w)(z i)(z i)
iw ( w)( w i)( w i) iw w(w i)(w i) Rs. (z w) w(w iw iw ) iw w(w iw ) y() w i i.i w 4 (w iw iw ) w(w i)(w i) w(w i)(w i) w w 4 Fo vificaio :- w w i iw (w i)(w i) iw i (w i)(w i) y () w w 4 iw iw w w w w (w i)(w i) (w i)(w i) y () 4w w 4 iw iw w i w i w w (w i)(w i) (w i)(w i) T y y y 4w w 4 iw w iw w (w i)(w i) iw iw iw 6w w w (w i)(w i) w 4 w (w i)(w i) iw w w (w i)(w i) w 4 iw iw w i wi w (w i)(w i) (w i)(w i) y y y (w iw i)(w i) [ i w w i] (w iw [i w w i] i)(w i) iw i (w i)(w (w i) iw i (w iw ) iw ) (w i)(w i) iw iw i iw iw ( ) i i
y y y H() si w Hc y y y si w. H() is vifid fo >. fo <, is o pol isid low alf pla so sid. y() y y y H() si w is vifid. Fo <. fo <, H() So R.H.S. of () is zo. Ad qal o L.H.S. Hc vify sl. Eampl: Fid FST of a Solio :- T si asfom of fcio f() a I F s [f()] f () si s d Diffiaio w... s, w g a si s d () di ds d ds F (f ()) s a. cos s d a cos s d a a ( a cos s ssi s) s d ds F [f ()] Igaig w g,. a a s s di ds d a s F s[f ()] ds a ds I F s [f()] s a A () a
w A is cosa of igaio. Fo s, w g fom (), I () A I A () w s, fom (), I () d fom () & (4), w g A Hc qid FST of giv fcio is (4) F s a a s a. Solio of paial diffial qaios (boday val poblms) Eampl: - Dmi disibio of mpa i smi-ifii mdim, w d is maiaid a zo mpa & iiial disibio of mpa is f(). Solio :- Ha qaio is giv by (, ) c (, ), >, > () w (, ) is disibio of mpa a ay poi ad im. W wa o dmi solio of () sbjc o iiial codiio (, ) f() () ad boday codiio (, ) () Sic () is giv, w apply F.S.T. Do F s [(, )] s (s,) s ad F c [(, )] c Takig FST of bo sids of (), w g si s d c si s d 4
d si s d c d si s d d d si s d Fs [(, )] d d c si s s cos s d d d s c s cos s d if as s cos s d c s [ (, )coss] c s si s d d d s c s(,) c s s ; assmig as d d s c s (, ) s s By sig (), (, ), w g d d c s s s (4) Also akig FST of (), w g F s [(, )] F s [f()] s (s, ) fs (s) (5) fom (4), w av (D s ) s Ailiay qaio is m c s m c s Solio of (4) is s s c s (s,) A ( ) To fid A, w s (5), fom ( ), w av s (s,) A 5
A f s (s) [sig (5)] Hc solio is (s,) f (s) s s c s Takig ivs FST, w g F s [ (s,)] F s s [f s (s) c s ] (, ) s si s ds (, ) fs (s) c s si s ds Tis is qid solio of PDE. Eampl:- T mpa i smi-ifii od < (o ) is dmid by qaio K sbjc o codiios (i) w, i.. (, ) (ii) μ (a cosa) w, > i.. (, ) μ o (, ) μ Solio :- Sic is giv, So akig Foi cosi asfom of bo sids of (), () cos s d K cos s d d d (s,) K c cos s K s si s d 6
d d c K Ks si s d By ass mi g as d d [ ] c K μ Ks sis Ks K μ Ks if as c s cos s d d d Ks c Kμ () c c μ ( s Ks ) (, ) μ cos s Ks ( ) s Tis is lia DE of Is od. ds I.F. ks d s k Solio of () is c. s k K.μ s k d A s k s k μ c A Kμ A s k s s k μ s s k c A () μ P c (s,) A (4) s fom codiio (i), c (s, ) (,) cos s d (5) μ (4), (5) A s 7
μ k s () c ( ) s Fii Foi Tasfom :- Fii Foi Sic Tasfom :- L f() do a fcio a is scioally coios ov som fii ival (, λ) of vaiabl. T fii foi si asfom of f() o ival is dfid as λ s f s (s) f ()si d w s is a ig λ Ivsio fomla fo si asfom f() s fs (s)si fo ival (, λ). λ s λ If (, ) is ival fo f s (s), f() fs (s) si s s Fii FT :- L f() do a fcio a is scioally coios ov som fii ival (, λ) of vaiabl. T fii Foi cosi asfom of f() o ival is dfid as λ s f c (s) f ()cos d w s is a ig, λ If (, ) is ival, f c (s) f () cos s d Ivsio fomla fo FT f() s fc () fc (s)cos λ λ s λ λ w f c () f () d If f(), (, ), f s (s) si s d coss s 8
[ coss cos] [ ( ) ] s s si s ad f c (s) cos s d if s,,,. s If s, f c (s). d Fid Fii FST ad fii FT of,, (, ) fo < < λ, >. Solio :- By dfiiio, fii FST of is λ s f s (s) si λ d λ s s s si (, ) λ (, )cos d λ λ λ f s (s) F s s λ λ (, ) cos s f s (s) F s F c[(,)] λ s λ d () λ s λ s λ F c cos d cos (, ) s λ F s [(, )] [(, ) (λ, ) cos s] To calcla fii FST & fii FT of, Rplac by i () ad (), w g λ λ s s (, )si d λ λ () F s s Fc λ 9
s s λ λ s λ F () { (, ) ( λ,)coss} s s λ s F s[] [(, ) (λ, ) cos s] λ () Similaly F c s Fs [ (, ) (λ, ) cos s] λ s s F c [(, )] { (, ) (λ, ) cos s] λ λ s λ F c [] { (, ) (λ, ) cos s} Eampl:- Us fii Foi asfom o solv (, ), (4, ) (, ) w < < 4, >. Solio :- () is giv, w apply FST. H λ 4. Takig Fii FST of (), w g (4) () Wiig 4 s 4 d si d si s F s [(, )], w g 4 s 4 d s s s s (, ) d 6 4 [Usig qaio () i pvios aicl ad giv codiio] d d d s s s 6 s s (s,) E 6 ( ) To fid A, w will ak fii FST of (, ), w g 4
(s,) s 4 s si d 4 4 s 4 4 s cos.. cos 4 s s 4 () s 6 s 8 s () s 6 8. () s s 4 s si. 4 () s cos s s s fom ( ), s (s, ) A A cos s s s 6 Hc s (s,) coss s Takig Ivs Fii Foi si asfom, w g 4 s 4 d (, ) 4 s coss s s 6 s si 4 6 (, ) s coss s Eampl :- Solv qaio s si 4 s 6 sbjc o codiios (i) (, ) (ii) (, ) (iii) (, ) is bodd w >, > Solio :- Sic () is giv, so akig FST, w g () 4
sis d si s d d d si s d si s d d d si s d si s d d d s sis s coss d () scos s d ass mig as d d s s((, )coss) s si s(, ) d s (, )si sd [assmig (, ) as ] d d d d s s (, ) si s d s A.E. is D s Solio is s s s s () s s (s,) A () To fid A, akig FST of codiio (ii) (, ) Takig FST, w g 4
(s,) s sis d s s a (4) Θ sig a a sib d (a si b b cos b) akig a, b s a b ad akig lim, w g s s Pig i (), w g s (s,) A (5) fom (4) ad (5), w g s A s Pig val of A i (), w g s s (s,) s Takig ivs FST, w g s (, ) s s s si s ds s s Eampl: Solv qaio s si s ds wic is qid solio. sbjc o codiios (i) (, ) (ii) (, ), <, > () 4
(iii) (, ) is bodd Solio :- Sic ( ) is giv, so akig FT of (), w g d d cos s d cos s d d d c cos s d cos s d coss s sis d [Θ as as ad also (, ) ] d d c [( ) s[si s(, )] s (,) cos s d d d c ( s ) cos s (, ) d d d c (s ) (, )cos s d s c d c s c () d A.E. is D s solio is c s (s,) A () Now o fid A, akig FT of codiio (ii) c (s,) (,) cos s d cos s d [Θ (, ), > ] sis s sis d s 44
sis coss s s sis s Pig i (), w g coss s a (4) sis coss c (s, ) A A s s fom () ad (4), w g sis coss c (s,) s s Takig ivs FT, w g s (, ) c (s,) cos s ds si s coss (, ) s s wic is qid solio. Eampl:- Solv qaio s cos s ds, >, y > () Sbjc o codiio (i) (, ) (ii) (, ), < < w, (iii) (, ) is bodd. Solio :- Sic () is giv, so akig FST of (), w g si s d si s d d d si s d si s d 45
d d si s d si s s cos s d d d s () s cos s d ass mig as s [ coss.(, ) ] s ( s si s)(, ) d d d s s (, ) s s s si s d [Θ (, ) giv] d d A.E. is D s Solio is s s s () s s (s, ) A () To fid A, w ak FST of codiio (ii), s (s,) (,) si s d () si s d [Θ (, ) fo ] cos s s cos s cos s s (s,) (4) s s s Fom (), w av s (s,) A (5) A cos s s () (s, ) Takig ivs FST, w g s cos s s s 46
(, ) cos s s s si s ds cos s s (, ) si s ds s wic is qid solio. 47
LESSON URVILINEAR O-ORDINATES Tasfomaio of coodias L cagla coodias (, y, z) of ay poi b pssd as fcios of (,, ) so a (,, ) y y(,, ) () z z(,, Sppos () ca b solvd fo,, i ms of, y, z i.. (, y,z) (, y,z) (, y,z) () H cospodc bw (, y, z) ad (,, ) is iq i.. if o ac poi P(, y, z) of som gio R, cospods o & oly o iad (,, ),,, ) a said o b cvilia coodias of poi P. T s of qaios () & () dfi a asfomaio of co-odias. z cv P(,y,z) (,, ) cv cv y o-odia sfacs ad cvs:- T sfacs c, c, c w c, c, c a cosas i.. sfacs wos qaios a (, y, z) c (, y, z) c (, y, z) c 48
a calld co-odia sfacs ad ac pai of s sfacs isc i cvs calld coodia cvs o lis. So c ad c givs cv Similaly c ad c givs cv ad c ad c givs cv So if c ad c ad if is oly vaiabls poi P dscib a cv kow as cv wic is a fcio of. z cv c c c cv cv y Ogoal vilia o-odias If co-odia sfacs isc a ig agls (oogoal), cvilia coodia sysm is calld oogoal i.. co-odias (,, ) a said o b oogoal cvilia co-odias. T,, co-odia cvs of a cvilia sysm a aalogos o, y, z coodias as of cagla sysm. Ui vcos i cvilia sysm L î y ĵ z kˆ b posiio vco of P, () ca b wi as (,, ) A ag vco o -cv a P(fo wic & a cosa) is T, A i ag vco i is dicio is 49
5 / o w Similaly if ad a i ag vcos o & cvs a P spcivly,, w, T qaiis,, a calld scal facos. Also, codiio fo oogoaliy of co-odia sfacs a.,.,. Also vcos,, a P a dicd alog omal o co-odia sfacs c, c, c spcivly. So i-vcos i s dicios a giv by Ê, Ê, Ê z y Ê cv Ê cv Ê cv
5 Ts, a ac poi P of a cvilia sysm, a, i gal, wo ss of i vcos,, ag o coodia cvs ad Ê, Ê. Ê omal o co-odia sfacs. T ss bcom idical if & oly if cvilia coodia sysm is oogoal. Ac lg ad Volm lm Fom ),, ( d d d d () d d d T, diffial of ac lg ds is dmid fom (ds) d.d Fo oogoal sysm,... o... () sig ( ) ad (ds) ) (d.. (d ). ) (d. d d. d d. d d (ds). ) (d. (d ). (d ) ) (d ) (d ) (d [sig ()]
(ds) w Now if (, y, z) T î ĵ kˆ y z (d) (d ) (d) cv d d d P cv cv Lg alog -cv Fo is & a cosa. c, c d, d If ds diffial of lg alog -cv (ds ) (d ) ds d Similaly lg alog -cv is ds d ad lg alog -cv is ds d T volm lm Fo a oogoal cvilia coodia sysm is giv by 5
5 dv ( d ) d )( d ).( [Θ volm ĉ) â.(bˆ dv d d d Diffial opaos i ms of oogoal cvilia coodias (,, ) GRADIENT L φ scala poi fcio ad ĉ f bˆ f â f f f f f i.. φ φ(,, ) φ[ (, y, z), (, y, z), (, y, z)] φ φ φ φ () y φ y φ y φ y φ () z φ z φ z φ z φ () Opaig î () ĵ () kˆ (), w g φ gad φ φ φ φ B,, gad φ φ φ φ φ Eampl:- If (,, ) a oogoal coodia pov a (i) p p, p,, (ii) p p Ê Poof :- (i) l φ,
Similaly φ, Similaly if φ, (ii) By dfiiio, Ê p p p Ê p p p p Rsls :- I. div (φfˆ ).(φfˆ ) φ div f fˆ. Gad φ φ. fˆ fˆ. φ II. div (fˆ g ). (fˆ g ) cl fˆ. g cl g.fˆ III. l gad φ φ div cl fˆ. fˆ IV. l (φfˆ ) (φ fˆ ) gad φ fˆ φ l fˆ DIVERGENE osid a vco fcio f (,, ) f f f w (,, ) a oogoal cvilia coodias. f ( ) f ( ) f ( ) f sig f f ( ) f ( ) f ( ) 54
55 dif f. f.[f ( )]. [f ( )]. [f ( )] ( ) Takig fis a,. [f ( )] f. ( ) ( ). (f ) [By sig. (φ f ) φ. f f. φ].[f ( )] f.( ) ( ). (f ) ().[( )] cl. cl. (cl gad ). ( ) (cl gad ). fom (), w g. [f ( )] ( ). (f ) ( ). (f ) (f ) ) (f ) (f ) (f ) (f,, Θ ) (f ) (f ) (f. ) (f.[f ( )]. (f ) Similaly.[f ( Δ )] (f )
56 ad.[f ( Δ )] (f ) So fom ( ), w g. ) (f ) (f ) (f f URL osid f f f f f f f f [Θ ] l f f ( f Δ ) ( f ) ( f ) () Takig fis a, w av by sig popy l (φ F ) φ F φ F w g ( f ) ( f ) f ( ) ( f ) Δ [Θ Δ Δ cl gad ] ( f ) ) (f ) (f ) (f [Θ By dfiiio of gadi of a fcio] [ f ] ) (f ) (f ) (f ( f ) ) (f. ) (f Similaly ( f ) ) (f ) (f ad ( f ) ) (f ) (f fom (),
57 l ) (f ) (f f ) (f ) (f ) (f ) (f f f f LAPLAIAN OF SALAR POINT FUNTION ( φ) Now φ.( φ) φ. φ φ φ. φ φ φ W kow a. ) (f ) (f ) (f f H p f φ, f φ f, φ Tfo φ φ φ φ
YLINDRIAL POLAR OORDINATES (,, z) L P is a poi avig asia co-odias (, y, z) OM, MN y( OQ), PN z z P(,y,z) y N M M y N z Q y OM ON cos cos () & y MN N si y sio () ad z z () so w av, y ad a y Dmi asfomaio fom cylidical o cagla coodias :- Opaig qaio () (), w g y y y ()/() a y a ad z z. (ds) (d) (dy) (dz) Now d cos d si d dy si d cos d (4) 58
59 dz dz (4) (ds) (d cos si d) (si d cos d (dz) (d) (si cos ) ( d) (si cos ) (dz) (ds) (d) (d) (dz) ompaig i wi (ds) ) (d ) (d ) (d W g,,,,,, z Tak z,, Usig s, w av gad Φ z z Φ Φ Φ w Φ is a scala poi fcio div ) (f z ) (f ) (f f l z f f f z f Φ z Φ z Φ Φ z Φ Φ Φ Φ SPHERIAL POLAR OORDINATES (,, φ) M y N z y φ z P(,y,z) (,,φ) O N P
OM, MN y, PN z, OP z PN cos ON si I Δ OMN, OM ON cos φ O φ M y N si cosφ y MN ON siφ y si si φ Now (ds) (d) (dy) (dz) () d si cosφ d cos cos φ d si si φ dφ dy si siφ d cos siφ d si cosφ dφ dz cos d si d () P val of () i () ad collcig cofficis of (d), (d), (dφ), w g (ds) (d) () (d) ( ) (dφ) ( si ) ompaig i wi, w g (ds) (d) (d) (d),, si,, φ,, φ So gad Φ Φ Φ φ Φ si φ div f (f si ) (f si (f ) si φ l f si si Φ () 6
6 l si f si f f f φ si f φ Φ φ Φ si φ Φ si Φ si si φ Φ si Φ si cos Φ Φ Φ Eampl:- Pov a cylidical coodia sysm is oogoal. Solio :- T posiio vco of ay poi i cylidical coodia is z kˆ y ĵ î kˆ z si ĵ î cos s z, s y, s (s) a i vco T ag vco o, ad z cvs a giv by z,, spcivly. w ĵ si cos î cos ĵ si î kˆ z Tfo i-vco i s ag dicios a ĵ si î cos si cos si ĵ cosî cos si cos ĵ siî /
si î cos ĵ si î cos ĵ / z z kˆ / z Now. (cos î si ĵ).( siî cos ĵ). cos si si cos (cos î si ĵ).kˆ z. z ( siî cos ĵ).kˆ.. (cos î si & j).(cos î si & j) cos si ( siî cos ĵ).( siî. si cos kˆ.kˆ z z Tis sows a z cos ĵ),, a mally ad fo coodia sysm is oogoal. Eampl:- (a) Fid i vcos î, ĵ, kˆ.,, of a spical co-odia sysm i ms of φ Solio :- T posiio vco of ay poi i spical coodia is R î y ĵ z kˆ ( si cos φ) î ( si si φ) ĵ ( cos) kˆ R R R W wa o fid,, φ Now So Also R si cosφ î si si φ ĵ cos kˆ R / sicosφ î si si φ ĵ coskˆ R / si (cos φ si φ) cos si cos φ î si siφ ĵ cos kˆ R cos cosφ î cos siφ ĵ si kˆ 6
So R / R / coscosφî cossi φ ĵ sikˆ cos (cos φ si φ) si Also So coscosφ î cossi φ ĵ si kˆ coscos φ î cos si φ ĵ si kˆ R si siφ î si cos φ ĵ. kˆ φ R / φ sisi φ î si cosφ ĵ.kˆ φ R / φ si (si φ cos φ) si si φ î si cos φ ĵ si φ si φ î cos φ ĵ Eampl:- Pov a a spical coodia sysm is oogoal. Solio :-. (sicos φ î si si φ ĵ cos kˆ ). (cos cos φ î cos si φ ĵ si kˆ ) cos φ si cos si φ si cos si cos si cos (si φ cos φ) si cos.. φ (cos cos φ î cos si φ ĵ si kˆ ). (siφ î cos φ ĵ ) si φ cos φ cos cos cosφ siφ Also. φ φ. (siφ î cosφ ĵ )(si cosφ î si siφ ĵ cos kˆ ) si siφ cosφ si siφ cosφ ad φ.. (si cosφ î si siφ ĵ cos kˆ ) (si cosφ î si siφ ĵ cos kˆ ) cos φ si si φ cos 6
si (cos φ si φ) cos ad.. (cos cosφ î cos siφ ĵ si kˆ ) (cos cosφ î cos siφ ĵ si kˆ ) cos φ cos cos si φ si cos (si φ cos φ) si ad. φ. φ (siφ î cosφ ĵ ). (siφ î cosφ ĵ ) si φ cos φ φ. φ Tis sows a,, a mally ad fo coodia sysm is oogoal. φ Eampl:- Rps vco A y î z ĵ kˆ () i spical coodias Solio :- H si cosφ ad y si siφ z cos si cosφ î si siφ ĵ cos kˆ cos cos φ î cos siφ ĵ si kˆ φ siφ î cos φ ĵ () () Solvig (), w g î si cosφ cos cosφ φ siφ ĵ si siφ siφ cos φ cosφ kˆ cos si (4) P () & (4) i (), w g A si siφ( si cosφ cos cosφ φ siφ) cos ( si siφ siφ cos φ cos) 64
si cosφ ( cos si ) A ( si cos φ si cos siφ si cosφ cos) ( si cos siφ cosφ cos siφ si cosφ) φ ( si siφ cos cosφ) A si (si cos φ cos siφ) (si cos siφ cosφ cos siφ si cosφ) φ ( si si φ cos cosφ) A A A A φ φ Eampl :- Pov a fo cylidical coodia sysm (,, z), d d d d & & Solio :-W av cos î si ĵ () si î cos ĵ () Also d d d d d d ( si )& î (cos)& ĵ (si î cos ĵ ) & & [sig ()] ( cos)& î ( si )& ĵ (cos î si ĵ) & d d & [sig ()] Eampl:- Epss vlociy v ad acclaio a of a paicl i cylidical coodias. Solio :- Posiio vco of a paicl P i cagla coodias î y ĵ z kˆ vco i cylidical coodia sysm is cos(cos si ) 65
cos z z d T vlociy v d si (si cos cos si d d dz v z z d d d ) z z si si cos z d d v & & d z& z Θ z d () dv d Diffiaig () agai, w obai acclaio a d d d a (& & z z ) d d d d & & && & & & & d d d zz z z sig d d d &, &, z d d d, w g a & && && && & ( & & a ( & & ) ( & ) && z & z Eampl: - Pov a i spical coodias (,, φ) d d d d d d φ Poof: - Now & & & & φ& si φ& φ cosφ& φ z ) & z si φ & cosφ& (*) si cosφi si si φĵ cos kˆ () z z d d coscosφ& î si ( si φ)φ& î cos cosφ î si cosφφ& ĵ (si )& kˆ 66
sig & (coscosφ î cossi φ ĵ si kˆ ) si φ( & si φ î cosφ ĵ) () coscosφî cossi φ ĵ sikˆ () P φ si φ î cosφ ĵ (4) () & (4) i (), w g d d Also fom(), d d & si φ& si cosφ& î cossi φφî & φ si si φ& î cossi φφ& ĵ cos kˆ & &(si cosφ î isi φ ĵ cos kˆ) cos φ( & si φ î cosφ ĵ) d d Also fom (4), & cos d d Takig R.H.S of qaio (*) R.H.S si φ & cosgq φ& φ φ φ& cosφφî & si φφ& si φφ& ĵ (5) si φ(si & cosφ î si si φ ĵ cos kˆ ) cosφ(coscosφ & î cossi φ ĵ si kˆ ) [sig() & ()] si cosφ φ& î cos cosφ φsi & si φ ĵ φ& si cos kˆ φ& î cos si φ φ & ĵ si cos φ& kˆ φ& î cos φ(si cos ) φ& ĵsi φ (si cos ) φ& cos φ î si φ ĵ L.H.S. [fom (5)] So d d L.H.S. R.H.S. φ si φ& cos φ& Eampl:- Epss vlociy v ad acclaio a of a paicl i spical coodias. Solio : Posiio vco of a paicl P i cagla coodias 67
î y ĵ z kˆ vco i spical coodia sysm is cos φ si ( si cosφ coscosφ si φ) si si φ ( si si φ cossi φ cosφ) cos ( cos si ) (si cos φ si si φ cos ) (sicoscos φ si si φcos cossi ) φ φ φ ( cosφsi si φ si si φcosφ) [si (si φ cos φ) cos ] [si cos (si φ cos φ) cos si] (si cos ) (si cos si cos) () d d T vlociy v is v ( ) d d d d v ( ) d d & (& si φ& ) [fom pvios g.] φ φ & & si φ& () Diffiaig () agai, w obai acclaio dv d a d d d a (& & si φ& φ ) d d d & & ( ) && & & ( ) d d d & siφ & cos && φ si φ& si φ& (φ ) d φ φ φ & ( && siφ& ) && & φ & ( & cos φ& φ ) si & φ& φ 68
cos φ & φ & si φ& φ si φ& ( si φ& cos φ& ) a ( & & φ& si ) [& & & sicosφ& ] φ [& φsi & φ& cos si φ&] a ( & & si φ& ) d d ( ) & si cos φ& d φ ( s φ) & si d () d ( ) & ( & ) & d Θ && & ad ( si φ& ) d si d & [ si φ& si & φ& φ& si (cos)] si si φ & && φ si & φcos & So () is qid pssio fo acclaio of a paicl i spical coodia. Eampl : d d φ φ& si φ& cosφ& Solio :- Now φ si φ î cosφ ĵ d d φ cosφ φ& î si φ φ& ĵ (5) P vals of d d φ î ad ĵ i(5), w g cosφ φ( & si cosφ φ coscosφ φ si φ) si φ φ& ( si si φ.cossi φ cosφ) φ si φ& (cos φ si φ) cos φ& (cos φ si φ) cosφsi φ φ& cosφsi φφ& φ φ 69
si φ & cos φ&. Hc Povd. Eampl:- Rps vco A z î ĵ y kˆ i cylidical coodias. Solio :- I cylidical coodias, cos, y si, z z () ad cos î si ĵ siî cos () z kˆ () T posiio vco of a paicl P i cagla coodias is A z î ( )ĵ y kˆ A z î cos ĵ si kˆ (4) Solv () fo î ad ĵ, w g o opaig si cos cos si î si ĵ si cos î cos ĵ si cos (si cos )j ĵ (5) Similaly si cos î cos ( si î cossi ĵ) sicos ĵ cos si (si cos )î sicos ĵ si cos ĵ cos si î (6) P val of î, ĵ, kˆ fom (), (5), (6) i qaio (4), w g A z( cos si) cos( si cos) si z (z cos cossi) w (zsi cos ) z ( si ) A A A A A z cos cos si A z si cos si A z si z z () 7
Eqaio () is qid vco A i cylidical coodias. Eampl:- Rps vco A z î ĵ y kˆ i spical coodias (,, φ) Solio :- I spical coodias, ad si cosφ y si siφ z cos si cosφ î si siφ ĵ cos kˆ cos cos φ î cos si φ ĵ si kˆ φ siφ î cos φ ĵ () () Solvig sysm () fo î, ĵ, kˆ by am s l, fo is w av D si cosφ cos cosφ si φ sisi φ cossi φ cosφ cos si si cosφ(si cosφ) si siφ (si siφ) cos(cos cos φ cos si φ) si cos φ si si φ cos (cos φ si φ) si (cos φ si φ) cos () si cos D ad D φ si si φ cossi φ cosφ cos si (si cosφ) si siφ (si φ ) cos cosφ cossi φ) ( φ si cosφ cos cosφ φ si φ(si cos ) D si cosφ cos cosφ φ siφ 7
D si cosφ coscosφ si φ φ cos si si cos φ(si φ ) (si siφ ) cos ( φ cos cos si φ) si siφ cos siφ cosφ φ (si cos ) D si siφ cos si φ cosφ φ D si cosφ coscosφ si φ si si φ cossi φ cosφ φ si cosφ (cos siφ φ cosφ ) si siφ(siφ φ cos cosφ) (cos cos φ cos si φ) cos(si φ cos φ) ( sicos φ si si φ) φ (si cos siφ cosφ si cos siφ cosφ) D cos si T D si cosφ cos cosφ φ siφ D î D ĵ si siφ cos siφ cos φ φ D D kˆ cos si D fom (), () ad (4), w g A( cos) ( si cos φ cos cosφ φ siφ) si cosφ ( si siφ cos siφ φ cosφ) si siφ ( cos si) 7
A (cos si cosφ si cosφ siφ si cos siφ) (cos cosφ si cos siφ cosφ) φ (cos siφ si cos φ) [si cos (cosφ siφ) si cosφ siφ] A A A (cos cosφ si cos siφ cosφ) φ (cos siφ si cos φ) A φ φ wic is qid pssio. Eampl:- Fid sq of lm of ac lg i cylidical & spical coodias Solio: - I cylidical coodias, cos, y si, z z Now (ds) (d) (dy) (dz) () as d d cos si d dy d si cos d dz dz P () i (), w g (ds) (cos d si dz) (si d cos d) (dz) (d) (si cos ) (d) ( si cos ) si cos d d si cos d d (dz) () (ds) (d) (d) (dz) (d) (d) (dz) I spical coodias (,, φ) si cosφ, y si siφ, z cos as (ds) (d) (dy) (dz) sig (), w g (ds) (d) (d) si (dφ) ampaig i wi, () (ds) (d ) (d ) (d ), w g 7
,, si,, φ w g (ds) (d) (d) (dφ) Eampl : - Fid pssio fo lms of aa i oogoal cvilia coodias. Solio: - d d P d Now aa lm is giv by da d d d d d d d d [ Θ ] da d d d d d d [ Θ ] da d d d d d d d d d d [ Θ Eampl:- Fid volm lm dv i cyldical & spical coodia sysm. Solio: - W kow a volm lm dv i oogoal cvilia coodias is dv d d d () I cylidical coodias (,, φ),,,, z 74
So () dv d d dz I spical coodias (,, φ),, si,, φ So () dv si d d dφ Eampl:- If,, a gal coodias, sow a cipocail sysm of vcos. Solio: - W kow a if φ φ (, y, z) φ φ φ dφ d dy dz d y z,, &,, a φ φ φ î ĵ kˆ.( d î dy ĵ dz kˆ ) y z dφ φ.d Rplacig φ wi p, w g d p p. d d. d d. d d. d Now (,, ) d d d d Takig do podc wi, w g.d. d. d. d d. d. d. ompaig lik cofficis o bo sids,.,.,. () () () [sig ()] 75
76 Similaly akig do podc of () wi ad, w g,..,., So w g qid sl i.. p. q p if q p if q W (p, q) (,, ) Eampl: - Pov a wi simila qaio fo ad w,, a oogoal coodias. Solio: - W kow a,, T Similaly k ad Hc povd. Eampl: - If,, a oogoal cvilia coodias, sow a jacobia of, y, z w...,, is
77 J z y z y z y ),, ( y,z) (, y,z, Poof: - w kow a A A î A ĵ A kˆ ),, ( ),,B,B (B B T B B B kˆ ĵ î A. ) A.(B (A î A ĵ A kˆ ).[ î (B B ) ĵ (B B kˆ (B B )] ) A.(B A (B B ) A (B B ) A (B B ) B B B A A A ).(B A Tfo kˆ z ĵ y î. kˆ z ŷ y î z y z y z y kˆ z ĵ y î ]. ).( ) (..
.. [ Θ. ] Hc povd. oavaia ompos of A ad covaia compos of A A i ms of i bas vcos,, o Ê,Ê, Ê ca b wi as A A A A aê a Ê a Ê W A, A, A & a, a, a a compos of A i ac a,, ad,, cosii cipocal sysm of vcos. W ca also ps A i ms of bas vcos,, o,, wic a calld iay bas vcos & a o i vcos. I gal, A A α α α w αp,p,, & A β β β w β p, p,, p w,, a calld coavaia compo of A ad,, a calld covaia compos of A. Eampl: - L A b a giv vco dfid w... wo gal cvilia coodias sysm (,, )& (,,). Fid laio bw coavaia compos of vco i wo coodia sysm. (Fid laio bw p ad p ) Solio: - Sppos asfomaio qaio fom cagla (, y, z) sysm o wo gal cvilia coodiaios sysms (,, ) ad (,,) a giv by (,, ), y y (,, ), z z (,, ) p 78
(,,), y y (,,), z z (,,) () T I a asfomaio dicly fom (,, ) sysm o (,,) sysm is dfid by (,,), (,,), (,,) (i) L (,, ), (,,), (,,) () î y ĵ z kˆ T s () (,, ), (,, ) d d d d αd αd αd () & d d d d α d α d α d (4) w αp, αp, p,, p p fom () & (4), w g αd α d αd αd αd αd (5) fom (i) sic p p,, ) ( d d d d d d d d d d d d d d d d d d sig s i L.H.S of (5) ad qaig coffici of d,d,d, w g α d αd αd d d d α d d 79
8 α d d d d d α d d d d d d α d α d α W g α α α α α α α α α α α α (6) Also α α α A α α α (7) w,, ad,, a coavaia compos of A i wo sysms (,, ) ad ),, (. Sbsiig (6) i (7), α α α α α α α α α α α α α α α Eqaig cofficis of,,α,α α w g
8 (8) o p p p p, p,, (9) o p q p q q, p,, () Similaly by icagig coodias, W ca g, q q p q p p,, () Eqaio (8), (9), (), () givs laio bw coavaia compos of vco i wo coodias sysms. Eampl: - L A b giv vco dfid w... wo cvilia coodias sysm (,, ) ad ),, (. Fid laio bw covaia compos of vcos i wo co od sysm. Solio: - L covaia compo of A i sysm (,, ) ad ),, ( a,, ad,, spcivly A () Sic ),, ( p p, p,, p p p p y y y y p p p p z z z z p p p p ()
8 Also kˆ z ĵ y î kˆ z ĵ y î kˆ z ĵ y î î ĵ y y y kˆ z z z () ad î ĵ y y y kˆ z z z (4) Eqaig cofficis of kˆ ĵ,, î i () & (4), W g y y y y y y z z z z z z (5) Sbsiig qaio () wi p,, o R.H.S i cospodig qaio of (5) ad qaig cofficis of,, o y, y, y o y, y, y o ac sid, W ca g,
8 Takig fis qaio of (5), H qaig cofficis of,, W g (6) o p p p p, p,, o q q p q q, p,, Similaly, w cam sow a q q q q p, p,, (7) Eqaio (6) ad (7) a qid laio. Eampl: - Sow a sqa of lm of ac lg i gal cvilia coodia ca by ds q q q p pq d d g Solio: - (ds) m d d d d.d
w α p. d d (ds) ( αd αd αd).(αd αd αd) p α.αd α.α dd α.α dd α.αdd α.α d α.α d d α.αdd α.α dd α.α d d (ds) q q g pq d p d q, g pq α. α p q Tos is calld fdamal qadaic fom o Mic fom. T qaiis g pq a calld mic cofficis ad s a symmic i.. g pq g qp If g pq p q, coodia sysm is oogoal. ad i is cas g H also α, g, α, g, α Eampl: - (a) Pov a i gal coodiay (,, ) g g g g g g g g g g.. w g pq a cofficis of d p d q i ds Solio: - w kow a g pq αp.α q. p q p q y p y q z p z, p,, q 84
85 z y z y z y z y z y z y.. z z z y y y z y z y z y g g g g g g g g g g Eampl (b) sow a volm lm i gal coodia is g d d d. Solio: - T volm lm is giv by dv d d. d d. d d g d d d
LESSON 4 RANDOM VARIABLE AND MATHEMATIAL EXPETATION Sampl spac T s pois psig possibl ocoms of a pim is calld sampl spac of pim. Eampl: - () I ossig o coi, sampl spac is S {H, T} () Two cois a ossd, S {HH, HT, TH, TT} () I ow a dic, S {(, ), (, ), (, ), (, 4), (, 5), (, 6) (, ), (, ), (, ), (, 4), (, 5), (, 6) Μ (6, ), (6, ), (6, ), (6, 4), (6, 5), (6, 6) Toal ocom 6 Rodom Vaiabl A adom vaiabl is a al vald fcio dfid o a sampl spac. W a pim is pfomd, sval ocoms a possibl cospodig o ac ocom, a mb ca b associad. Eampl: - If wo cois a ossd, possibl ocoms a TT, TH, HT, HH L X dos mb of ads T mb associad wi ocom a : TT, TH, HT, HH No. of ads, X :,,, T vaiabl X is said o b adom vaiabl. ad may b dfid as L S b sampl spac associad wi a giv pim. A al vald fcio dfid o S & akig vals i R(, ) ( al o.) is calld a adom vaiabl (o cac vaiabl o Socasic vaiabl o vaia). 86
Disc ad oios Sampl Spac A sampl spac a cosiss of a fii mb o a ifii sqc, of pois is calld a disc sampl spac ad a cosis of o o mo ivals of pois is calld a coios sampl spac. Disc ad oioss Radom Vaiabl A adom vaiabl dfid o a disc sampl spac is calld disc.v. o If a.v. aks a mos a coabl o. vals, i is calld Disc.v. A.v. is said o b oios if i ca aks al possibl val bw cai limis o cai ival. Eampl: - ) If X pss sm of pois o wo dic, X is a disc.v. ) If X pss ig o wig of sds i a class, i is a coios.v. ) If X pss amo of aifall, i is a coios.v. Dsiy fcio (d.f) o pobabiliy dsiy fcio(p.d.f) A fcio associad wi disc.v. X s.. f() pob[x ] is calld dsiy fcio of X. Eampl: - I ossig wocois, Ocoms {HH, TH, HT, TT} X [,, ] f() P[X ] 4 f() P[X ] /4 ½ f() P[X ] ¼ Eampl: - I owig wo dic, sampl spac of sm of pois o wo dic is S [,,,,] f() P[X ] (, ) f() (6,6) 6 f() 6 f() [Θ (, ), (, )] f(4) f() 6 87
f(5) f(9) 6 4 f(6) f(8) 6 5 f(7) P[X 7] 6 6 Also ( ) f i X i,, Disibio fcio Fo a.v. X, fcio F() P(X ) is calld disibio fcio of X o mlaiv disibio. Sic F() P(X ), f() p.d.f. W av If F() f () P(X ), f () f() f() f() Eampl: - A.v. X as followig disibio X : 4 5 6 7 8 f() : k k 5k 8k 9k k k 4k 7k () Fid k, As f () k k. 7k 8k k 8 () Fid P(X < ) P(X ) P(X ) P() P() k k 4k 9 P(X < ) P() P() P() 8 4 8 88
7 () P(X ) P() P(4) P(8) 8 Also 9 7 P(X ) P(X < ) 8 8 (4) P( < X < 5) P() P() P() P(4) 5/8 Disibio fcio F() is obaid F() : 8 4 8 9 8 7 8 6 8 7 8 49 8 6 8 8 8 Joi dsiy fcio L X, Y b wo.v., Joi d.f. givs pobabiliy a X will assm Y will ak a val y i.. f(, y) P(X, Y y) Eampl: - 5 cads ad cads a daw X o. of spad i s daw Y o. of spad i d daw Wio placig s cad daw. 9 8 f(, ) P[X, Y ] 5 5 X f(, ) P[X, Y ] Y 9 5 5 Toal 9/4 /68 5/68 /68 /7 7/68 Toal 5/68 7/68 f(, ) P[X, Y ] f(, ) P[X, Y ] L A & B b wo vs, P(A B) P(B/A) P(A) 9 5 5 5 5 89
P(A B) P(A/B) P(B) odiioal dsiy fcio f(y/) is dfid as f(y/) P[Y y X ] givs disibio of Y w X is fid f(y/) ad f(/y) f (, y) f () f (, y) f () Magial dsiy fcio H f(, y) f(y/).f() () odidioal disac Y y w X is fid. f (y / ) y Smmig ov all possibl vals of y o bi sids of (), w g f (, y) y f (, y) f () y y f (y / ) f () Tis f() is kow as Magial dsiy fcio of X. Similaly g(y) f (, y) Tis givs Magial fcio of Y. Eampl: - wi & 4 black balls fid pobabiliy of avig wo wi ball. fo black ball X, Y fo wi black Solio: - f(, ) f() f(/) 4. 6 5 f(, ) f() f(/) 6 5 4. 6 5 9
f(,) f() f(/) 4. 6 5 f(, ) f() f(/). 6 5 Magial dsiy fcio of X is f() f (,Y) y f() y 6 f) 5 & f() y f (, y) f(, ) f(, ) f 4 5 (, y) 5 f(, ) f(, ) f() 8 () T codiioal dsiy fcio of Y fo fid ca b obaid fom (), f(y/) f(/) f(/) f (, y) f () f (,) 4/5 4 [sig ()] f () / 5 f (,) f () /5 / oios.v. :- < X < 5 Dsiy fcio A d.f. fo a coios.v. X is a fcio f() a posssss followig popis (i) f() (ii) f () d b (iii) f () d P[a < X < b] w a < b a 9
Disibio fcio F() f () d P[X ] Povidd igal iss. Magial dsiy fcio f() f (, y) dy Magial dsiy fcio of X f(y) f (, y) d Magial dsiy fcio of Y. Idpd Radom Vaiabl Two.v. X ad Y a said o b idpd if f(, y) f() f(y) Eampl: - L joi d.f. f(, y) of.v. s X ad Y b k(y ), <, y < f(, y), owis ) Dmi k ) Eami w X&Y a idpd. Solio: - ) Fo d.f. (, y) (y k ) d dy k y dy y k dy f d dy y k y y 4 9
k 4 k 4 k 4 4 () f() f(y) f f (, y) (, y) dy d cck w f(, y) f() f(y) Now f() k (y ) dy y k y k f(y) k (y ) d k y y k So f(,y) f() f(y) so y a o idpd., < <, < y < Eampl: - f(, y), owis ) Fid magial d.f. of X ad Y ) Fid codiioal d.f. of Y giv X Fid codiioal d.f. of X giv Y y 9
) ck w X&Y a idpd o o? Solio: -() f() f (, y) dy dy ( y), < < f(y) f (, y) d d ( y) y y y ( y), < y < So f(, y) f() f(y) X ad Y a o idpd. () f (, y) f(y/x ), f () < < f (, y) f(x/y y), < y < f (y) ( y) y Eampl: - A coios.v. X as p.d.f. f(),. Fid a abd b s.. () P(X a) P(X a) () P(X b).5 Solio: - Sic P(X a) P(X > a) So, ac ms b qal o ½ bcas oal pobabiliy is always o i.. P(X a) a f () d a a a d a a a a / Also a f () d a d 94
a a a a / () P(X > b).5 f () d.5 b b d.5.5 (b ) b 9 b b 9 / Eampl: - If X b a coios. v. wi a, a, f() a a,, lsw () Fid cosa a () Fid P(X.5) Solio: - () f ()d f ()d f ()d f ()d f ()d f ()d f ()d f ()d a d a d (a a) d a a() a 95
a 9 a a 9 6 a a a.5 () Now P(X.5) f ()d f d f ()d f ()d /.5 a d a d a / a ( ) a a Eampl: - Fom giv bivaia pobabiliy disibio () Obai magial disibio of X & Y. () T codiioal disibio of X giv Y Y X f (, y) f (y) f() y /5 /5 /5 4/5 /5 /5 /5 6/5 /5 /5 /5 5/5 f (, y) 6/5 5/5 4/5 Solio: () Magial disibio of X Fom abov abl f() y f (, y) Tfo f() P(X ) 6/5 96
f() P(X ) 5/5 f() P(X ) 4/5 Magial disibio of Y Fom abov abl f(y) f (, y) Tfo P(Y ) 4/5 P(Y ) 6/5 P(Y ) 5/5 () odiioal disibio of X giv Y, w g P(X /y ) P(X,Y ) P(Y ) Fo X, P(X /Y ) Similaly P(X /Y ) P(X /Y ) /5 5 /5 /5 5 /5 /5 5 /5 MATHEMATIAL EXPETATION:- 5 5 5 L X b a.v. wi p.d.f. f(), is mamaical pcaio (o is ma val) is dfid as E(X) f () I X assms vals Wi pobaliis,. f( ), f( ) f( ) E(X) i f ( i i ) Also E(X) Ma of disibio ad f (i ) T pcd val o Mamaical pcaio of fcio g() of disc.v. X, wos p.d.f. is f() is giv by E[g(X)] g (i )f (i ) i ad f( i ) P(X i ) Eampl: - W cois a ossd ad 97
X pss mb of ads is a. v., oal ocoms a {HHH, HTH, HHT, THH, THT, TTH, HTT, TTT} H X ca ak vals X,,, wi f(),,, 8 8 8 8 T E(X) f ().... 8 8 8 8 Rs. Rs..5 8 If g() T g(),,, E [g()] g( )f ( ) i 8 8 9 Rs.. 8 8 8 i 8 8 Mamaical Epcaio fo oios.v. L X b a coios.v. wi p.d.f. f(), is mamaical pcaio is E() f ()d Fo fcio g(), E[g()] g()f () d Tom: - If is a fii al mb & if E(X) iss, Poof: - fo coios.v. fo disc.v.; L E (X) f ()d f ()d E(X) E(X) f () f () E(X) 98
Rsl: - E[a X] E(a) E(X) a E(X) Poof: - Now E(a) af () a f () a. a () fo coios.v., E(a) af ()d E(a) a f ()d a. a Now By dfiiio, E[a X] ( a )f () E[a X] [ af () f() ] af() af () E(X) E(a) E(X) f () E(a X) a E(X) sig () Tom: - T pcaio of sm of wo.v. s is qal o sm of i pcaios, i.., E(X Y) E(X) E(Y) Poof: - Fo disc cas L X & Y b wo disc.v., f( i, y i ) is joi p.d.f. of X ad Y. (X Y) is also a.v. f( I, y i ) P(X i, Y y j ) Now by dfiiio, E(X Y) ( y i j i j)f (i, y j) f (i, y j) i j i y jf (i, y j) i j ( i, y j) y j f ( i i i, y j) j j i f ( i ) y jf (y E(X Y) E(X) E(Y) Fo coios.v. i E(X Y) i j) j ( y)f (, y)d dy 99
f (, y)d dy yf(, y)d dy f (, y)dy d y f (, y)d dy f ()d yf(y) dy E(X Y) E(X) E(Y) I gal, E(X X. X ) E(X ) E(X ). E(X ) Tom: - If Y a X a X..a X, w a s a cosas, E(Y) a E(X ) a E(X ).. a E(X ) Poof: - As E(Y) E(a X a X... a X ) E(a X ) E(a X ) E(a X ) a E(X ) a E(X ) a E(X ) [ ΘE(aX) ae(x)] Tom: - If X is a coios.v. ad a ad b a cosas, b) a E(X) b, povidd all pcaio iss. Poof: - By dfiiio, w av E(aX E(aX B) ( a b) f()d af ()d bf () d a f ()d b f () d E(aX b) ae(x) b as: - If b, E (ax) ae(x) Θ f ()d as: - If a, b X E(X), w g E(X X )
Eampl: - If f(, y) ( y) is joi p.d.f. of.v. X ad Y, fid P ( < X <, < Y < ) Solio: - Fis w cck fo:- f (, y)d dy L.H.S ( y) y [ ] y d dy d dy dy P ( < X <, < Y < ) ( y) y dyd dyd y d dy ( ) d ( ) ( ) ( )( ) ( ) 4 4 ( ), < (, ) < Eampl: - L f(, ), owis (i) Dmi (ii) Eami w X ad X a idpd o o.
Solio: - Fo dsiy fcio F(, ) w ms av f (, y)d dy d d 4 [ 4 4] 4 o 4 4 Tom: - Sow a pcaio of podc of idpd.v. is qal o podc of i pcaios. i.. E(XY) E(X).E(Y) Poof: - L X ad Y a wo idpd adom vaiabls, X:,,., Wi d.f: - f( ), f( ),..f( ) ad Y : y, y,, y m wi d.f: f(y ), f(y ), f(y m ) E(X) m i f (i ), E(Y) y jf (y j) i j L f( i,y j ) is joi p.d.f. of X & Y. ad Sic X & Y a idpd, so f( i, y j ) f( i ) f(y j ) Now E(XY) y m i j i j f (i, y j)
m i j i y jf (i )f (y j) f (i ) y jf (y i i j) j E(XY) E(X) E(Y) Similaly fo coios.v.: f(, y) f() f(y), sic X ad Y a idpd. Now E(XY) y f (, y)d dy E(XY) yf ()f (y)d dy f ()d yf(y)dy E(X) E(Y) I gal, w g E(X X X..X ) E(X ) E(X ) E(X ) Eampl: - L X pss mb o fac of dic X: 4 5 6 f(): 6 6 Now E(X) f () 6 6 7 (. 6) 6 6 Ad w X is sm of pois w wo dis a ow, i.. X: 4 5 6 7 8 9 f(): 4 5 6 5 4 6 6 6 6 6 6 6 6 6 6 6 E(X) f () [ 6 4 4 6 ] 6 6 6 E(X). 5 7 6
4 Eampl: - L X b pofi a a pso maks i a bsiss. H may a Rs. 8 wi a pobabiliy.5, may los Rs 55 wi pobabiliy. ad may i a o los wi a pobabiliy.. alcla E(X) Solio: - H P(X 8).5 P(X 55). P(X ). T E(X) ) ( f 8(.5) (55) (.) () (.) 4 65 5, may los Rs 5. Eampl: - A ad B i s ow a odiay dic fo a pic of Rs. 44. T fis o ow a si wis. If A as fis ow, wa is is pcaio?. Also calcla B s pcaio Solio: - T poblm of gig a si o dic is p() 6 A as s ow, so ca wi i s, d, 5 Hc A cac (pobabiliy) of wiig is 6 6 5 6 5 6 5 6 5 6 6 5 6 5 6... 6 5 6 5 6 4 6 6. 6 6 5. 6 Amo of A 6 44 Rs. 4 Similaly B ca wi i d, 4, 6,.. Hc B (cac) of wiig a 6. 6 5. 6 5. 6 5 6. 6 5... 6 5 6. 6 5
5 5 6. 6 5. 6 Amo of B 49 5 Rs. Eampl: - A bag coais a coi of val M ad a mb of o cois wos aggga val is m. a pso daws o a a im ill daws coi M, fid val of is pcaio. Solio: - L b K o cois ac of val m/k, so a i aggga val is m. may daw coi M a s daw o d o d o..o (K) daw wi pobabiliy, K, K, K,... K K K, K K, K Θ, K, K, K. T cospodig amo daw X is M, M k Km M, K )m (k M,..., K m M, K m E(X) M K Km... M K m M K m M K K)... ( K m ) (K M K M m M ) K(K K m. K
LESSON 5 MOMENTS AND MOMENT GENERATING FUNTIONS Moms: - L X is a. v., E [X ], if iss is calld mom of X abo oigi ad is dod by μ i. μ E [X ] () ad abo som poi a, i is dfid as μ (a) E[(X a) ] ad mom abo ma is μ E[(X E(X) ] E[(X μ) ] () Moms abo Ma a calld al Moms. I cas of disc.v.:- μ E[X ] f () & μ E[(X μ) ] ( i μ) f ( i ) i W μ E(X) ad f( i ) P(X i ) w fom (), w g μ E(X) f () Ma ad fom (), μ E (X μ) E (X) E(μ) μ μ μ [Θ E(μ) μ, E(X) μ] Moms abo Ma (μ ) i ms of moms abo ay poi a :- L X. v., E(X) X μ μ E [X μ] E [X a μ a] μ E(X a d) W d μ a o μ E [(X a) d(x a).. () d (X a) d () ] μ μ (a) dμ (a) () d E(X a) d () 6
Now μ [μ μ (a) d μ (a) μ (a) ] ad μ μ (a) d μ (a) d μ μ μ [Θ d E(X) a E(X a), d μ (a)] Tis μ is calld Vaiac. Similaly, μ ' ' ' μ μ μ μ ' ' ' 4 ad μ 4 μ 4 4μ 6μ μ μ If X is a coios.v. ' μ ( a) f () d μ ( μ) f () d ad E(X) μ Ma Also μ E(X μ). Tis E(X μ) is calld Vaiac ad is dod by. ovaiac bw X ad Y ov(x, Y) E [(X X) (Y Y)] E [(X E(X)) (Y E(y))] Eampl: - Fid E(X), E(X ), E(X E(X)) fom followig:- X: 8 6 f(): 8 6 Solio: - E(X) f () 8 4 8... 4 8 6 E(X) 6 Ma ad E(X ) f () 64 44 56... 576 8 6 8 E(X ) 76 ' μ μ E [X E(X)] E(X ) [E(X)] [Θμ μ ] 7
μ 76 (6) Tis μ is vaiac. Eampl: - Fid E(X), E(X ), E(X E(X)) X: 4 5 6 7 8 9 f(): 4 5 6 5 4 6 6 6 6 6 6 6 6 6 6 6 Solio: - E(X) f () [6 4 4 6 ] 6 7 Ma E(X ) E f (X) [484854964844] 6 6 (974) 9 6 μ E [X E(X)] E(X ) [E(X)] 974 9 49 49 6 6 9 94 5 6 6 L m pss mdia, P(X < m) P(X > m) I is g, m 7 mod 7 Also ma 7 So i is a vy good disibio. Eampl: - Fo disibio, (_ 6 (6 ) f() 6 ( ) 6,,,, < < < owis ck w i pss a pobabiliy disibio o o. Also fid is ma ad vaiac. 8
Solio: - Fis w pov a f ()d o f ()d Now f d f d f (d) ( 6 )d (6 6 )d ( )d 6 ( ) 6 6 6 ( ) 6 [8 ] 6 6 [ 8] 48 6 48 6 6 4 f ()d. 6 Also Ma E(X) f () d 6 E(X) f ()d ( ) d (6 )d 6 6 ( ) d 6 6 ( ) 6 ( ) () d ( ) ( ) () 6 d [Igad i scod igal is a odd fcio of, so is val is zo] 9
4 4 ( ) ( ) [()8 ] [,8] 48 48 4 48 48 4. (6 ). ( 6) 6 48 4 6 48 4 E(X)..6..6 48 4 48 4 Also Vaiac V(X) E(X E(X)) E(X ) [E(X)] Now E(X ) f ()d ( ) d (6 )d 6 6 ( ) d 6 E(X 4 ) (9 6)d (6 )d 6 6 (9 6 6) 6 9 5 4 6 5 4 6 6 5 5 6 6 5 9 8 5 5 4 5 6 4 4 8 6 4 8 8 8 6 5 5 4 4 4. 8 4 6 5 5 6 5 4 4 6 78. 8 5 6 5 4 4 8 5 5 5 5
E(X 4 ). 8 5 5 5 5 Hc Vaiac E(X ) [E(X)] Eampl: - Sow a val of cov(x, Y) fo pobabiliy disibio f(, y) y / 9,, y is 9. owis Solio: - W Kow a ov.(x,y) E(XY) E(X) E(Y) Ts w av o fid E(X), E(Y) ad E(XY) B E(X) f ()d E(Y) yf (y)dy Fis w av o fid f() ad f(y). Now magial dsiy fcio of X is giv by g() f (, y)dy f (, y)dy f (, y) dy 9 y / dy 9 y / ( ) g() y /, Similaly Magial dsiy fcio of Y will b (y) y y / y y / 9 d 9, y E(X) f ()d /. d / d
/ [. ( ) ] / [ [ ( ) ] / d () E(Y) y (y)dy 9 y y / dy y / { y ( ) } 9 y / [ y ( ) ] y / [ ] E(XY) 6 y y / / dy dy ( ) 6 () y y yf (, y)d dy y. 9 y / d dy y / y dy d 9 y y / [ y ( ) ] 9 / / ( () ) 9 / / [ 9 ] 9 d y / d dy d / I II d / II I d [54 9]
Hc povd 54 7 8 7 ov(x, Y) E(XY) E(X)E(Y) 7 6 7 8 9 Eampl: - If f() a b, b a < a. Fid ma ad vaiac. b Solio: - f() a a Now b < a a < b < a f() Similaly b a < < a b f() a a ( a ( a b) b) fo b a < < b b ( b) fo (b a) < < b fo b < < b a as b b fo b < < b a Ma E(X) f ()d b ba b a ( b). d. a a b ( b) d a a b ba b a b d a a a a b b d a a a b a b ba a a b a b b a b a b a b a b a ab (b a) a a b a (b a) a b a ab (b a) a b(b a) a b b a a b a
b b a a b a b a a a ab a b a a a ab a ( b a ab a b) b a b ab a a ( b a ab a b) b a b ab a a b b a a Vaiac V(X) E[X E(X)] E(X b) ( b) f () d b a b b ( b). d ba a b b a a b a ( b) d ba ( b) a b ( b) a b a b ( b) (a b) d a a ( b) a d ( b) a ( b) 4a 4 b ba ( b) a ( b) 4a 4 b a b. a a 4 a 4a 4 a a a 4a a a 4 a a a a a 6 Eampl: - Fid ma ad Vaiac fo followig disibio f() ( ) 4,, owis Solio: - ck fo f ()d As ma E(X) f ()d ( ) 4. d 4
4 ( )d 4 5 Vaiac V(X) E[X E(X) ] EX 6 5 4 5 f () d ( ) 4 d 4 5 4 4 4 5 ( ) 4 4 ( ) 5 5 ( ) 5 5 4 d ( ) 5 6 6 4 5 4 5 4 5 8 5 5 4 5 4 5 4 5 Tom: - If X is a.v., V(aX b) a V(X), w a ad b a cosas. Poof: - L Y ax b, E(Y) ae(x) b T V(aX b) V(Y) E[Y E(Y)] E[aX b ae(x) b] 4 5 8 75 a E(X ) a [E(X)] a [E(X)] a E(X ) a [E(X)] a [E(X ) [E(X)] ] a V(X) o(i) If b, V(a) a V(X) (ii) If a, V(b) (iii) If a, V(X b) V(X). 5
Tom: - Pov a V(X ± Y) V(X) ± V(Y) ± ov(x, Y) ad V(X ± Y) V(X) V(Y) povidd X ad Y a idpd.v. Poof: - V(X Y) E[(X Y) E(X Y)] E[(X Y) E(X) E(Y)] E[{X E(X)} {Y E(Y)}] E[X E(X)] [Y E(Y)] E[(X E(X)][Y E(Y)] E[X E(X)] E[Y E(Y)] E[{X E(X)}{Y E(Y)}] V(X) V(Y) cov(x, Y) Similaly V(X Y) V(X) V(Y) cov(x, Y) V(X ± Y) V(X) V(Y) ± cov(x, Y) If X ad Y a idpd, cov(x, Y) V(X ± Y) V(X) V(Y) ±. V(X) V(Y). ovaiac: - If X ad Y a wo.v., cov bw is dfid as ov(x, Y) E[[X E(X)[Y E(Y)]] E[XY XE(Y) YE(X) E(X) E(Y)] E(XY) E(X)E(Y) E(Y)E(X) E(X)E(Y) E(XY) E(X)E(Y) W ca also pss i as ov(x, Y) E[[X E(X)][Y E(Y)]] X)(y i j ( fo disc cas i j Y)f ( i, y j) ov(x, Y) E[XY X Y XY X Y ] E(XY) X E(Y) YE(X) X Y E(XY) X Y Y X X Y E(XY) X Y If X ad Y a idpd.v., E(XY) E(X) E(Y) ov(x, Y) E(X) E(Y) X Y X Y X Y 6
Rmak ov(x, Y) if X ad Y a idpd. i) ov(a, by) E[(aX E(a))(bY E(bY))] E[(aX ae(x)) (by BE(Y))] E[a[X E(X)]. b[y E(Y)]] ab[e(x E(X)) (Y E(Y))] ab ov(x, Y) ovaiac is o idpd of cag of scal. ii) ov(x a, Y b) E[{(X a) E(X a)} {Y b E(Y b)}] E[{(X a) E(X) a} { Y b E(Y) b} E[{X E(X)} { Y E(Y)}] ov(x, Y) Ts ov(x, Y) is idpd of cag of oigi b is o idpd of cag of scal. MEAN DERIVATION FOR ONTINUOUS ASE E[ X a ] a f()d Vaiac E[X E(X)] ( X) f()d Absol mom L X b. v. wi p.d.f f(), is absol mom abo ay poi a is giv by E[ X a ] a f() d Fo Vaiac i.. μ E[X E()] E[X XE(X) E (X)] E(X ) E(X) E(X) E (X) [Θ E(X) μ ] μ E(X ) [E(X)] E (X) E(X ) [E(X)] μ μ ' ' μ Effc of cag of oigi & scal o mom 7