UNIT III STANDARD DISTRIBUTIONS
|
|
|
- Linette Byrd
- 5 years ago
- Views:
Transcription
1 UNIT III STANDARD DISTRIBUTIONS Biomial, Poisso, Normal, Gomric, Uiform, Eoial, Gamma disribuios ad hir roris. Prard by Dr. V. Valliammal Ngaiv biomial disribuios Prard by Dr.A.R.VIJAYALAKSHMI
2 Sadard Disribuios Biomial Disribuio Assumios. Th radom rim corrsods o oly wo ossibly oucoms.. Th umbr of rials is fii.. Th rials ar idd. 4. Th robabiliy of succss is a cosa from rial o rial.
3 Noaios umbr of rials robabiliy of succss robabiliy of failur A radom variabl which rrss h umbr of succsss Biomial Disribuio A discr radom variabl is said o follow Biomial disribuio if i s robabiliy mass fucio is P C,,,,...,
4 Mom Graig Fucio Th M.G.F of a Biomial varia is P ] E[ M C C M
5 Raw Moms d d d d M [ ] Hc ma of Biomial disribuio =
6 Variac of Biomial Disribuio ] [ ] [ ] [ ] [ ] [ d d M d d
7 Addiiv Prory of Biomial Disribuio L follow biomial disribuio wih aramrs ad. L follow Biomial disribuio wih aramrs ad. Furhr l ad b idd. M Cosidr M M M M [ ad ar idd] Which is o of h form +. Hc + is o a biomial varia.
8 Bu if = =, h = =. Euaio bcoms M Which is of h form +. Hc + follows Biomial disribuio wh = =. i.., Biomial disribuio has addiiv rory wh = =.
9 Problms Basd O Biomial Disribuio. I has b claimd ha i 6 % of all solar ha isallaio h uiliy bill is rducd by alas ohird. Accordigly wha ar h robabiliis ha h uiliy bill will b rducd by alas o-hird i alas four of fiv isallaio. Soluio Giv = 5, = 6% =.6 ad = - =.4 4 5c [ 4.7 4].6 4 [ ] 5c
10 . Th ma ad variac of a biomial varia ar 6 ad rscivly. Fid P. Soluio Giv ha E = 6 i.., = 6 ad V = = Dividig Euaio by Euaio givs, 6
11 Us i Euaio C C P C P P P 9 9
12 . A ubiasd di is rolld ims. Gig a oucom grar ha 4 i a di is rmd as a succss. Wha is h chac of gig a las 8 succsss? Soluio : umbr of succsss : P umbr of succss = P oucom is grar C ha 4 i a di 6 C = umbr of rials =
13 P alas 8 succsss = P 8,,,..., C C P C C C C P
14 Poisso Disribuio Th alicaio of Biomial disribuio will b ivalid wh ad. Hc i h rsc of h abov wo codiios w d a horical disribuio which ovrcoms h drawbac of Biomial disribuio. Th Biomial disribuio ds o oisso disribuio wh i. Th umbr of rials is idfiily larg i.., ii. Th robabiliy of succss is vry small i.., iii. is a cosa i.., =.
15 Poisso Disribuio mf A discr radom variabl is said o follow Poisso disribuio wih aramr if is robabiliy mass fucio is P Mom Graig Fucio!,,,,..., M E[ ] P!! a! a
16 Raw Moms of Poisso Disribuio i.., ma of Poisso disribuio =. d d M d d
17 Scod Ordr Raw Mom ] [ ] [ ] [ d d d dm. d d d M d
18 Variac of Poisso disribuio Variac = Scod ordr cral mom Thus i Poisso disribuio ma = Variac =.
19 Addiiv Prory of Poisso Disribuio L ad b wo idd Poisso varias wih aramr ad rscivly. Th M ad M Cosidr h varia + Now M M...M [ ad ar idd] Which is of h form. Hc + f ollows Poisso disribuio wih aramr +. i.., h Poisso disribuio has addiiv rory.
20 Prov ha oisso disribuio is h limiig cas of Biomial disribuio. or Poisso disribuio is a limiig cas of Biomial disribuio udr h followig codiios i, h o. of rials is idfiily larg, i., ii, h cosa robabiliy of succss i ach rial is vry small, i. iii is if ii or ad, is osiiv ral Soluio If is biomial r.v wih aramr &, h
21 c...!....!!...!!!...!!!,,,...,
22 Taig limi as o boh sids ad i is oisso disb. Hc h roof.,...,,,!....!...!...! lim lim lim lim lim,,,...,!
23 Prov ha h sum of wo idd oisso varias is a oisso varia, whil h diffrc is o a oisso varia. Soluio L ad b idd r.v.s ha follow oisso disb. wih Paramrs ad rscivly. L = +
24 This is oisso wih aramr + r r r r r r r r r r r r r r c r r r r r r aridd c r r!.!!!!..!!!.!.!.!..!. & si.
25 ii Diffrc is o oisso L = I is o oisso. E E E E E E E E E E E
26 Prov ha oisso disribuio is h limiig cas of Biomial disribuio. or Poisso disribuio is a limiig cas of Biomial disribuio udr h followig codiios i, h o. of rials is idfiily larg, i., ii, h cosa robabiliy of succss i ach rial is vry small, i. iii is if ii or ad, is osiiv Soluio If is biomial r.v wih aramr &, h ral
27 c...!....!!...!!!...!!!,,,...,
28 Taig limi as o boh sids ad i is oisso disb. Hc h roof.,...,,,!......!...!...! lim lim lim lim lim,,,...,!
29 Problms Basd o Poisso Disribuio. Th o. of mohly bradows of a comur is a r.v. havig oisso disb wih ma.8. Fid h robabiliy ha his comur will fucio for a moh wih oly o bradow. Soluio, giv.8!.8.8!.975
30 . I is ow ha h robabiliy of a im roducd by a crai machi will b dfciv is.5. If h roducd ims ar s o h mar i acs of, fid h o. of acs coaiig alas, acly, amos dfcivs i a cosigm of acs usig oisso. Soluio Giv =, =.5, N = Ma = = L do h o. of dfcivs.
31 Thrfor, ou of acs, h o. of acs coaiig alas dfcivs ii Ou of acs,=n *[=] =84 acs iii For acs = *.9975=9 acs aroimaly. 64.!!,...,,!.!.!. acs N *. 895.! 9975.!!! ] [ ] [ ] [
32 . Th aoms of radio aciv lm ar radomly disigraig. If vry gram of his lm, o avrag, mis.9 alha aricls r scod, wha is h robabiliy durig h scod h o. of alha aricls mid from gram is i amos 6 ii alas iii alas ad amos 6? Soluio Giv =.9 L do h o. of alha aricls mid i !.9.9!.9.9! ! 6.898
33 !.!... ii.645 6!.9 5!.9 4!.9! iii
34 Gomric Disribuio A discr radom variabl which rrss h umbr of failurs rcdig h firs, succss is said o follow Gomric disribuio if is robabiliy mass fucio is P =, =,,, Mom Graig Fucio M E[ ] P [ [ ] ]
35 Raw Moms of Gomric Disribuio d d M d d
36 ] [ ] [ d d d M d
37 Variac of Gomric Disribuio ] [
38 Ecaio of Gomric Disribuio wihou usig MGF ] [...] [...] [ P E
39 Variac of Gomric Disribuio [E] E V ] [...] 4 [...] 8 6 [ P P ]P [ P ] E[ 4
40 ] [ [E] E V
41 Esablish h mmorylss rory of gomric disb. Soluio If is a discr r.v. followig a gomric disb. Now......,...,, m m m m m m ad m m m m m
42 Esablish h mmorylss rory of gomric disb. Soluio If is a discr r.v. followig a gomric disb. Now......,,..., m m m m m m ad m m m m m
43 Problm Basd o Gomric Disribuio. Suos ha a rai soldir shoos a arg i a idd fashio. If h robabiliy ha h arg is sho o ay o sho is.7. i Wha is h robabiliy ha h arg would b hi i h am? ii Wha is h robabiliy ha i as him lss ha 4 shos? iii Wha is h robabiliy ha i as him a v o. of shos? iv Wha is h avrag o. of shos dd o hi h arg? Soluio L do h o. of shos dd o hi h arg ad follows gomric disribuio wih mf,,,...
44 i ii iii is avumbr iv Avrag o.of shos E.7.486
45 Ngaiv biomial disribuio If rad idd rials ca rsul i a succss wih robabiliy, ad a failur wih robabiliy =-, h h robabiliy disribuio of h radom variabl, h o. of failurs rcdig h h succss is giv by c,,,,... Mom Graig Fucio of Ngaiv Biomial Disribuio M E c
46 c c c c c c...!!.!!....!!!!!!....!!!!! M
47 d d M d d Ma Ma Ma ad Variac of Ngaiv biomial
48 d d M d d Var
49 Var i. Variac Problm If h robabiliy is.4 ha a child osd o a crai coagious disas will cach i, wha is h robabiliy ha h h child osd o h disas will h hird o cach i? Soluio : Third succss i h rial. so, l us aly gaiv biomial disribuio. R uird robabiliy 7 7 9C
50 Uiform Disribuio Coiuous Uiform Disribuio or Rcagular Disribuio A coiuous radom variabl dfid i h irval a,b is said o follow uiform disribuio if is robabiliy dsiy fucio is giv by f b, a, a b ohrwis No: ad b ar h aramrs of h disribuio.
51 Mom Graig Fucio of a Uiform Disribuio M E b a b a fd d b a b a b a b a MGF b a
52 Raw Moms of Rcagular Disribuio Wh r =, Euaio bcoms Ma r a b a b r a b d a b fd r r r b a r b a r b a r r a b a b a ab b a b a b
53 Wh r =, Euaio bcoms Variac of Uiform Disribuio a ab b a ab ab b a b a b a b b a ab b a a 6ab b 4a 4ab 4b a b a ab b [E] E V
54 Problms Basd o Uiform Disribuio.Show ha for h uiform disribuio f, a a a Soluio: Giv sih a, h mgf abou origi is. f, a a a a MGF M E M a a f d a a d a a a a sih a a sih a a a a d a a sih a a
55 . Th umbr of rsol comur c sold daily a a comur world is uiformly disribud wih a miimum of c ad a maimum of 5 c. Fid Th robabiliy ha daily sals will fall bw 5 ad c Wha is h robabiliy ha h comur world will sll alas c s? Wha is h robabiliy ha h comur world will sll acl c s? Soluio L ~Ua, b, h h df is giv by
56 f, a b b a, 5 5, i. i is aricular oi, h valu is zro f d f d d d
57 . Sarig a 5. am vry half a hour hr is a fligh from Sa Frasisco airor o Losagls. Suos ha o of hr las is comlly sold ou ad ha hy always hav room for assgrs. A rso who was o fly o Losagls arriv a a radom im bw 8.45 am ad 9.45 am. Fid h robabiliy ha sh wais a Amos mi b alas 5 mi Soluio L b h uiform r.v. ovr h irval, 6. Th h df is giv by f b a, 6, a b 6
58 . Sarig a 5. am vry half a hour hr is a fligh from Sa Frasisco airor o Losagls. Suos ha o of hr las is comlly sold ou ad ha hy always hav room for assgrs. A rso who was o fly o Losagls arriv a a radom im bw 8.45 am ad 9.45 am. Fid h robabiliy ha sh wais a Amos mi b alas 5 mi Soluio L b h uiform r.v. ovr h irval, 6. Th h df is giv by f b a, 6, a b 6
59 a Th assgrs will hav o wai lss ha mi. if sh arrivs a h airor b Th robabiliy ha sh has o wai alas 5 mi d d d d 6 45
60 Eoial Disribuio A coiuous radom variabl is said o follow oial disribuio if is df is giv by Ma ad variac of a Eoial disribuio:, f d d d f E Ma
61 Ma d d d f E E E Variac
62 Mom Graig Fucio of a Eoial Disribuio: ] [ d d fd ] E[ M
63 Esablish h mmory lss rory of a oial disribuio. Soluio If is oially disribud, h Th df of oial dis is giv by for ay s s s, ohrwis f,, d d f s s s s ad s s s s s for ay s s s,
64 Problms basd o Eoial Disribuio Th im i hours ruird o rair a machi is oially disribud wih aramr = /. awha is h robabiliy ha h rair im cds hrs? bwha is h codiioal robabiliy ha a rair as alas hrs giv ha Is dircio cds 8 hrs? Soluio If rrss h im o rair h machi, h dsiy fucio Of is giv by f,
65 679. d d d f 8. d d d f
66 Gamma Disribuio A coiuous radom variabl is said o follow Gamma disribuio wih aramr if is robabiliy dsiy fucio is f ohrwis Mom graig fucio of gamma disribuio: M E[ ] fd d d
67 Raw moms of gamma disribuio Variac of gamma disribuio Hc i gamma disribuio ma = variac = d d M d d d d d M d
68 Variac of gamma disribuio V E [E] Addiiv rory of Gamma Disribuio Cosidr wo idd gamma varias ad wih aramrs ad rscivly. M M M M M Which is of h form. Hc + is also a Gamma varia wih aramr +. Hc Gamma disribuio has addiiv rory. [ ad ar idd]
69 Problms Basd o Gamma Disribuio. I a crai ciy h daily cosumio of lcric owr i millios of ilowa hrs ca b rad as cral gamma dis wih,. If h owr la has a daily caaciy of millio ilowa hours. Wha is h robabiliy ha h owr suly will b iadua o ay giv day. Soluio L b h daily cosumio of lcric owr Th h dsiy fucio of is giv by
70 [ h owr suly is iadua]=[>] 6! 8 f d d d f
71 . Th daily cosumio of mil i a ciy i css of, lirs is aroimaly disribud as a Gamma dis wih aramr,. Th ciy has a daily soc of, lirs. Wha is h robabiliy ha h soc is isuffici o a aricular day. Soluio L b h daily cosumio, so, h r.v. Y=-. Th y f Y y y y y y y y y!
72 [ isuffici soc]=[>] =[Y>].757,, ] [ y u By subsiuio mhod dy y dy y dy y f Y y y
73 Normal Disribuio Th Eglish mahmaicia D-Moivr, obaid a coiuous disribuio as a limiig cas of biomial disribuio i h yar 7. This disribuio was amd ormal disribuio. Th firs rso who mad rfrc o his disribuio was Gauss who usd i o sudy h disribuio of rrors i Asroomy. Probabiliy Dsiy Fucio A coiuous radom variabl is said o follow ormal disribuio wih aramrs ma ad variac, i is dsiy fucio is giv by h robabiliy law: f σ σ,, ad σ
74 Rmars A radom variabl which follows Normal disribuio wih ma ad variac is rrsd as ~ N,. If is a ormal varia, z is calld as a sadard σ ormal varia. If ~ N,, h z ~ N,. Th df of a sadard ormal varia Z is giv by, z π Z /, Z
75 Mom Graig Fucio of Normal Disribuio If follows ormal disribuio wih ma ad variac, is mom graig fucio is drivd as follows: M E[ ] fd σ π σ d L z d σdz σ z z
76 dz π dz π dz π σdz π σ M σz z z σz z σz z σz dz π dz π dz π M σσ z σ σ σ σz z σ σ σz z
77 z σ A dz da z A z A M π σ A da A π da bcaus A π is h df of a sadard ormal varia. M σ
78 Graig h raw moms usig MGF σ d dm M σ σ σ σ σ σ σ σ σ σ σ σ σ d M d
79 Problms basd o Normal Disribuio. Fid P z > -.9 Soluio P z > -.9 = P-.9 < z < +.5 = P < z < = =.95
80 . Fid P z Soluio P z = P z P z = =.59
81 I a ormal disribuio % of h ims ar udr 45 ad 8% ar ovr64, fid h ma ad h sadard dviaio. Sol: L ma ad sadard dviaio of h giv ormal disribuio b, Th ara lyig o h lf of h ordia a =45 is..h corrsodig valu of Z is gaiv. h ara lyig o h righ of h ordias u o h ma is,.5-.=.9 Th valu of Z corrsodig o h ara.9 is.5 arly, 45 = Ara o h lf of ordia a =64 is, =.5-.8 =.4
82 SOLVING AND.,
83 FORMULA Sl. No. Discr radom variabl Coiuous radom variabl.. i i F P[ ] fd F P[ ] fd. Ma E[] i i i Ma E[] fd 4. E[ ] i i i E[ ] fd 5. Var E [E] Var E [E] Mom E[ M.G.F. M E[ r ] i r i ] i Mom E[ M.G.F M E[ r ] ] r fd fd
84 s Mom abou origi = E = [M ] = Ma d Mom abou origi = E = [M ] = r r Th co-ffici of E[ ] r h Mom abou h origi Limiaio of M.G.F: r! i A radom variabl may hav o moms alhough is m.g.f iss. ii A radom variabl ca hav is m.g.f ad som or all moms, y h m.g.f dos o gra h moms. iii A radom variabl ca hav all or som moms, bu m.g.f dos o is c rhas a o oi.
85 .P.D.F, M.G.F, Ma ad Variac of all h disribuios: Sl. No. Disribuio P.D.F. P = M.G.F Ma Variac. Biomial C +. Poisso!
86 Sl. No. Disribuio P.D.F. P = M.G.F Ma Variac. Gomric or 4 uiform f, b a, a b ohrwis b b a a a b b a 5 Eoial f,,, ohrwis 6 Gamma f Γ,,.
MA6451-PROBABILITY AND RANDOM PROCESSES
MA645-PROBABILITY AND RANDOM PROCESSES UNIT I RANDOM VARIABLES Dr. V. Valliammal Darm of Alid Mahmaics Sri Vkaswara Collg of Egirig Radom variabl Radom Variabls A ral variabl whos valu is drmid by h oucom
Poisson Arrival Process
1 Poisso Arrival Procss Arrivals occur i) i a mmorylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = 1 λδ + ( Δ ) P o P j arrivals durig Δ = o Δ for j = 2,3, ( ) o Δ whr lim =
Poisson Arrival Process
Poisso Arrival Procss Arrivals occur i) i a mmylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = λδ + ( Δ ) P o P j arrivals durig Δ = o Δ f j = 2,3, o Δ whr lim =. Δ Δ C C 2 C
Chapter 3 Linear Equations of Higher Order (Page # 144)
Ma Modr Dirial Equaios Lcur wk 4 Jul 4-8 Dr Firozzama Darm o Mahmaics ad Saisics Arizoa Sa Uivrsi This wk s lcur will covr har ad har 4 Scios 4 har Liar Equaios o Highr Ordr Pag # 44 Scio Iroducio: Scod
1973 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
( A) ( B) ( C) ( D) ( E)
d Smsr Fial Exam Worksh x 5x.( NC)If f ( ) d + 7, h 4 f ( ) d is 9x + x 5 6 ( B) ( C) 0 7 ( E) divrg +. (NC) Th ifii sris ak has h parial sum S ( ) for. k Wha is h sum of h sris a? ( B) 0 ( C) ( E) divrgs
Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia
Par B: rasform Mhods Profssor E. Ambikairaah UNSW, Ausralia Chapr : Fourir Rprsaio of Sigal. Fourir Sris. Fourir rasform.3 Ivrs Fourir rasform.4 Propris.4. Frqucy Shif.4. im Shif.4.3 Scalig.4.4 Diffriaio
What Is the Difference between Gamma and Gaussian Distributions?
Applid Mahmaics,,, 85-89 hp://ddoiorg/6/am Publishd Oli Fbruary (hp://wwwscirporg/joural/am) Wha Is h Diffrc bw Gamma ad Gaussia Disribuios? iao-li Hu chool of Elcrical Egirig ad Compur cic, Uivrsiy of
Response of LTI Systems to Complex Exponentials
3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will
Continous 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
4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b
![4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b](/thumbs/71/65779443.jpg "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
MAT3700. Tutorial Letter 201/2/2016. Mathematics III (Engineering) Semester 2. Department of Mathematical sciences MAT3700/201/2/2016
MAT3700/0//06 Tuorial Lr 0//06 Mahmaics III (Egirig) MAT3700 Smsr Dparm of Mahmaical scics This uorial lr coais soluios ad aswrs o assigms. BARCODE CONTENTS Pag SOLUTIONS ASSIGNMENT... 3 SOLUTIONS ASSIGNMENT...
Practice papers A and B, produced by Edexcel in 2009, with mark schemes. Practice Paper A. 5 cosh x 2 sinh x = 11,
Prai paprs A ad B, produd by Edl i 9, wih mark shms Prai Papr A. Fid h valus of for whih 5 osh sih =, givig your aswrs as aural logarihms. (Toal 6 marks) k. A = k, whr k is a ral osa. 9 (a) Fid valus of
(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is
[STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs
F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mathematics
![F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mathematics](/thumbs/89/97771684.jpg "F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mathematics")
F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mahemaics Prelim Quesio Paper Soluio Q. Aemp ay FIVE of he followig : [0] Q.(a) Defie Eve ad odd fucios. [] As.: A fucio f() is said o be eve fucio if
Numerical Simulation for the 2-D Heat Equation with Derivative Boundary Conditions
IOSR Joural of Applid Chmisr IOSR-JAC -ISSN: 78-576.Volum 9 Issu 8 Vr. I Aug. 6 PP 4-8 www.iosrjourals.org Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios Ima. I. Gorial parm of Mahmaics
) and furthermore all X. The definition of the term stationary requires that the distribution fulfills the condition:
Assigm Thomas Aam, Spha Brumm, Haik Lor May 6 h, 3 8 h smsr, 357, 7544, 757 oblm For R b X a raom variabl havig ormal isribuio wih ma µ a variac σ (his is wri as ~ (,) X. by: R a. Is X ) a urhrmor all
Fourier Series: main points
BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca
Chapter 2 The Poisson Process
Chapr 2 Th oisso rocss 2. Expoial ad oisso disribuios 2... Th Birh Modl I scods, a oal of popl ar bor. Sarig a ay poi i im, wha is h waiig im for h firs birh? I milliscods, a oal of lpho calls arriv a
Mixing time with Coupling
Mixig im wih Couplig Jihui Li Mig Zhg Saisics Dparm May 7 Goal Iroducio o boudig h mixig im for MCMC wih couplig ad pah couplig Prsig a simpl xampl o illusra h basic ida Noaio M is a Markov chai o fii
Moment Generating Function
1 Mome Geeraig Fucio m h mome m m m E[ ] x f ( x) dx m h ceral mome m m m E[( ) ] ( ) ( x ) f ( x) dx Mome Geeraig Fucio For a real, M () E[ e ] e k x k e p ( x ) discree x k e f ( x) dx coiuous Example
Note 6 Frequency Response
No 6 Frqucy Rpo Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada. alyical Exprio
Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120
Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,
Dr. Junchao Xia Center of Biophysics and Computational Biology. Fall /21/2016 1/23
BIO53 Bosascs Lcur 04: Cral Lm Thorm ad Thr Dsrbuos Drvd from h Normal Dsrbuo Dr. Juchao a Cr of Bophyscs ad Compuaoal Bology Fall 06 906 3 Iroduco I hs lcur w wll alk abou ma cocps as lsd blow, pcd valu
Discussion 06 Solutions
STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P
Probability & Statistics,
Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said
EEE 303: Signals and Linear Systems
33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d =
Exercise 3 Stochastic Models of Manufacturing Systems 4T400, 6 May
Exercise 3 Sochasic Models of Maufacurig Sysems 4T4, 6 May. Each week a very popular loery i Adorra pris 4 ickes. Each ickes has wo 4-digi umbers o i, oe visible ad he oher covered. The umbers are radomly
Institute of Actuaries of India
Isiue of cuaries of Idia Subjec CT3-robabiliy ad Mahemaical Saisics May 008 Eamiaio INDICTIVE SOLUTION Iroducio The idicaive soluio has bee wrie by he Eamiers wih he aim of helig cadidaes. The soluios
2 Dirac delta function, modeling of impulse processes. 3 Sine integral function. Exponential integral function
Chapr VII Spcial Fucios Ocobr 7, 7 479 CHAPTER VII SPECIAL FUNCTIONS Cos: Havisid sp fucio, filr fucio Dirac dla fucio, modlig of impuls procsss 3 Si igral fucio 4 Error fucio 5 Gamma fucio E Epoial igral
Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.
Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%
EXERCISE - 01 CHECK YOUR GRASP
DEFNTE NTEGRATON EXERCSE - CHECK YOUR GRASP. ( ) d [ ] d [ ] d d ƒ( ) ƒ '( ) [ ] [ ] 8 5. ( cos )( c)d 8 ( cos )( c)d + 8 ( cos )( c) d 8 ( cos )( c) d sic + cos 8 is lwys posiiv f() d ( > ) ms f() is
Mathematical Preliminaries for Transforms, Subbands, and Wavelets
Mahmaical Prlimiaris for rasforms, Subbads, ad Wavls C.M. Liu Prcpual Sigal Procssig Lab Collg of Compur Scic Naioal Chiao-ug Uivrsiy hp://www.csi.cu.du.w/~cmliu/courss/comprssio/ Offic: EC538 (03)5731877
Double Slits in Space and Time
Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an
Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series
Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris
CS 688 Pattern Recognition. Linear Models for Classification
//6 S 688 Pr Rcogiio Lir Modls for lssificio Ø Probbilisic griv modls Ø Probbilisic discrimiiv modls Probbilisic Griv Modls Ø W o ur o robbilisic roch o clssificio Ø W ll s ho modls ih lir dcisio boudris
ELECTROMAGNETIC COMPATIBILITY HANDBOOK 1. Chapter 12: Spectra of Periodic and Aperiodic Signals
ELECTOMAGNETIC COMPATIBILITY HANDBOOK Chapr : Spcra of Priodic ad Apriodic Sigals. Drmi whhr ach of h followig fucios ar priodic. If hy ar priodic, provid hir fudamal frqucy ad priod. a) x 4cos( 5 ) si(
1.7 Vector Calculus 2 - Integration
cio.7.7 cor alculus - Igraio.7. Ordiary Igrals o a cor A vcor ca b igrad i h ordiary way o roduc aohr vcor or aml 5 5 d 6.7. Li Igrals Discussd hr is h oio o a dii igral ivolvig a vcor ucio ha gras a scalar.
EE415/515 Fundamentals of Semiconductor Devices Fall 2012
3 EE4555 Fudmls of Smicoducor vics Fll cur 8: PN ucio iod hr 8 Forwrd & rvrs bis Moriy crrir diffusio Brrir lowrd blcd by iffusio rducd iffusio icrsd mioriy crrir drif rif hcd 3 EE 4555. E. Morris 3 3
Chapter (8) Estimation and Confedence Intervals Examples
Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit
th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)
![th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)](/thumbs/80/81319353.jpg "th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)")
1 Trasform Techiques h m m m m mome : E[ ] x f ( x) dx h m m m m ceral mome : E[( ) ] ( ) ( x) f ( x) dx A coveie wa of fidig he momes of a radom variable is he mome geeraig fucio (MGF). Oher rasform echiques
Microscopic Flow Characteristics Time Headway - Distribution
CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,
STK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random
Overview. Review Elliptic and Parabolic. Review General and Hyperbolic. Review Multidimensional II. Review Multidimensional
Mlil idd variabls March 9 Mlidisioal Parial Dirial Eaios arr aro Mchaical Egirig 5B iar i Egirig Aalsis March 9 Ovrviw Rviw las class haracrisics ad classiicaio o arial dirial aios Probls i or ha wo idd
Web-appendix 1: macro to calculate the range of ( ρ, for which R is positive definite
Wb-basd Supplmary Marials for Sampl siz cosidraios for GEE aalyss of hr-lvl clusr radomizd rials by Sv Trsra, Big Lu, oh S. Prissr, Tho va Achrbrg, ad Gorg F. Borm Wb-appdix : macro o calcula h rag of
Mathematical Statistics. Chapter VIII Sampling Distributions and the Central Limit Theorem
Mahmacal ascs 8 Chapr VIII amplg Dsrbos ad h Cral Lm Thorm Fcos of radom arabls ar sall of rs sascal applcao Cosdr a s of obsrabl radom arabls L For ampl sppos h arabls ar a radom sampl of s from a poplao
15. Numerical Methods
S K Modal' 5. Numrical Mhod. Th quaio + 4 4 i o b olvd uig h Nwo-Rapho mhod. If i ak a h iiial approimaio of h oluio, h h approimaio uig hi mhod will b [EC: GATE-7].(a (a (b 4 Nwo-Rapho iraio chm i f(
Solution to 1223 The Evil Warden.
Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud
Mathematical Statistics
ahmaical Saisics 4 Cha IV Disc Disibuios Th obabili modls fo adom ims ha ill b dscibd i his ad chas occu ful i alicaios Coiuous disibuios ill b sd i cha This cha ill ioduc som disc disibuios icludig Boulli
Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1
Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida
Chapter4 Time Domain Analysis of Control System
Chpr4 im Domi Alyi of Corol Sym Rouh biliy cririo Sdy rror ri rpo of h fir-ordr ym ri rpo of h cod-ordr ym im domi prformc pcificio h rliohip bw h prformc pcificio d ym prmr ri rpo of highr-ordr ym Dfiiio
Mathematical Statistics. 1 Introduction to the materials to be covered in this course
Mahemaical Saisics Iroducio o he maerials o be covered i his course. Uivariae & Mulivariae r.v s 2. Borl-Caelli Lemma Large Deviaios. e.g. X,, X are iid r.v s, P ( X + + X where I(A) is a umber depedig
The Development of Suitable and Well-founded Numerical Methods to Solve Systems of Integro- Differential Equations,
Shiraz Uivrsiy of Tchology From h SlcdWorks of Habibolla Laifizadh Th Dvlopm of Suiabl ad Wll-foudd Numrical Mhods o Solv Sysms of Igro- Diffrial Equaios, Habibolla Laifizadh, Shiraz Uivrsiy of Tchology
Review Topics from Chapter 3&4. Fourier Series Fourier Transform Linear Time Invariant (LTI) Systems Energy-Type Signals Power-Type Signals
Rviw opics from Chapr 3&4 Fourir Sris Fourir rasform Liar im Ivaria (LI) Sysms Ergy-yp Sigals Powr-yp Sigals Fourir Sris Rprsaio for Priodic Sigals Dfiiio: L h sigal () b a priodic sigal wih priod. ()
ECEN620: Network Theory Broadband Circuit Design Fall 2014
ECE60: work Thory Broadbad Circui Dig Fall 04 Lcur 6: PLL Trai Bhavior Sam Palrmo Aalog & Mixd-Sigal Cr Txa A&M Uivriy Aoucm, Agda, & Rfrc HW i du oday by 5PM PLL Trackig Rpo Pha Dcor Modl PLL Hold Rag
82A Engineering Mathematics
Class Nos 5: Sod Ordr Diffrial Eqaio No Homoos 8A Eiri Mahmais Sod Ordr Liar Diffrial Eqaios Homoos & No Homoos v q Homoos No-homoos q ar iv oios fios o h o irval I Sod Ordr Liar Diffrial Eqaios Homoos
AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
8. Queueing systems. Contents. Simple teletraffic model. Pure queueing system
8. Quug sysms Cos 8. Quug sysms Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs lc8. S-38.45 Iroduco o Tlraffc Thory Srg 5 8. Quug sysms 8.
K3 p K2 p Kp 0 p 2 p 3 p
Mah 80-00 Mo Ar 0 Chaer 9 Fourier Series ad alicaios o differeial equaios (ad arial differeial equaios) 9.-9. Fourier series defiiio ad covergece. The idea of Fourier series is relaed o he liear algebra
Math 6710, Fall 2016 Final Exam Solutions
Mah 67, Fall 6 Fial Exam Soluios. Firs, a sude poied ou a suble hig: if P (X i p >, he X + + X (X + + X / ( evaluaes o / wih probabiliy p >. This is roublesome because a radom variable is supposed o be
Calculus BC 2015 Scoring Guidelines
AP Calculus BC 5 Scorig Guidelies 5 The College Board. College Board, Advaced Placeme Program, AP, AP Ceral, ad he acor logo are regisered rademarks of he College Board. AP Ceral is he official olie home
Chapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
ln x = n e = 20 (nearest integer)
H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77
Institute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 4 9/16/2013. Applications of the large deviation technique
MASSACHUSETTS ISTITUTE OF TECHOLOGY 6.265/5.070J Fall 203 Lecure 4 9/6/203 Applicaios of he large deviaio echique Coe.. Isurace problem 2. Queueig problem 3. Buffer overflow probabiliy Safey capial for
Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
Law of large numbers
Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs
APPENDIX: STATISTICAL TOOLS
I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.
LINEAR 2 nd ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
Diol Bgyoko (0) I.INTRODUCTION LINEAR d ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS I. Dfiiio All suh diffril quios s i h sdrd or oil form: y + y + y Q( x) dy d y wih y d y d dx dx whr,, d
Some Applications of the Poisson Process
Applid Maaics, 24, 5, 3-37 Publishd Oli Novbr 24 i SciRs. hp://www.scirp.org/oural/a hp://dx.doi.org/.4236/a.24.59288 So Applicaios of Poisso Procss Kug-Ku s Dpar of Maaics, Ka Uivrsiy, Uio, USA Eail:
Decline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.
Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.
MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI
MAHALAKSHMI EGIEERIG COLLEGE TIRUCHIRAALLI 6 QUESTIO BAK - ASWERS -SEMESTER: V MA 6 - ROBABILITY AD QUEUEIG THEORY UIT IV:QUEUEIG THEORY ART-A Quesio : AUC M / J Wha are he haraerisis of a queueig heory?
z 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z
Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist
07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Pag ( Aswrs at h d of all qustios ) ( ) If = a, y = b, z = c, whr a, b, c ar i A.P. ad = 0 = 0 = 0 l a l
Chapter 3 Moments of a Distribution
Chaper 3 Moes of a Disribuio Epecaio We develop he epecaio operaor i ers of he Lebesgue iegral. Recall ha he Lebesgue easure λ(a) for soe se A gives he legh/area/volue of he se A. If A = (3; 7), he λ(a)
2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)
Mah PracTes Be sure o review Lab (ad all labs) There are los of good quesios o i a) Sae he Mea Value Theorem ad draw a graph ha illusraes b) Name a impora heorem where he Mea Value Theorem was used i he
1985 AP Calculus BC: Section I
985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b
3(8 ) (8 x x ) 3x x (8 )
Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6
Chapter 5 Special Discrete Distributions. Wen-Guey Tzeng Computer Science Department National Chiao University
Chatr 5 Scal Dscrt Dstrbutos W-Guy Tzg Comutr Scc Dartmt Natoal Chao Uvrsty Why study scal radom varabls Thy aar frqutly thory, alcatos, statstcs, scc, grg, fac, tc. For aml, Th umbr of customrs a rod
Testing Goodness-of-Fit in Autoregressive Fractionally Integrated Moving- Average Models with Conditional Hetroscedastic Errors of Unknown form
Rsarch Joural of Rc Scics ISSN 77-5 Vol. (5, 9-4, May ( Rs.J.Rc Sci. Tsig Goodss-of-Fi i Auorgrssiv Fracioally Igrad Movig- Avrag Modls wih Codiioal roscdasic Errors of Uow form Absrac Ali Amad, Salahuddi
Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is
39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h
Lecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
Boyce/DiPrima 9 th ed, Ch 7.9: Nonhomogeneous Linear Systems
BoDiPrima 9 h d Ch 7.9: Nohomogou Liar Sm Elmar Diffrial Equaio ad Boudar Valu Prolm 9 h diio William E. Bo ad Rihard C. DiPrima 9 Joh Wil & So I. Th gral hor of a ohomogou m of quaio g g aralll ha of
10. Joint Moments and Joint Characteristic Functions
0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi
Right Angle Trigonometry
Righ gl Trigoomry I. si Fs d Dfiiios. Righ gl gl msurig 90. Srigh gl gl msurig 80. u gl gl msurig w 0 d 90 4. omplmry gls wo gls whos sum is 90 5. Supplmry gls wo gls whos sum is 80 6. Righ rigl rigl wih
Actuarial Society of India
Acuarial Sociey of Idia EXAMINAIONS Jue 5 C4 (3) Models oal Marks - 5 Idicaive Soluio Q. (i) a) Le U deoe he process described by 3 ad V deoe he process described by 4. he 5 e 5 PU [ ] PV [ ] ( e ).538!
PURE MATHEMATICS A-LEVEL PAPER 1
-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
![S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]](/thumbs/92/108666103.jpg "S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]")
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
10.3 Autocorrelation Function of Ergodic RP 10.4 Power Spectral Density of Ergodic RP 10.5 Normal RP (Gaussian RP)
ENGG450 Probabiliy ad Saisics for Egieers Iroducio 3 Probabiliy 4 Probabiliy disribuios 5 Probabiliy Desiies Orgaizaio ad descripio of daa 6 Samplig disribuios 7 Ifereces cocerig a mea 8 Comparig wo reames
Hadamard Exponential Hankel Matrix, Its Eigenvalues and Some Norms
Math Sci Ltt Vol No 8-87 (0) adamard Exotial al Matrix, Its Eigvalus ad Som Norms İ ad M bula Mathmatical Scics Lttrs Itratioal Joural @ 0 NSP Natural Scics Publishig Cor Dartmt of Mathmatics, aculty of
Improved estimation of population variance using information on auxiliary attribute in simple random sampling. Rajesh Singh and Sachin Malik
Imrovd imaio of oulaio variac uig iformaio o auxiliar ariu i iml radom amlig Rajh igh ad achi alik Darm of aiic, Baara Hidu Uivri Varaai-5, Idia (righa@gmail.com, achikurava999@gmail.com) Arac igh ad Kumar
On the approximation of the constant of Napier
Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of
Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2
MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm
Boyce/DiPrima/Meade 11 th ed, Ch 4.1: Higher Order Linear ODEs: General Theory
Bo/DiPima/Mad h d Ch.: High Od Lia ODEs: Gal Tho Elma Diffial Eqaios ad Boda Val Poblms h diio b William E. Bo Rihad C. DiPima ad Dog Mad 7 b Joh Wil & Sos I. A h od ODE has h gal fom d d P P P d d W assm
Lecture 15 First Properties of the Brownian Motion
Lecure 15: Firs Properies 1 of 8 Course: Theory of Probabiliy II Term: Sprig 2015 Isrucor: Gorda Zikovic Lecure 15 Firs Properies of he Browia Moio This lecure deals wih some of he more immediae properies
Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
N! AND THE GAMMA FUNCTION
N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio