Chapter 7. Bounds for weighted sums of Random Variables
|
|
|
- Ronald Martin
- 5 years ago
- Views:
Transcription
1 Chpter 7. Bouds for weghted sums of Rdom Vrbles 7. Itroducto Let d 2 be two depedet rdom vrbles hvg commo dstrbuto fucto. Htczeko (998 d Hu d L (2000 vestgted the Rylegh dstrbuto d obted some results bout the Schur-cocvty of dstrbuto fucto of weghted sum of two rdom vrbles. Hu d L ( b exteded the prevous results for the geerlzed gmm dstrbuto d the geerlzed Rylegh dstrbuto. Yu (20 compred weghted sums of..d. postve rdom vrbles ccordg to the usul stochstc order. The m equltes re derved by usg mjorzto techques uder cert log-cocvty ssumptos. My results bout lkelhood rto orderg d orderg propertes of covoluto of some dstrbuto fuctos hve ppered the lterture for exmple Fthmesh d Khled (2008 hve cosdered these for geerlzed Rylegh rdom vrbles d Zho d Blkrsh ( hve studed these propertes for heterogeeous expoetl d geometrc rdom vrbles d heterogeeous Erlg d Pscl rdom vrbles d the Chpter two Chpter three d Chpter fve of ths thess. 92
2 Bouds for weghted sum of Rdom Vrbles I ths chpter we study some stochstc equltes for the weghted sums of depedet rdom vrbles. Weghted sums of depedet ch squre vrbles rse turlly multvrte sttstcs s qudrtc forms orml vrbles. Stochstc comprsos betwee such weghted sums re therefore sttstclly terestg d c led to bouds o the dstrbuto fuctos. Suppose the compoet lfetmes of redudt stdby system (wthout reprg re modeled by scle fmly of dstrbutos. The the totl lfetme s of the form. Whe the s re..d. expoetl vrbles Bo d Plte (999 hve obted comprsos of the totl lfetme wth respect to severl stochstc orders For the usul stochstc order smlr results hold for brod clss of dstrbutos cludg the commoly used gmm Webull d log-orml dstrbutos. Whe s re..d. expoetl vrbles d the qutty E(log( ppers cert wreless commuctos problems (Jorsweck d Boche (2007. By the mootocty of log( x we hve st b E (log( E (log( b. I the ext secto we compre weghted sums of..d. postve rdom vrbles ccordg to the usul stochstc order. The m equltes re derved usg mjorzto techques. 7.2 Stochstc order of weghted sum We use the followg theorem of Yu (20. Theorem 7. (Yu (20. Let... be..d. rdom vrbles wth desty fucto f ( x 93
3 Bouds for weghted sum of Rdom Vrbles If f x (e s log-cocve (or log( hvg log-cocve desty x the for ech t 0 P( t s Schur-cocve fucto of log( (log(...log(. Equvletly f b the log( log( b b. (7. st Let p f such tht the fucto p x f x p m 02 log log ( hs log-cocve desty x or equvletly f p... p hs log-cocve desty x the for ech fucto of t 0 P( t s Schur-covex q q q (... where p q. Equvletly f b the q q b b st. (7.2 If... hs log-cocve desty tht s symmetrc bout zero o the P( t s Schur-cocve fucto of. Equvletly f b the 94
4 Bouds for weghted sum of Rdom Vrbles b b. (7.3 st Remrk 7.. By usg the Theorem 7. we c fd the bouds o the dstrbuto fucto of terms of the dstrbuto fucto of. Let ( the log( = log( d hece log( lo g( where (... (... d ow usg (7. we obt. (7.4 st Now let ( q q q the q q where (... d By usg (7.2 (... st. (7.5 From (7.4 d (7.5 we get 95
5 Bouds for weghted sum of Rdom Vrbles. (7.6 P( t P( t P( t The equlty (7. holds for ech of the followg dstrbutos; ( Uform o the tervl (0 s s 0 ; (2 gmm( 0 ; (3 Ay log-orml dstrbuto; (4 The Webull dstrbuto wth prmeter k 0 whose desty s k x f ( x kx e x 0; k (5 The geerlzed Rylegh dstrbuto wth prmeter 0 For the bove dstrbutos d for ( the equlty P ( x t P ( x t holds. The equlty ( 7.2 holds for the followg dstrbutos where p d p q ( Webull dstrbuto wth prmeter k ; (2 pq 2 d geerlzed Rylegh vrble ; I these cses equlty (7.6 holds. 96
6 Bouds for weghted sum of Rdom Vrbles Next we obt the boud for survvl fucto of for the bove dstrbuto frst we cosder weghted sum of geerlzed Rylegh rdom vrbles. 7.3 Bouds for weghted sum of geerlzed Rylegh rdom vrbles Below we obt bouds for the survvl fucto of weghted sum of Rylegh rdom vrbles. Theorem 7.2. Let... be oegtve depedet rdom vrbles from dstrbuto fucto GRE( umbers..... d let... be sequeces of rel The the followg bouds for the survvl fucto of S( hold: t t P( P( t P( t P( t P( t P( t P(. Proof. Let GRE( by prevous studes we hve below stochstc orderg for
7 Bouds for weghted sum of Rdom Vrbles Hu d L (2000 cosdered the problem of stochstc comprsos of covolutos of depedet geerlzed Rylegh rdom vrbles d showed tht for fxed st (... (.... (7.7 Fthmesh d Khled (2008 showed tht for fxed lr st (... (... (. (7.8 Usg Chpter fve we hve the reltos (... (... (... ( ( lr st (7.9 o the other hd; (... ( ( st... (.... (7.0 Let 2 d the survvl fucto of ; ( 2 the we hve followg bouds for ( st hr st b ( st hr c st. d st ( hr st (7. 98
8 Bouds for weghted sum of Rdom Vrbles Now by usg Remrk 7. we c obt the followg boud wth respect to ch mjorzto. Let GRE(... d let... be sequeces of rel umbers. By ( 7.6 the bouds for the survvl fucto of ( q q re x where ( d. (7.2 ( ( ( P t P t P t By usg (7. we get t t P( P( t P(. (7.3 Smlrly from (7.b d (7.c t P( t P( (7.4 d t P( t P(. ( Bouds for weghted sum of gmm rdom vrbles Let... be depedet rdom vrbles from gmm dstrbuto fuctos wth prmeters d.... From Chpter two for fxed we hve 99
9 Bouds for weghted sum of Rdom Vrbles lr hr s t ( 2 ( hr st d for fxed 3 lr hr s t ( 4 ( lr hr st. For dfferet s ( d s 5 ( lr hr st 6 ( lr hr st 7 ( hr st 8 ( hr st. Let ( the from (7.4 for the gmm dstrbuto. st By usg (7.4 d bove relto we c fd the useful boud for survvl fucto of weghted sum of gmm rdom vrble defferet cses for exmple 00
10 Bouds for weghted sum of Rdom Vrbles For fxed usg prt d 2 of the bove reltos the followg boud for the survvl fucto of c be obted by tkg : (7.6 st st d by tkg (. (7.7 st st For fxed usg prt 3 d 4 the followg boud for the survvl fucto of c be obted by lettg : (7.8 st st d by lettg (. (7.9 st st I the geerl cse for dfferet s ( d s s by prt fve to eght the followg boud for survvl fucto of c be obted by tkg d (7.20 st st by tkg usg ( d : 0
11 Bouds for weghted sum of Rdom Vrbles (7.2 st st by tkg d ( (7.22 st st d by tkg ( d (. (7.23 st st Exmple 7.. Suppose stdby system comprses of 8 compoet. Let Gmm (...8 be the lfe-tme of the th compoet. Let (... 8 re dexes of mportce of the compoet. The vector of the scles d shpe prmeters d the vector re gve the followg tble Tble 7.. The vector of the scles d shpe prmeters d wegh Ex we hve d ( ( Cse. from (
12 Bouds for weghted sum of Rdom Vrbles st. We obt the followg lower bouds for the survvl fucto of S( ths cse the followg tble. Tble 7.2 Lower boud for weghted sum of gmm r.v. ccordg to Cse. x PS ( ( x Cse 2. Smlrly to Cse from ( st d lower bouds for the survvl fucto of S( ths cse re: Tble 7.3. Lower boud for weghted sum of gmm r.v. ccordg to Cse 2. x PS ( ( x Cse 3. From ( st d lower bouds for the survvl fucto of S( ths cse re: Tble 7.4. Lower boud for weghted sum of gmm r.v. ccordg to Cse 3. x PS ( ( x
13 Bouds for weghted sum of Rdom Vrbles Cse 4. From ( st d lower bouds for the survvl fucto of S( ths cse re: Tble 7.5. Lower boud for weghted sum of gmm r.v. ccordg to Cse 4. x PS ( ( x Wth comprg the Tble 7.2 Tble 7.3 Tble 7.4 d Tble 7.5 t s cler tht boud for survvl fucto of S( Cse 4 s the shrpest. I ths cse bouds re term of by (the hrmoc me of the s d (the geometrc me of the s. If the shpe (scle prmeter of the gmm dstrbuto ws fxed we c use the other reltos to obt the bouds. I the sme mer we c fd smlr bouds for the survvl fucto of weghted sum of rdom vrbles wth other dstrbutos. 04
Introduction to mathematical Statistics
Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs
Available online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
SUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
A Brief Introduction to Olympiad Inequalities
Ev Che Aprl 0, 04 The gol of ths documet s to provde eser troducto to olympd equltes th the stdrd exposto Olympd Iequltes, by Thoms Mldorf I ws motvted to wrte t by feelg gulty for gettg free 7 s o problems
Analytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases
Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN 2349-7874 (Prt) & ISSN 2349-7882 (Ole) www.rcourls.org Alytcl Approch for the Soluto of Thermodymc Idettes
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
DISCRETE TIME MODELS OF FORWARD CONTRACTS INSURANCE
G Tstsshvl DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE (Vol) 008 September DSCRETE TME MODELS OF FORWARD CONTRACTS NSURANCE GSh Tstsshvl e-ml: gurm@mdvoru 69004 Vldvosto Rdo str 7 sttute for Appled
A Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares
Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com
PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS
PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 17, Number 4/2016, pp
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 7 Number 4/06 pp 87 9 UPER AND LOWER BOUNDS FOR A FINITE-TYPE RUIN PROBABILITY IN A NONHOMOGENEOUS RISK PROCESS
On a class of analytic functions defined by Ruscheweyh derivative
Lfe Scece Jourl ;9( http://wwwlfescecestecom O clss of lytc fuctos defed by Ruscheweyh dervtve S N Ml M Arf K I Noor 3 d M Rz Deprtmet of Mthemtcs GC Uversty Fslbd Pujb Pst Deprtmet of Mthemtcs Abdul Wl
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Fll 4 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
Sequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
Random variables and sampling theory
Revew Rdom vrbles d smplg theory [Note: Beg your study of ths chpter by redg the Overvew secto below. The red the correspodg chpter the textbook, vew the correspodg sldeshows o the webste, d do the strred
CHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
Chapter 3 Supplemental Text Material
S3-. The Defto of Fctor Effects Chpter 3 Supplemetl Text Mterl As oted Sectos 3- d 3-3, there re two wys to wrte the model for sglefctor expermet, the mes model d the effects model. We wll geerlly use
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
Chapter 4: Distributions
Chpter 4: Dstrbutos Prerequste: Chpter 4. The Algebr of Expecttos d Vrces I ths secto we wll mke use of the followg symbols: s rdom vrble b s rdom vrble c s costt vector md s costt mtrx, d F m s costt
DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
Z = = = = X np n. n n. npq. npq pq
Stt 4, secto 4 Goodess of Ft Ctegory Probbltes Specfed otes by Tm Plchowsk Recll bck to Lectures 6c, 84 (83 the 8 th edto d 94 whe we delt wth populto proportos Vocbulry from 6c: The pot estmte for populto
St John s College. UPPER V Mathematics: Paper 1 Learning Outcome 1 and 2. Examiner: GE Marks: 150 Moderator: BT / SLS INSTRUCTIONS AND INFORMATION
St Joh s College UPPER V Mthemtcs: Pper Lerg Outcome d ugust 00 Tme: 3 hours Emer: GE Mrks: 50 Modertor: BT / SLS INSTRUCTIONS ND INFORMTION Red the followg structos crefull. Ths questo pper cossts of
ICS141: Discrete Mathematics for Computer Science I
Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll
POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS
IRRS 9 y 04 wwwrppresscom/volumes/vol9issue/irrs_9 05pdf OWERS OF COLE ERSERIC I-RIIGOL RICES WIH COS I-IGOLS Wg usu * Q e Wg Hbo & ue College of Scece versty of Shgh for Scece d echology Shgh Ch 00093
Regression. By Jugal Kalita Based on Chapter 17 of Chapra and Canale, Numerical Methods for Engineers
Regresso By Jugl Klt Bsed o Chpter 7 of Chpr d Cle, Numercl Methods for Egeers Regresso Descrbes techques to ft curves (curve fttg) to dscrete dt to obt termedte estmtes. There re two geerl pproches two
MTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
Patterns of Continued Fractions with a Positive Integer as a Gap
IOSR Jourl of Mthemtcs (IOSR-JM) e-issn: 78-578, -ISSN: 39-765X Volume, Issue 3 Ver III (My - Ju 6), PP -5 wwwosrjourlsorg Ptters of Cotued Frctos wth Postve Iteger s G A Gm, S Krth (Mthemtcs, Govermet
this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
![this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]](/thumbs/78/76749440.jpg "this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]")
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
Section 7.2 Two-way ANOVA with random effect(s)
Secto 7. Two-wy ANOVA wth rdom effect(s) 1 1. Model wth Two Rdom ffects The fctors hgher-wy ANOVAs c g e cosdered fxed or rdom depedg o the cotext of the study. or ech fctor: Are the levels of tht fctor
Xidian University Liu Congfeng Page 1 of 22
Rdom Sgl rocessg Chpter Expermets d robblty Chpter Expermets d robblty Cotets Expermets d robblty.... Defto of Expermet..... The Smple Spce..... The Borel Feld...3..3 The robblty Mesure...3. Combed Expermets...5..
An Extended Mixture Inverse Gaussian Distribution
Avlble ole t htt://wwwssstjscssructh Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty A Eteded Mture Iverse Guss Dstrbuto Chookt Pudrommrt * Fculty o Scece d Techology,
Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:
Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1
Itō Calculus (An Abridged Overview)
Itō Clculus (A Abrdged Overvew) Arturo Ferdez Uversty of Clfor, Berkeley Sttstcs 157: Topcs I Stochstc Processes Semr Aprl 14, 211 1 Itroducto I my prevous set of otes, I troduced the cocept of Stochstc
Probabilistic approach to the distribution of primes and to the proof of Legendre and Elliott-Halberstam conjectures VICTOR VOLFSON
Probblstc pproch to the dstrbuto of prmes d to the proof of Legedre d Ellott-Hlberstm cojectures VICTOR VOLFSON ABSTRACT. Probblstc models for the dstrbuto of prmes the turl umbers re costructed the rtcle.
In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
![Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]](/thumbs/83/87614710.jpg "Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]")
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
On Several Inequalities Deduced Using a Power Series Approach
It J Cotemp Mth Sceces, Vol 8, 203, o 8, 855-864 HIKARI Ltd, wwwm-hrcom http://dxdoorg/02988/jcms2033896 O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty
14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y
4. Le Itegrls I ths secto we defe tegrl tht s smlr to sgle tegrl except tht sted of tegrtg over tervl [ ] we tegrte over curve. Such tegrls re clled le tegrls lthough curve tegrls would e etter termology.
Chapter Unary Matrix Operations
Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt
Metric Spaces: Basic Properties and Examples
1 Metrc Spces: Bsc Propertes d Exmples 1.1 NTODUCTON Metrc spce s dspesble termedte course of evoluto of the geerl topologcl spces. Metrc spces re geerlstos of Euclde spce wth ts vector spce structure
The definite Riemann integral
Roberto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4 The defte Rem tegrl Wht you eed to kow lredy: How to ppromte the re uder curve by usg Rem sums. Wht you c ler here: How to use
D KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
Functions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek
Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt
Density estimation II
CS 750 Mche Lerg Lecture 6 esty estmto II Mlos Husrecht mlos@tt.edu 539 Seott Squre t: esty estmto {.. } vector of ttrute vlues Ojectve: estmte the model of the uderlyg rolty dstruto over vrles X X usg
Mathematics HL and further mathematics HL formula booklet
Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Publshed Jue 0 Itertol Bcclurete Orgzto 0 5048 Mthemtcs HL d further mthemtcs formul boolet
Chapter Gauss-Seidel Method
Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos
under the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
Applied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
Bond Additive Modeling 5. Mathematical Properties of the Variable Sum Exdeg Index
CROATICA CHEMICA ACTA CCACAA ISSN 00-6 e-issn -7X Crot. Chem. Act 8 () (0) 9 0. CCA-5 Orgl Scetfc Artcle Bod Addtve Modelg 5. Mthemtcl Propertes of the Vrble Sum Edeg Ide Dmr Vukčevć Fculty of Nturl Sceces
Bounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
Special Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
Inference on Parameters of a Shifted and Rescaled Wiener Process Based on Nonidentically Distributed Observations
WDS'09 Proceedgs of Cotruted Ppers, Prt I, 95 0, 009. ISBN 978-80-7378-0-9 MATFYZPRESS Iferece o Prmeters of Shfted d Rescled Weer Process Bsed o Nodetclly Dstruted Oservtos A. Kvtkovčová Chrles Uversty,
Differential Entropy 吳家麟教授
Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set
Differential Method of Thin Layer for Retaining Wall Active Earth Pressure and Its Distribution under Seismic Condition Li-Min XU, Yong SUN
Itertol Coferece o Mechcs d Cvl Egeerg (ICMCE 014) Dfferetl Method of Th Lyer for Retg Wll Actve Erth Pressure d Its Dstrbuto uder Sesmc Codto L-M XU, Yog SUN Key Lbortory of Krst Evromet d Geologcl Hzrd
MATH2999 Directed Studies in Mathematics Matrix Theory and Its Applications
MATH999 Drected Studes Mthemtcs Mtr Theory d Its Applctos Reserch Topc Sttory Probblty Vector of Hgher-order Mrkov Ch By Zhg Sho Supervsors: Prof. L Ch-Kwog d Dr. Ch Jor-Tg Cotets Abstrct. Itroducto: Bckgroud.
12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the
(EXPONENTIAL) TELETRAFFIC MODELS
(EXPONENTIAL) TELETRAFFIC MODELS Zdzsłw PAPIR Dept. of Telecommuctos TELETRAFFIC STREAMS I computer scece, prtculr etworkg, sesso s sempermet terctve formto terchge, lso kow s dlogue, coversto or meetg,
Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
ρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
The z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
Parameter, Statistic and Random Samples
Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose
Abstrct Pell equto s mportt reserch object elemetry umber theory of defte equto ts form s smple, but t s rch ture My umber theory problems ce trsforme
The solvblty of egtve Pell equto Jq Wg, Lde C Metor: Xgxue J Bejg Ntol Dy School No66 Yuqu Rod Hd Dst Bejg Ch PC December 8, 013 1 / 4 Pge - 38 Abstrct Pell equto s mportt reserch object elemetry umber
Qualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
EVALUATING COMPARISON BETWEEN CONSISTENCY IMPROVING METHOD AND RESURVEY IN AHP
ISAHP 00, Bere, Stzerld, August -4, 00 EVALUATING COMPARISON BETWEEN CONSISTENCY IMPROVING METHOD AND RESURVEY IN AHP J Rhro, S Hlm d Set Wto Petr Chrst Uversty, Surby, Idoes @peter.petr.c.d sh@peter.petr.c.d
Mathematics HL and further mathematics HL formula booklet
Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Edted 05 (verso ) Itertol Bcclurete Orgzto 0 5048 Cotets Pror lerg Core 3 Topc : Algebr
Systems of second order ordinary differential equations
Ffth order dgolly mplct Ruge-Kutt Nystrom geerl method solvg secod Order IVPs Fudzh Isml Astrct A dgolly mplct Ruge-Kutt-Nystróm Geerl (SDIRKNG) method of ffth order wth explct frst stge for the tegrto
Random Variate Generation ENM 307 SIMULATION. Anadolu Üniversitesi, Endüstri Mühendisliği Bölümü. Yrd. Doç. Dr. Gürkan ÖZTÜRK.
adom Varate Geerato ENM 307 SIMULATION Aadolu Üverstes, Edüstr Mühedslğ Bölümü Yrd. Doç. Dr. Gürka ÖZTÜK 0 adom Varate Geerato adom varate geerato s about procedures for samplg from a varety of wdely-used
18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
Chapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
β (cf Khan, 2006). In this model, p independent
Proc. ICCS-3, Bogor, Idoes December 8-4 Vol. Testg the Equlty of the Two Itercets for the Prllel Regresso Model Bud Prtko d Shhjh Kh Dertmet of Mthemtcs d Nturl Scece Jederl Soedrm Uversty, Purwokerto,
TiCC TR November, Gauss Sums, Partitions and Constant-Value Codes. A.J. van Zanten. TiCC, Tilburg University Tilburg, The Netherlands
Tlburg ceter for Cogto d Coucto P.O. Box 953 Tlburg Uversty 5 LE Tlburg, The Netherlds htt://www.tlburguversty.edu/reserch/sttutes-d-reserch-grous/tcc/cc/techcl-reorts/ El: tcc@uvt.l Coyrght A.J. v Zte,
Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests
JASA js v.003/0/0 Pr:8/0/004; 3:59 F:jstm09r3.tex; R p. Heteroscedstc Oe-Wy ANOVA d Lck-of-Ft Tests 60 6 M. G. AKRITAS d N. PAPADATOS 3 6 5 Recet rtcles hve cosdered the symptotc behvor of the oe-wy lyss
Mathematics HL and further mathematics HL formula booklet
Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Publshed Jue 0 Itertol Bcclurete Orgzto 0 5048 Cotets Pror lerg Core Topc : Algebr Topc
Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants
Rochester Isttute of echology RI Scholr Wors Artcles 8-00 bocc d ucs Nubers s rdgol trx Deterts Nth D. Chll Est Kod Copy Drre Nry Rochester Isttute of echology ollow ths d ddtol wors t: http://scholrwors.rt.edu/rtcle
Roberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums
Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego
INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
Lecture 4 Sep 9, 2015
CS 388R: Radomzed Algorthms Fall 205 Prof. Erc Prce Lecture 4 Sep 9, 205 Scrbe: Xagru Huag & Chad Voegele Overvew I prevous lectures, we troduced some basc probablty, the Cheroff boud, the coupo collector
means the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
Predicting Survival Outcomes Based on Compound Covariate Method under Cox Proportional Hazard Models with Microarrays
Predctg Survvl Outcomes Bsed o Compoud Covrte Method uder Cox Proportol Hzrd Models wth Mcrorrys PLoS ONE 7(10). do:10.1371/ourl.poe.0047627. http://dx.plos.org/10.1371/ourl.poe.0047627 Tkesh Emur Grdute
Linear Algebra Concepts
Ler Algebr Cocepts Ke Kreutz-Delgdo (Nuo Vscocelos) ECE 75A Wter 22 UCSD Vector spces Defto: vector spce s set H where ddto d sclr multplcto re defed d stsf: ) +( + ) (+ )+ 5) l H 2) + + H 6) 3) H, + 7)
Section l h l Stem=Tens. 8l Leaf=Ones. 8h l 03. 9h 58
Secto.. 6l 34 6h 667899 7l 44 7h Stem=Tes 8l 344 Leaf=Oes 8h 5557899 9l 3 9h 58 Ths dsplay brgs out the gap the data: There are o scores the hgh 7's. 6. a. beams cylders 9 5 8 88533 6 6 98877643 7 488
Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information
Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst
X ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
1 Solution to Problem 6.40
1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we
Acoustooptic Cell Array (AOCA) System for DWDM Application in Optical Communication
596 Acoustooptc Cell Arry (AOCA) System for DWDM Applcto Optcl Commucto ml S. Rwt*, Mocef. Tyh, Sumth R. Ktkur d Vdy Nll Deprtmet of Electrcl Egeerg Uversty of Nevd, Reo, NV 89557, U.S.A. Tel: -775-78-57;
A METHOD FOR THE RAPID NUMERICAL CALCULATION OF PARTIAL SUMS OF GENERALIZED HARMONICAL SERIES WITH PRESCRIBED ACCURACY
UPB c Bull, eres D, Vol 8, No, 00 A METHOD FOR THE RAPD NUMERAL ALULATON OF PARTAL UM OF GENERALZED HARMONAL ERE WTH PRERBED AURAY BERBENTE e roue o etodă ouă etru clculul rd l suelor rţle le serlor roce
Homework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015
Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bg-o ) otato. I ths problem, you wll prove some basc facts
Introduction to Probability
Itroducto to Probablty Nader H Bshouty Departmet of Computer Scece Techo 32000 Israel e-mal: bshouty@cstechoacl 1 Combatorcs 11 Smple Rules I Combatorcs The rule of sum says that the umber of ways to choose
Preliminary Examinations: Upper V Mathematics Paper 1
relmr Emtos: Upper V Mthemtcs per Jul 03 Emer: G Evs Tme: 3 hrs Modertor: D Grgortos Mrks: 50 INSTRUCTIONS ND INFORMTION Ths questo pper sts of 0 pges, cludg swer Sheet pge 8 d Iformto Sheet pges 9 d 0
CS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorthms I Lecture 3 Solvg Recurreces Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet Solvg Recurreces The lyss of merge sort Lecture requred us to solve recurrece. Recurreces re lke solvg
Chapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS RELATED TO GENERALIZED POWER AND ALTERNATING SUMS
Bullet of Mthemtcl Alyss d Applctos ISSN: 181-191, URL: http://www.mth.org Volume 4 Issue 4 01, Pges 76-90. SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS
Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector
evel-2 BS trx-vector opertos wth 2 opertos sequetlly BS-Notto: S --- sgle precso G E geerl mtrx V --- vector defes SGEV, mtrx-vector product: r y r α x β r y ther evel-2 BS: Solvg trgulr system x wth trgulr
Strategies for the AP Calculus Exam
Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst
hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations
HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several
EECE 301 Signals & Systems
EECE 01 Sgals & Systems Prof. Mark Fowler Note Set #9 Computg D-T Covoluto Readg Assgmet: Secto. of Kame ad Heck 1/ Course Flow Dagram The arrows here show coceptual flow betwee deas. Note the parallel
6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder