t. Neymann-Fisher factorization: where g and h are non-negative function. = is a sufficient statistic. : t( y)
|
|
- Αντιγόνη Σωφρονία Κορομηλάς
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Statstcs et Y ~ p( ; ( ; ( ; ( ( ; ( p Y t p g t h Y suffcet statstc (for the parametrc faml p( ; s depedet of t Nema-Fsher factorzato: where g ad h are o-egatve fucto p( ; Smple bar hpotheses: {, } : t( p( ; p( ; p( ; For {,, }, t (,, p( ; p( ; Suffcet a (measurable fucto h ( such that t( ht ( s a suffcet statstc s a suffcet statstc t s a mmal suffcet statstc f, for a other suffcet t, there s Suppose there exsts a fucto p( x; the rato p( ; he t s complete f A statstc ( Y such that for two sample pots x ad, s a costat as a fucto of f ad ol f ( x ( Y s a mmal suffcet statstc for Prg t Y Λ ( Eg t Y g t p ; d ( Y complete ad suffcet mmal suffcet ad uque Ex X ~ (, t( x max{ x} d U, K-parameter expoetal faml: A faml of dstrbutos s sad to be a K- parameter expoetal faml f ( ; exp K c t + d + s IA K p c( t( e f ( h( IA ( If c {( c,, c K, Λ} has a teror pot, the t ( ( t (,, tk ( s complete ad suffcet mmal suffcet ad uque Setup Parameter space { } Pr where IA ( s the dcator fucto ot related to Λ Λ P( Θ Λ Λ Pr [ Θ ] Θ Θ Λ pothess: : ~ ( ; Observato space Γ d Y p Λ,,,, -
2 A determstc detector : {,, -} Parttos the observato space Γ to K dsjot subsets Γ wth Θ Whe Θ { }, : Λ { } ( : pdf/pmf o {,,, } ( ( Γ ad detf A radomzed detector ( D Y Pr ( he detecto D d s a realzato accordg to D ( Cost: C(, Cost C(, j Cost j Uform cost: [ ], Assume Cj > C for all, j he Baesa Detector Baesa / Baes Rs: R E p R R p R d ( [ Cost] ( ( ( Codtoal rs: Cost R E Y (, Pr Θ (, ( ; R C D C p d Γ C, C j, Λ j C, Pr Γ Θ C, p ; d for determstc detector R R d Λ Baesa Detector B argm R o the detector, the Baesa detector for a gve : B ( argmr( argme Cost Y Smple potheses: { } he Baesa detector s determstc: Because the dstrbuto of does t depeds,,, Pr D Y, o/ w d argme Cost D Y, argm Cj jp ; j where E Cost D Y, C p Θ Y C p ; j j j j j j j p j (, Pr Θ (, ( ; R C D C p d Γ,
3 R( j Cj p( ; j d Wth uform cost, t s the probablt of j Γ error, Baesa detector for pror p(, τ ( C C d ( p( where τ ( C C, otherwse ( C p ( C p ( C p ( C p ( + + c Γ : p τp Γ Bar Smple potheses: Pr[ ] B, { ( } ( ( ; ( ; ( ; ( ; R C p d C p d C p d C p d Γ Γ Γ Γ Ex : Y ~ N (, σ, : Y ~ (, σ N < + σ C C he, Γ : γ + l For uform cost, ( CC γ γ R( Q + Q σ σ a b a τ b + a + b Uform cost wth dett cost matrx Composte Bar pothess estg Pr[ Θ ] p( Θ Λ p( Pr Θ Θ Λ d p( d Λ p( Pr[ ] d Θ Λ Pr[ Θ ] Λ Pr Θ Θ Λ Λ, ( τ ( C C B τ where, ( < τ ( C C P( Θ Λ Pr [ Θ ] d Λ Λ
4 ( p p ( Θ Λ ( Θ Λ Λ Λ p d Pr[ Θ ] ( C C τ ( Pr[ Θ ] p d R Pr D Θ Λ + Pr D Θ Λ C C p dp Λ d + p dp Λ d Λ Γ Λ Γ p p dd+ p p dd Γ Λ Γ Λ p p dd+ p p dd Λ Γ Λ Γ max Detecto max detector/rule/crtero: If we ow Θ ~ Λ mmax R Λ R R d Smple bar hpothess testg: the mmax detector s mmax R Rs for gve pror : r(, R ( ( R ( r(, r(, B, max detector s mmax { R, R} mum Baesa rs V( r(, B, ( ( V r, B, + (ear wrt R ( B, V ( r (, B, R ( ( ( V R R ( B, Cocave ad cotuous [,]
5 argmaxv ( least favorable pror If V ( exsts at ( ( B, B,, the r (, B, V R R V at s a taget le of ( Cost (, ( ; + (, ( ( ; B, R E C p d C p d ( Cost (, ( ; + (, ( ( ; R E C p d C p d the Baesa detector desged at ( B, argm r, mum Baesa rs gve pror: r r ν,,, Solvg for max detector for bar smple hpotheses: Equalzer rule: If there exsts a pror s the least favorable pror V( V ( B, If ot the, If or, the B, B, such that R ( R ( B, B,, the, ad the mmax detector Otherwse, (f V ( s lear the small eghborhood of ( wth probablt q ad B, B, R +, ( ( R ( B, + B, ( ( + ( +,, ( R + R +,, ( R( B, R(, B B B, cosder wth q from wth probablt q + + ( V ( R R B B V q V ( (, B, V r (, (, B, r r ( (, B, V r r (, B, r (, r (, B, +
6 he mmax rs V( R ( B, R ( B, A detector there exsts a pror Radomzed Baesa Detector: ( ( ( for all cases s mmax f t satsfes the equalzer rule R ( R such that V( ( wth probablt q wth probablt q R qr + q R R r (, R( R( R R( R( ad he equalzer rule remas vald for composte hpotheses If the m Baesa rs s dfferetable at the least favorable pror, the the Baesa detector s the mmax detector Nema-Pearso Detector m ( subject to R K R α, < K Bar potheses: : Y p( ; alteratve, Λ,, : ull hpothess : the Smple bar hpotheses: : Y p( ; ; : Y p( ; rego; Γ : rejecto rego Γ : acceptace False alarm (I P ( ; Pr[ D ; ] ( p( ; F d E ( ( he sze / level of a detector α( sup P F ( ; Λ Λ ss detecto (II P ( ; Pr[ D ; ] ( ( p( ; d E ( ( Λ he power of a detector P ( ; Pr[ D ; ] ( p( ; d E ( ( Λ P ( ; P ( ; D D F D has the same formula but usg from dfferet sets Uforml most powerful (UP: A sze α detector UP s UP f of sze α, P ( ; P ( ; D UP D Λ For smple bar hpotheses, (If Λ P ( ; ad P ( ; Λ ( α( α( P ( ; P ( ; UP ( ; UP D UP D ( arg max P ; E Pr D ; α UP UP s a sze α NP detector for : Y p ( ; ; : Y p ; for a Λ? For smple bar hpotheses, NP detector s arg max P arg m P arg m R wth uform cost PF ( α D P PF α R α α D, the
7 Nema-Pearso emma for smple bar hpotheses: Optmalt A p( > ηp(, ( ; ( ; p( ηp(, ; ; γ p ηp, ; < ; γ, s the best of ts sze [ ] ( ( η ( ( D D F F P P > P P for all Exstece α [,] η, there exsts a detector of the form above m η γ ( Pr ( > η; α for some η ad γ, Pr η ; arbtrar, otherwse Pr ( > η ; s a complmetar dstrbuto fucto, rght cotuous, ad mootocall decreasg γ ( α Pr > η ; Pr η ; ( α P Pr ( η ; γ Pr ( η ; F > + 3 Uqueess If s a sze α NP detector, the ( has the form above except perhaps for a set of wth zero probablt uder both ad P Pr D ; Pr η ; γ Pr η ; D > + Sce Note: [ ] > η Pr ; s also mootocall decreasg, we wat low η to get hgh P elpful to plot Pr D ( > η ; stead of ( whe the trasformato s :, creasg vs η 3 Ca wor wth t ( UP detector et be a real parameter he real-parameter faml ( ; lelhood rato ( ( f <, p( ; ad ( ; p( ; ( ;, s a odecreasg fucto of some real valued p( ; Ex Ce µ σ ( u ( Q ( c h e d Beroull, ( Oe-sded potheses estg: Y p( > : ; p has mootoe p are dstct ad whe Q ( s mootoe 3 Y p( : ; > (or he Kal Rub heorem: et be a real parameter ad let p( ; have mootoe lelhood rato ( For testg the oe-sded hpotheses, there
8 exsts a sze α UP detector of the form ( are determed b the sze costrat τ ( ;, > τ γ, τ, < τ m τ, Pr > τ ; α E p d α τ ad γ are fuctos of Gve a <, ad P ; P ; where τ ad γ wth NP form he F F E ( Pr D s a odecreasg ( fucto of So, for Λ (, s α( sup PF( ; PF( ;, the sze (false alarm of wo-sded pothess: ( ; Λ a ( b wth odecreasg p h e UP detector for : or, : (, of the form, c < < c ( γ, c where c < c ad γ are determed b, otherwse E ( ( α E If s surrouded b, the suspect o UP detector Baesa Estmao Estmate radom ~ p( a Θ from Y ~ p( he cost s EC( ΘΘ mmze EC( ( Y Θ Baesa ( argm R ( R( ( Θ ( C p( d estmator E C Y SE: ( argm Baesa SE p d EΘ Y SE, µ +Λ Y Λ YY µ N Θ Θ Y I addto, f µ µ are zero, the ( Λ Λ, lear For For jotl Gaussa Θ ad Y,, Θ Y example, let Y a N SE, N, Θ Y YY d Θ+,,, Θ ~ N (,σ Θ N ~ N (, σn σ Θ SE, N, ( a where γ If a, the σ N aa+ γ γ SE, N, ( + γ ear observato model: et Y S + W, S W, S ~ N ( µ, Σ S S, W ~ N (, ΣW, the the SE estmator s he
9 E SY µ +Σ S S ( Σ +Σ S W ( Y µ S Error covarace matrx: CovSY Σ Σ ( Σ +Σ Σ S S S W S C C x C x, ad covex, Gve, assume s smmetrcal, e Θ, the p( +, the ( ( let E m ( ( m ( m p p s smmetrcal wth respect to m, e Baesa SE m ( argm p( d argm p AE p d p d Θ Y For C u, ( argmax + +, >, (, ( p d et d + + u, argm p( d, the AP ( argmax p( SE For E X X ubased, Cov( X X E ( X X( X X SE( X X X X trace( ( X E Cov X X E ear SE: EΘ? Gve zero mea radom varables Y,,,, Θ f Y fy f EΘY ( YY E mmzes SE ΘΘ E Θ E ΘY E YY E YΘ E to Θ s the orthogoal projecto of Θ oto { Y Y } E ΘΘ For Y C, X C, m F XY E E YY spa,, X s the lear SE estmate of X X s also the optmal lear estmate usg the weghted cost fucto m E ( X F Y Λ( X F Y for a Λ F
10 ear observato model: Y S + W S, E, W S S Σ SS Cov, W W Σ, the the lear SE estmate of S s gve b WW S ( Y ΣSS Σ +Σ SS WW Affe SE estmator: Gve radom vector Y If E Y ad E X are ow, the the SE affe estmator of X b Y s gve b X µ +Σ Σ X XY Y ( Y µ Y ( Cov X X Σ Σ Σ Σ E X X X XY Y YX (, (, Cov X X Cov X X Σ Σ Σ XY Y YX If s osgular ad u, the the SE affe estmator of x usg s the same as that usg u If W X ad are ucorrelated, ad V BX + W, the the SE affe Y V Y BX Bµ + BΣ Σ Y µ, estmator of V usg Y s gve b X XY Y ( Y Cov V V Pot Estmato ( ( BCov X X B affe,se +Σ g E gy g for a fxed E gy g gy gy + gy g E E E ĝ of g( s ubased f E g ( Y g( he, ( g gy gy E E ; For ubased, ( trace{ Cov( ( Y } Cov Y Cov Y Cov Y + a alwas Crtero: mze SE (rs ( A estmator Note that ( ( UVU A estmator ĝ( of g( s UVU (uforml mmum varace ubased f ubased For all ubased ĝ, ( g ( g Rao-Blacwell heorem: Suppose that estmator for g( wth Eg ( W Y s suffcet for ad that < for all ĝ s a
11 et g ( Eg( ( Y ( ( ( YY he g g p d, g ( Y g E E gy g( 3 If compoets of ĝ have fte varaces, the the strct equalt holds uless Prg ( Y g( Y E gy g gy g ( Y + g ( Y g( E E Furthermore, f ĝ( s ubased, the 4 ĝ ( s ubased for g( 5 ( ca be wrtte as Varg ( Y Varg( Y ĝ h g g If, the ehma-scheffé heorem: If ( Y s complete suffcet, ad ubased estmator of g( he g ( ( g( Y ( Y ( estmator Shortcut: Kowg ĝy s a E s a UVU Y s complete suffcet, tr fdg ( Y E For oe-parameter expoetal faml, (Y s complete ad suffcet, f t s ubased, the t s UVU CRB: Cramér-Rao ower Boud he score fucto s( l p( ; ; l p( ; E sy; l p( ; K I Es Y; s Y; Cov s Y; Fsher Iformato atrx: Ij E l py ( ; l py ( ; E l p( Y; j j ( l ( ; ( l ( ; l ( ; I E py py py E For scalar, I E l p( ; l p( ; E l p ( ; he Cramér-Rao Boud: et be a scalar ubased estmator of he, CRB: ( Var( Var( ( Y E ( Y I wth equalt ff ( ; l ( ; ( s p I
12 Iformato lower boud: For based estmator, E ( Y Φ(, the ( Φ ( Var ( Y wth equalt ff s( x; I I ( ( Φ If ( Y acheves formato lower boud, the t has mmum varace amog all s effcet ad UVU Y s estmators ( Y satsfg E ( Y E ( Y Furthermore, f ubased, the ( Y Oe parameter expoetal faml: et Λ be a ope terval, ad p( ; g ( C( e h( ( Y f a ol f ( Y ( Y Wth regulart, the formato lower boud s acheved b Also, (Y s complete ad suffcet 3 If (Y s ubased, the t s UVU ad effcet A ubased estmator s effcet f t acheves CRB A effcet estmator s UVU but a UVU estmator ma ot be effcet (whe CRB s ot achevable Y acheves CRB, the t s the soluto to the lelhood equato If l p( ; effcet estmator dstrbuto of the observato must belog to the expoetal faml he effcet estmator ca be foud b the estmator CRB ubased estmator of, the E ff l p( ; I ( et E wth equalt ( Y ( ( Y I ( g ĝ be a ubased estmator of ( gy g( gy g( ( (, the ( dg I ( ( dg(, wth equalt ff g g dg I l p ; µ Y ~ N µ, Σ µ Σ Σ he, I( Σ + tr Σ Σ j j µ µ ( µ Σ Σ j where,,, et Y ~ N( µ, Σ, the I( ( d µ Σ d µ l p ; dµ Σ µ et Also,
13 ear model: X + W, W ~ N (, Σ he X ~ N (, Σ, ad I Σ l p ; Σ Σ Σ Σ Σ s UVU, effcet, Gaussa,, east-square Need full colum ra for detfablt ~, ( Σ et Y ~ CN ( µ, Σ, real he I ( j Σ Σ tr Σ Σ j State Estmato N µ µ Re Σ + State Estmato: states: S AS + + U observato: Y S + W Kow dstrbuto of S, put sequece { U }, observato ose { W } E S s, VAR S Σ Fd the SE estmator of S gve Y, Y,, e, s E S,, Dscrete-me Kalma Buc X FX GU Y X + V Q Cov( U, R Cov( V +, + t t t t X E [ X ], t Σ Σ Cov( X Σ Cov tt ( Xt Y Kalma ga matrx K t Σ t tt tσ tt t + R t t X E X ( tt t Y X + K tt t Yt X t tt Σ Σ K tt tt t tσ t X E X tt t t t+ Y FX + t Σ F tt tσ F t t tt t + GQG + t t t Kalma Flterg t Notato: {,, } t t the past samples Σ ( ( tt t tt t tt s the SE predcto of S from t t E S tt t t E S s S s t s E S tt t (the SE flter t Σ ( ( tt E St stt St stt u, { } Λ U Euu Λ W Eww [ ] s ~ N( s, Λ depedet of { u }, { w } Italzato: ŝ ES, Σ VARS Σ, s SS Gaussa odel: { } w are zero mea, depedet, Gaussa [ ] easuremet Update: flterg: K Σ Σ +Σ W s s + K ( Σ Σ K Σ
14 me Update: predcto: s As + Σ A Σ A + +Σ U s Same formula for lear SE of o Gaussa Example Normal d: µ ; σ ( µ based Expoetal d: p( ; e I( > e I > ( d X ~ P ( λ ( λ s UVU ( based x λ x λ e λ ; λ ; λ p x e x h x ( xj! ( xj! j j p x; I max x < I m x > d X ~ U (, ( { } g ( max { x }, + max complete ad suffcet ( { } Bar d: p( x; UVU p ; g ( Bomal: Estmator ( { } t( x max{ x } s UVU he estmator of parameter from Y ~ ( ; argmax p( ; argmaxl p( ; Θ Θ s complete ad suffcet ; s complete No ubased estmator for p he best lear ubased estmator (BUE s BUE ABUE E AY subject to [ AY ] Θ s ( ABUE where argm E + wth zero mea ose, For lear model X W BUE s
15 For the K-parameter expoetal faml, let C be the teror of the rage of {( c,, c, } K Λ If E t( Y t(,,, K have a soluto ( for whch c (,, c ( K C, the s the uque estmator of oto Ivarace: et g ( : Θ Φ, g ( φ : Φ { A: A } mage Defe ( ; φ sup p( ; g ( φ Θ be the verse l If s the estmate of, the φ argsup l ; φ ( φ Φ ( g ( py ( ; p( ; D E l ( ; p l d p( Y; p( ; p( ; p( ; ae If s detfable, the s the global mmum of D wth equalt ff D D, ad m ( max l py ( ; Θ E For d Y, argmax l p( ; argmax l py ( ; o solve for : Θ N N E s ; l p ; ( + ( Newto-Raphso: J ; s ; J ; l p ; Scorg ethod: et the complete data ( + s( ; + I S Z Y ~ p ( z ; Ol Y ~ ( ; ( ( ; E ( l ( ;, + Q, > Q, l p ; l p ; p s observed E: Q pz Y argmax Q ; If dstrbuto of S does ot deped o, ( Q( ; E ( l pys ( ; Y + costat lme Y Asmptotcall ubased Cosstec: (d dstrbuto lm p ( p (, (p wea lmpr( Y >, (wp strog ( Y Pr lm, (ms mea square lme ( Y (wp (p (d (ms p p ad bouded Θ (ms ~ N, Σ Asmptotcall Normal:
16 Var ( Y Asmptotcall effcet (BAN: best asmptotcall ormal lm CRB ( ~ (, N I ; I lm I Y d Uder regulart codtos, (wea ( s asmptotcall Gaussa ad asmptotcall effcet Sequetal Detecto Fxed sample sze (FSS detector Example: Cosder the -sample smple bar d hpotheses vs : Y ~ ( µ, σ µ µ µ µ l Y ( Y σ σ ( µ µ ( µ µ N,, σ σ l Y ( ~ ( µ µ ( µ µ N,, σ σ N,,,,,, µ > µ he he mmum such that the σ µ µ optmal detector has sze < α, ad power > β s Q ( α Q ( β d ~ ; Smple bar hpotheses vs Y p(,,, A sequetal detector ( φ, s defed b stoppg rule sequece [ ] φ ( R { } φ : stop data collecto & mae decso :, termal decso cotue data collecto : ( Pr D Y Stop N φ m : φ Y rule sequece [ ] { } tme: ( he sequetal probablt rato test: p( ; SPR(A,B: (, p( ; ( B, stop, φ( or ( A,, ( ( AB,, ( B (, ( A ( B B A A Choose Choose
17 β SPR s optmal the Baesa problem SPR satsfes A ad B β α α he Wald-Wolfowtz heorem: he SPR(A,B detector ( φ, has the mmum stop tme amog all detectors (cludg FSS detector wth sze o larger ad power PF ( PF ( o less tha those of ( φ, E N N(,, φ φ PD( PD( E py ( ; p ( ; Z l z l l ( z py; p ; Wald s Approxmatos : Gve α ad β, the optmal SPR(A,B ca be β β approxmated b SPR ( A, B wth A ad B α α β β? β β A A < B B α < β α α α α AB, A, B the approxmato requres more samples P ( + P ( α + ( β α + ( β P ( + P ( F F he Wald s Equato: et Z be depedet ad d wth EZ et N be a E Z + + Z E Z E N stoppg tme, the, [ ] [ ] [ ] E [ N ] [ Z] N β β αlb+ α l A αl + ( α l α α E E [ Z] β βlb + ( β l A βl + ( β l E [ N ] α α E [ Z] E [ Z] E p( ; [ Z] ( ; p l d p( ; [ ] p( ; E Z ; p l d p( ; Revew b( x ( + ( + ( +,, f ( x g( x, d, f ( x g b ( x g( xb, ( x a ( x g( xax, + ( xd,, x! N, N + a λx xe dx + λx e λ 6 ax e dx a λx β bx a x a x x e ax b 4ac ( ax + bx+ c 4a e dx e α ( Γ + dx + a, for
18 ( f x ( df ( x f f x df ( (,, ( d x x x x d( Ax + b A ( d a x a d ( f ( x g ( x f ( x dg ( x + g ( x df ( x d( f ( x Qf ( x f ( x Qdf ( x ( d f x f ( x df ( x x x x, x ( Ax + b A, x ( a x a, x ( f ( x g( x g x ( x f x + x ( f ( x g( x x ( f ( x Qf ( x x ( f ( x Qf ( x ( x Qx Qx x f x f x f x x ( x f XY, Y f x, pxy xdx E f XY, Y E f ( X g( Y g ( Y f ( X Y g (, f(x x, the have E [ X] XY X X µ X Λ Λ XX XY Z s jotl Gaussa ~ N, Y Y he, µ Λ Λ Y YX YY p( x ~ ( EX Λ, X ( EX µ µ +Λ Λ, X XY YY Y Λ Λ Λ Λ Λ X XX XY YY YX Defe K Λ Λ he E XY YY X µ + K X ( µ Y, Λ Λ K X XX ΛYX ( XYW,, jotl Gaussa W ( XY, BX + W he V ~ N( E V, Λ V where EV BEX + E W, E a costat ad Λ BΛ B +Λ ( V X WW aa + ci I aa c c a a c ( + Schur complemet of A A AA A I A A I A A A A A I A A ; I A AA A A ( A A ad ( A A A A A A ( + ( + A BCD A A B C DA B DA ( + ( + A BCD A A B I CDA B CDA
19 N(m,σ : X ( det( Λ ( σ ( x m σ f x e σ e, Λ ( x m ( x m E e jvx e jmv v σ x µ σ ; d exp σ µ µ exp x + x σ σ σ det ( Λ e ( x m ( x m Λ ; CN ( µ, Σ N ( µ, Σ Q (, Q( z Q( z xm Q ( Q( z z P[ X > x] Q σ xm xm P[ X < x] Q Q σ σ λ λ ux Posso P(λ, e ; Ω N, λ, EX λ, VAR(X λ, Φ X ( u Ee ( e e λ u! Bomal p ( p ; p, p(-p, (pe u +-p a+ b Uform U(a,b,, b a b+ a s u ( b a u, e Expoetal E(α,, b a u α α, α aplaca α u αx α x (α, e x < e α ;α >,,, αx e x α, α α + u q x Gamma fucto: Γ ( q x e dx ; q > Γ ( (! for N ( N! Γ λ x Γ q, λ p( x Γ f { } dstrbuto: E ( λ s (,λ Γ Beta dstrbuto: p( z β q, q ( x x ( x q q λx ( q Γ + Γ Gamma e Γ +! q > x Expoetal dstrbuto ( q q ( q Γ( q Γ + q ( q z z z Γ ; z (,
20 et X ~ p( x λ x Γ e q q λ x ( q X X, depedet he Z X + X are depedet Z β ( z Z X X ~ ( q q, λ ~ Γ q, λ X ~ q, q X + X Cetral ch-square dstrbuto: X ~ (, σ σ e, ( u p X d σ ( σ ~ N, Y ( Φ + Γ + N Y juσ X he ad Z X + X ch-square (or gamma: X he σ p e, σ ( σ Φ u j uσ E [ Y] σ, Var[ Y] Decorrelato If A s the covarace matrx ( E[ xx ] Γ, Γ 4 σ x of a zero mea radom vector x x he vector x x ca be decorrelated va trasform x x AA x wth A covarace Cov( E[ ] A A A A Cov ( x A A x wth equalt ff Pr x A A x
CS 1675 Introduction to Machine Learning Lecture 7. Density estimation. Milos Hauskrecht 5329 Sennott Square
CS 675 Itroducto to Mache Learg Lecture 7 esty estmato Mlos Hausrecht mlos@cs.tt.edu 539 Seott Square ata: esty estmato {.. } a vector of attrbute values Objectve: estmate the model of the uderlyg robablty
Διαβάστε περισσότεραEstimators when the Correlation Coefficient. is Negative
It J Cotemp Math Sceces, Vol 5, 00, o 3, 45-50 Estmators whe the Correlato Coeffcet s Negatve Sad Al Al-Hadhram College of Appled Sceces, Nzwa, Oma abur97@ahoocouk Abstract Rato estmators for the mea of
Διαβάστε περισσότεραExam Statistics 6 th September 2017 Solution
Exam Statstcs 6 th September 17 Soluto Maura Mezzett Exercse 1 Let (X 1,..., X be a raom sample of... raom varables. Let f θ (x be the esty fucto. Let ˆθ be the MLE of θ, θ be the true parameter, L(θ be
Διαβάστε περισσότεραExamples of Cost and Production Functions
Dvso of the Humates ad Socal Sceces Examples of Cost ad Producto Fuctos KC Border October 200 v 20605::004 These otes sho ho you ca use the frst order codtos for cost mmzato to actually solve for cost
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραHomework for 1/27 Due 2/5
Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where
Διαβάστε περισσότεραDiscrete Fourier Transform { } ( ) sin( ) Discrete Sine Transformation. n, n= 0,1,2,, when the function is odd, f (x) = f ( x) L L L N N.
Dscrete Fourer Trasform Refereces:. umercal Aalyss of Spectral Methods: Theory ad Applcatos, Davd Gottleb ad S.A. Orszag, Soc. for Idust. App. Math. 977.. umercal smulato of compressble flows wth smple
Διαβάστε περισσότεραLast Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing.
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 19 Likelihood Ratio Test Hyu Mi Kag March 26th, 2013 Describe the followig cocepts i your ow words Hypothesis Null Hypothesis Alterative Hypothesis
Διαβάστε περισσότεραMulti-dimensional Central Limit Theorem
Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t tme
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότερα( ) S( x ) 2 ( ) = ( ) ( ) = ( ) ( )
Ορίζουμε την πληροφορία κατά Fsher ( σαν το ποσό της πληροφορίας που περιέχει η παρατήρηση για την παράμετρο Συμβολίζοντας με S( την λογαριθμική παράγωγο της πιθανοφάνειας ως προς την παράμετρο (score
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραMulti-dimensional Central Limit Theorem
Mult-dmensonal Central Lmt heorem Outlne () () () t as () + () + + () () () Consder a sequence of ndependent random proceses t, t, dentcal to some ( t). Assume t 0. Defne the sum process t t t t () t ();
Διαβάστε περισσότεραΝΕΑ ΑΠΟΤΕΛΕΣΜΑΤΑ ΣΧΕΤΙΚΑ ΜΕ ΤΗΝ ΥΠΑΡΞΗ ΕΚΤΙΜΗΤΩΝ ΜΕΓΙΣΤΗΣ ΠΙΘΑΝΟΦΑΝΕΙΑΣ ΓΙΑ ΤΗΝ 3-ΠΑΡΑΜΕΤΡΙΚΗ ΓΑΜΜΑ ΚΑΤΑΝΟΜΗ
Ελληνικό Στατιστικό Ινστιτούτο Πρακτικά ου Πανελληνίου Συνεδρίου Στατιστικής 008, σελ 9-98 ΝΕΑ ΑΠΟΤΕΛΕΣΜΑΤΑ ΣΧΕΤΙΚΑ ΜΕ ΤΗΝ ΥΠΑΡΞΗ ΕΚΤΙΜΗΤΩΝ ΜΕΓΙΣΤΗΣ ΠΙΘΑΝΟΦΑΝΕΙΑΣ ΓΙΑ ΤΗΝ 3-ΠΑΡΑΜΕΤΡΙΚΗ ΓΑΜΜΑ ΚΑΤΑΝΟΜΗ Γεώργιος
Διαβάστε περισσότεραMarkov Processes and Applications
Markov rocesses ad Applcatos Dscrete-Tme Markov Chas Cotuous-Tme Markov Chas Applcatos Queug theory erformace aalyss ΠΜΣ524: Μοντελοποίηση και Ανάλυση Απόδοσης Δικτύων (Ι. Σταυρακάκης - ΕΚΠΑ) Dscrete-Tme
Διαβάστε περισσότερα6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Διαβάστε περισσότερα8.1 The Nature of Heteroskedasticity 8.2 Using the Least Squares Estimator 8.3 The Generalized Least Squares Estimator 8.
8.1 The Nature of Heteroskedastcty 8. Usng the Least Squares Estmator 8.3 The Generalzed Least Squares Estmator 8.4 Detectng Heteroskedastcty E( y) = β+β 1 x e = y E( y ) = y β β x 1 y = β+β x + e 1 Fgure
Διαβάστε περισσότεραLecture 17: Minimum Variance Unbiased (MVUB) Estimators
ECE 830 Fall 2011 Statistical Sigal Processig istructor: R. Nowak, scribe: Iseok Heo Lecture 17: Miimum Variace Ubiased (MVUB Estimators Ultimately, we would like to be able to argue that a give estimator
Διαβάστε περισσότεραp n r.01.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.95
r r Table 4 Biomial Probability Distributio C, r p q This table shows the probability of r successes i idepedet trials, each with probability of success p. p r.01.05.10.15.0.5.30.35.40.45.50.55.60.65.70.75.80.85.90.95
Διαβάστε περισσότεραCytotoxicity of ionic liquids and precursor compounds towards human cell line HeLa
Cytotoxcty of oc lqud ad precuror compoud toward huma cell le HeLa Xuefeg Wag, a,b C. Adré Ohl, a Qghua Lu,* a Zhaofu Fe, c Ju Hu, b ad Paul J. Dyo c a School of Chemtry ad Chemcal Techology, Shagha Jao
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραOutline. Detection Theory. Background. Background (Cont.)
Outlie etectio heory Chapter7. etermiistic Sigals with Ukow Parameters afiseh S. Mazloum ov. 3th Backgroud Importace of sigal iformatio Ukow amplitude Ukow arrival time Siusoidal detectio Classical liear
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραArticle Multivariate Extended Gamma Distribution
axoms Artcle Multvarate Exteded Gamma Dstrbuto Dhaya P. Joseph Departmet of Statstcs, Kurakose Elas College, Maaam, Kottayam, Kerala 686561, Ida; dhayapj@gmal.com; Tel.: +91-9400-733-065 Academc Edtor:
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραAnswers - Worksheet A ALGEBRA PMT. 1 a = 7 b = 11 c = 1 3. e = 0.1 f = 0.3 g = 2 h = 10 i = 3 j = d = k = 3 1. = 1 or 0.5 l =
C ALGEBRA Answers - Worksheet A a 7 b c d e 0. f 0. g h 0 i j k 6 8 or 0. l or 8 a 7 b 0 c 7 d 6 e f g 6 h 8 8 i 6 j k 6 l a 9 b c d 9 7 e 00 0 f 8 9 a b 7 7 c 6 d 9 e 6 6 f 6 8 g 9 h 0 0 i j 6 7 7 k 9
Διαβάστε περισσότεραΠανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών. ΗΥ-570: Στατιστική Επεξεργασία Σήµατος. ιδάσκων : Α. Μουχτάρης. εύτερη Σειρά Ασκήσεων.
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 2015 ιδάσκων : Α. Μουχτάρης εύτερη Σειρά Ασκήσεων Λύσεις Ασκηση 1. 1. Consder the gven expresson for R 1/2 : R 1/2
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραLatent variable models Variational approximations.
CS 3750 Mache Learg Lectre 9 Latet varable moel Varatoal appromato. Mlo arecht mlo@c.ptt.e 539 Seott Sqare CS 750 Mache Learg Cooperatve vector qatzer Latet varable : meoalty bary var Oberve varable :
Διαβάστε περισσότεραL.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:
Διαβάστε περισσότεραSUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6
SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραThe Equivalence Theorem in Optimal Design
he Equivalece heorem i Optimal Desig Raier Schwabe & homas Schmelter, Otto vo Guericke Uiversity agdeburg Bayer Scherig Pharma, Berli rschwabe@ovgu.de PODE 007 ay 4, 007 Outlie Prologue: Simple eamples.
Διαβάστε περισσότεραLatent variable models Variational approximations.
CS 3750 Mache Learg Lectre 9 Latet varable moel Varatoal appromato. Mlo arecht mlo@c.ptt.e 539 Seott Sqare CS 750 Mache Learg Cooperatve vector qatzer Latet varable : meoalty bary var Oberve varable :
Διαβάστε περισσότεραStatistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
Διαβάστε περισσότεραSolve the difference equation
Solve the differece equatio Solutio: y + 3 3y + + y 0 give tat y 0 4, y 0 ad y 8. Let Z{y()} F() Taig Z-trasform o both sides i (), we get y + 3 3y + + y 0 () Z y + 3 3y + + y Z 0 Z y + 3 3Z y + + Z y
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραMinimum density power divergence estimator for diffusion processes
A Ist Stat Math 3) 65:3 36 DOI.7/s463--366-9 Mmum desty power dvergece estmator for dffuso processes Sagyeol Lee Jumo Sog Receved: 3 March 7 / Revsed: Aprl / ublshed ole: July The Isttute of Statstcal
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραExercise 2: The form of the generalized likelihood ratio
Stats 2 Winter 28 Homework 9: Solutions Due Friday, March 6 Exercise 2: The form of the generalized likelihood ratio We want to test H : θ Θ against H : θ Θ, and compare the two following rules of rejection:
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραPolitical Science 552
emedal easres egresso Aprl, 4 Poltcal Scece 55 emedal easres Otlers Dscardg Otlers Trcatg Otlers obst estmato o A/AD east Absolte esdals/east Absolte Devatos (qreg Stata) o S east eda east Sqares o IS
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραMATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra
MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutios to Poblems o Matix Algeba 1 Let A be a squae diagoal matix takig the fom a 11 0 0 0 a 22 0 A 0 0 a pp The ad So, log det A t log A t log
Διαβάστε περισσότερα1. For each of the following power series, find the interval of convergence and the radius of convergence:
Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.
Διαβάστε περισσότεραParts Manual. Trio Mobile Surgery Platform. Model 1033
Trio Mobile Surgery Platform Model 1033 Parts Manual For parts or technical assistance: Pour pièces de service ou assistance technique : Für Teile oder technische Unterstützung Anruf: Voor delen of technische
Διαβάστε περισσότεραMarkov Processes and Applications
Markov Processes ad Alcatos Dscrete-Tme Markov Chas Cotuous-Tme Markov Chas Alcatos Queug theory Performace aalyss Dscrete-Tme Markov Chas Books - Itroducto to Stochastc Processes (Erha Clar), Cha. 5,
Διαβάστε περισσότεραSolutions: Homework 3
Solutios: Homework 3 Suppose that the radom variables Y,, Y satisfy Y i = βx i + ε i : i,, where x,, x R are fixed values ad ε,, ε Normal0, σ ) with σ R + kow Fid ˆβ = MLEβ) IND Solutio: Observe that Y
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραMean-Variance Analysis
Mean-Variance Analysis Jan Schneider McCombs School of Business University of Texas at Austin Jan Schneider Mean-Variance Analysis Beta Representation of the Risk Premium risk premium E t [Rt t+τ ] R1
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότερα557: MATHEMATICAL STATISTICS II RESULTS FROM CLASSICAL HYPOTHESIS TESTING
Most Powerful Tests 557: MATHEMATICAL STATISTICS II RESULTS FROM CLASSICAL HYPOTHESIS TESTING To construct and assess the quality of a statistical test, we consider the power function β(θ). Consider a
Διαβάστε περισσότεραLAD Estimation for Time Series Models With Finite and Infinite Variance
LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite
Διαβάστε περισσότεραStudies on Properties and Estimation Problems for Modified Extension of Exponential Distribution
Iteratoal Joural of Computer Applcatos (975 8887) Volume 5 No. September 5 Studes o Propertes ad Estmato Problems for odfed Exteso of Expoetal Dstrbuto.A. El-Damcese athematcs Departmet Faculty of Scece
Διαβάστε περισσότεραP P Ó P. r r t r r r s 1. r r ó t t ó rr r rr r rí st s t s. Pr s t P r s rr. r t r s s s é 3 ñ
P P Ó P r r t r r r s 1 r r ó t t ó rr r rr r rí st s t s Pr s t P r s rr r t r s s s é 3 ñ í sé 3 ñ 3 é1 r P P Ó P str r r r t é t r r r s 1 t r P r s rr 1 1 s t r r ó s r s st rr t s r t s rr s r q s
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραrs r r â t át r st tíst Ó P ã t r r r â
rs r r â t át r st tíst P Ó P ã t r r r â ã t r r P Ó P r sã rs r s t à r çã rs r st tíst r q s t r r t çã r r st tíst r t r ú r s r ú r â rs r r â t át r çã rs r st tíst 1 r r 1 ss rt q çã st tr sã
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραΕδώ θα θέσουμε τα θεμέλια της εκτίμησης κατά Bayes αρχίζοντας με τα μονοπαραμετρικά μοντέλα δηλαδή όταν ϑ : Ω Θ.
Μονοπαραμετρικά Μοντέλα Εδώ θα θέσουμε τα θεμέλια της εκτίμησης κατά Bayes αρχίζοντας με τα μονοπαραμετρικά μοντέλα δηλαδή όταν : Ω Θ Εκτίμηση πιθανότητας από boal data Έστω δεδομένα που δίδονται με την
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότερα1. Matrix Algebra and Linear Economic Models
Matrix Algebra ad Liear Ecoomic Models Refereces Ch 3 (Turkigto); Ch 4 5 (Klei) [] Motivatio Oe market equilibrium Model Assume perfectly competitive market: Both buyers ad sellers are price-takers Demad:
Διαβάστε περισσότεραSOLVING CUBICS AND QUARTICS BY RADICALS
SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραFORMULAS FOR STATISTICS 1
FORMULAS FOR STATISTICS 1 X = 1 n Sample statistics X i or x = 1 n x i (sample mean) S 2 = 1 n 1 s 2 = 1 n 1 (X i X) 2 = 1 n 1 (x i x) 2 = 1 n 1 Xi 2 n n 1 X 2 x 2 i n n 1 x 2 or (sample variance) E(X)
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότερα12. Radon-Nikodym Theorem
Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say
Διαβάστε περισσότεραThree-Dimensional Experimental Kinematics
Notes_5_3 o 8 Three-Dmesoal Epermetal Kematcs Dgte locatos o ladmarks { r } o bod or pots to at gve tme t All pots must be o same bod Use ladmark weghtg actor = pot k s avalable at tme t. Use = pot k ot
Διαβάστε περισσότεραAn Introduction to Signal Detection and Estimation - Second Edition Chapter II: Selected Solutions
An Introduction to Signal Detection Estimation - Second Edition Chapter II: Selected Solutions H V Poor Princeton University March 16, 5 Exercise : The likelihood ratio is given by L(y) (y +1), y 1 a With
Διαβάστε περισσότεραΓιάννης Σαριδάκης Σχολή Μ.Π.Δ., Πολυτεχνείο Κρήτης
2 η Διάλεξη Ακολουθίες 29 Νοεµβρίου 206 Γιάννης Σαριδάκης Σχολή Μ.Π.Δ., Πολυτεχνείο Κρήτης ΑΠΕΙΡΟΣΤΙΚΟΣ ΛΟΓΙΣΜΟΣ, ΤΟΜΟΣ Ι - Fiey R.L. / Weir M.D. / Giordao F.R. Πανεπιστημιακές Εκδόσεις Κρήτης 2 Όρια Ακολουθιών
Διαβάστε περισσότεραPost Graduate Diploma in Applied Statistics (PGDAST)
FORMULAE AD STATISTICAL TABLES BOOKLET for Post Graduate Dploma Appled Statstcs (PGDAST) IMPORTAT The Formulae ad Statstcal Tables Boolet cotas the ma formulae of the courses of the PGDAST programme ad
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότερα: Monte Carlo EM 313, Louis (1982) EM, EM Newton-Raphson, /. EM, 2 Monte Carlo EM Newton-Raphson, Monte Carlo EM, Monte Carlo EM, /. 3, Monte Carlo EM
2008 6 Chinese Journal of Applied Probability and Statistics Vol.24 No.3 Jun. 2008 Monte Carlo EM 1,2 ( 1,, 200241; 2,, 310018) EM, E,,. Monte Carlo EM, EM E Monte Carlo,. EM, Monte Carlo EM,,,,. Newton-Raphson.
Διαβάστε περισσότεραThe ε-pseudospectrum of a Matrix
The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems
Διαβάστε περισσότεραMÉTHODES ET EXERCICES
J.-M. MONIER I G. HABERER I C. LARDON MATHS PCSI PTSI MÉTHODES ET EXERCICES 4 e édition Création graphique de la couverture : Hokus Pokus Créations Dunod, 2018 11 rue Paul Bert, 92240 Malakoff www.dunod.com
Διαβάστε περισσότεραΕΙ Η ΠΑΛΙΝ ΡΟΜΗΣΗΣ. ΑΠΛΗ ΓΡΑΜΜΙΚΗ ΠΑΛΛΙΝ ΡΟΜΗΣΗ (Simple Linear Regression) ΓΡΑΜΜΙΚΗ ΠΑΛΙΝ ΡΟΜΗΣΗ (Regression) ΠΑΛΙΝ ΡΟΜΗΣΗ.
ΑΠΛΗ ΓΡΑΜΜΙΚΗ ΠΑΛΛΙΝ ΡΟΜΗΣΗ (Smple Lear Regresso) Να κατανοηθεί η έννοια της παλινδρόµησης Ποιες οι προϋποθέσεις για να εφαρµοσθεί η γραµµική παλινδρόµηση; Τι είναι το γραµµικό µοντέλο και πως εκτιµούνται
Διαβάστε περισσότεραTheorem 8 Let φ be the most powerful size α test of H
Testing composite hypotheses Θ = Θ 0 Θ c 0 H 0 : θ Θ 0 H 1 : θ Θ c 0 Definition 16 A test φ is a uniformly most powerful (UMP) level α test for H 0 vs. H 1 if φ has level α and for any other level α test
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραSuppose Mr. Bump observes the selling price and sales volume of milk gallons for 10 randomly selected weeks as follows
Albert Ludwgs Unverst Freburg Department of Emprcal Research and Econometrcs Appled Econometrcs Dr Kestel ummer 9 EXAMPLE IMPLE LINEAR REGREION ANALYI uppose Mr Bump observes the sellng prce and sales
Διαβάστε περισσότεραOptimal stopping under nonlinear expectation
Avalable ole at www.scecedrect.com SceceDrect Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 www.elsever.com/locate/spa Optmal stoppg uder olear expectato Ibrahm Ekre a, Nzar Touz b, Jafeg
Διαβάστε περισσότεραOn Hypersurface of Special Finsler Spaces. Admitting Metric Like Tensor Field
It J otem Mat Sceces Vo 7 0 o 9 99-98 O Hyersurface of Seca Fser Saces Admttg Metrc Lke Tesor Fed H Wosoug Deartmet of Matematcs Isamc Azad Uversty Babo Brac Ira md_vosog@yaoocom Abstract I te reset work
Διαβάστε περισσότεραΗΥ537: Έλεγχος Πόρων και Επίδοση σε Ευρυζωνικά Δίκτυα,
ΗΥ537: Έλεγχος Πόρων και Επίδοση σε Ευρυζωνικά Δίκτυα Βασίλειος Σύρης Τμήμα Επιστήμης Υπολογιστών Πανεπιστήμιο Κρήτης Εαρινό εξάμηνο 2008 Economcs Contents The contet The basc model user utlty, rces and
Διαβάστε περισσότεραss rt çã r s t Pr r Pós r çã ê t çã st t t ê s 1 t s r s r s r s r q s t r r t çã r str ê t çã r t r r r t r s
P P P P ss rt çã r s t Pr r Pós r çã ê t çã st t t ê s 1 t s r s r s r s r q s t r r t çã r str ê t çã r t r r r t r s r t r 3 2 r r r 3 t r ér t r s s r t s r s r s ér t r r t t q s t s sã s s s ér t
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραFormal Semantics. 1 Type Logic
Formal Semantics Principle of Compositionality The meaning of a sentence is determined by the meanings of its parts and the way they are put together. 1 Type Logic Types (a measure on expressions) The
Διαβάστε περισσότερα