Dr. Shalabh. Indian Institute of Technology Kanpur

Transcription

1 Aalyss of Varace ad Desg of Expermets-I MODULE -I LECTURE - SOME RESULTS ON LINEAR ALGEBRA, MATRIX THEORY AND DISTRIBUTIONS Dr. Shalabh Departmet t of Mathematcs t ad Statstcs t t Ida Isttute of Techology Kapur

2 Quadratc forms If A s a gve matrx of order m ad X ad Y are two gve vectors of order m ad respectvely, the the quadratc form s gve by X ' AY m = = j= a x y j j where a ' s are the ostochastc elemets of A. j If A s square matrx of order m ad X = Y, the X ' AX = a x a x + ( a + a ) x x ( a + a ) x x. mm m m, m m, m m m If A s symmetrc also, the X ' AX = a x a x + a x x a x x mm m m, m m m m = axx j j = j= s called a quadratc form m varables x, x,, x m or a quadratc form X. To every quadratc form correspods a symmetrc matrx ad vce versa. The matrx A s called the matrx of quadratc form. The quadratc form X ' AX ad the matrx A of the form s called Postve defte f X ' AX > 0 for all x 0. Postve sem defte f X ' AX 0 for all x 0. Negatve defte f X ' AX < 0 for all x 0. Negatve sem defte f X ' AX 0 for all x 0.

3 3 If A s postve sem defte matrx the a 0 ad f a = 0 the a = 0 for all j, ad a j = 0 for all j. If P s ay osgular matrx ad A s ay postve defte matrx (or postve sem-defte matrx) the P' AP s also a postve defte matrx (or postve sem-defte matrx). A matrx A s postve defte f ad oly f there exsts a o-sgular matrx P such that A= P' P. A postve defte matrx s a osgular matrx. If A s m matrx ad ra ( A) = m < the AA' s postve defte ad A' A s postve semdefte. If A m matrx ad ra ( A) = < m <, the both AA ' ad AA' are postve semdefte. ( ) j

4 4 Smultaeous lear equatos The set of m lear equatos uows x, x,..., x ad scalars aj a x + a x a x = b a x + a x a x = b a x a x a x b m + m m = m ad b, =,,..., m, j =,,..., of the form ca be formulated as AX = b where A s a real matrx of ow scalars of order vector of ow scalars gve by m called as coeffcet matrx, X s real vector ad b s real a a... a a a... a A=, s a m am am... am real matrx called as coeffcet matrx, x b x b =, s a vector of varables ad = s a real vector. x b m X b m

5 5 If A s osgular matrx, the AX = b has a uque soluto. Let B = [A, b] s a augmeted matrx. A soluto to AX = b exst tff ad oly f ra(a) = ra(b). If A s a m matrx of ra m, the AX = b has a soluto. Lear homogeeous system AX = 0 has a soluto other tha X = 0 f ad oly f ra (A) <. If AX = b s cosstet the AX = b has a uque soluto f ad oly f ra (A) = If a s the th dagoal elemet of a orthogoal matrx, the. Let the matrx be parttoed as A= [ a where s a vector of the elemets of th, a,..., a ] a colum of A. A ecessary ad suffcet codto that A s a orthogoal matrx s gve by the followg: ' ( ) = =,,..., aa for ' ( ) aa j = 0 for j =,,...,. a Orthogoal matrx A square matrx A s called a orthogoal matrx f A orthogoal matrx s o-sgular. If A s orthogoal, the AA' s also orthogoal. A ' A = AA' = I or equvaletly f A = A'. If A s a matrx ad let P s a orthogoal matrx, the the determats of A ad PAP ' are the same.

6 6 Radom vectors Let Y be radom varables the s called a radom vector., Y,..., Y Y = ( Y, Y,..., Y )' The mea vector of Y s EY = EY EY EY ( ) (( ( ), ( ),..., ( )) '. The covarace matrx or dsperso matrx of Y s Var( Y ) Var( Y) Cov( Y, Y)... Cov( Y, Y ) Cov( Y, Y ) Var( Y )... Cov( Y, Y ) Cov( Y, Y) Cov( Y, Y)... Var( Y) = whch s a symmetrc matrx. If Y, Y,..., Y are par-wse ucorrelated, the the covarace matrx s a dagoal matrx. If Var( Y ) = σ for all =,,, the Var Y I ( ) = σ.

7 7 Lear fucto of radom varable If Y, Y,..., Y are radom varables ad,,.., are scalars, the Y s called a lear fucto of radom varables Y, Y,..., Y. = If Y = ( Y, Y,..., Y)', K = (,,..., )' the K ' Y = Y, = the mea K ' Y s E( KY ' ) KEY ' ( ) EY ( ) ad the varace of s ' = = = K Y Var ( K Y ) = ( ) ' K ' Var Y K. Multvarate t ormal ldstrbuto tb t A radom vector Y = ( Y, Y,..., Y )' has a multvarate ormal dstrbuto wth mea vector μ = ( μ, μ,..., μ ) ad dsperso matrx Σ f ts probablty desty fucto s f Y Y Y ( π ) Σ ( μ, Σ ) = exp ( μ)' Σ ( μ) / / assumg Σ s a osgular matrx.

8 Ch-square dstrbuto 8 If Y, Y,...,, Y are detcally ad depedetly dstrbuted radom varables followg the ormal dstrbuto wth commo mea 0 ad commo varace, the the dstrbuto of Y s called the χ - dstrbuto wth degrees of freedom. = The probablty desty fucto of χ - dstrbuto wth degrees of freedom s gve as x f ( x ) = x exp ; 0 x. χ / < < Γ( /) If Y, Y,..., Y are depedetly dstrbuted followg the ormal dstrbuto wth commo meas 0 ad commo = varace σ, the has - dstrbuto wth degrees of freedom. Y χ σ If the radom varables Y, Y,..., Y are ormally dstrbuted wth o-ull meas μ, μ,..., μ but commo varace YY, the the dstrbuto b t of has o-cetral χ - dstrbuto b t wth degrees of freedom ad o-cetralty parameter λ = μ = = If Y, Y,..., Y are depedetly dstrbuted followg the ormal dstrbuto wth meas μ, μ,..., μ but commo varace σ the Y has o-cetral χ -dstrbuto wth degrees of freedom ad ocetralty parameter λ = μ. σ = σ =

9 9 If U has a Ch-square dstrbuto wth degrees of freedom the EU ( ) = ad Var( U ) =. If U has a ocetral Ch-square dstrbuto wth degrees of freedom ad ocetralty parameter λ the ( ) EU = + λ ad Var( U ) = + 4 λ. If U, U,..., U are depedetly dstrbuted radom varables wth each U havg a ocetral Ch-square dstrbuto wth degrees of freedom ad o cetralty parameter λ, =,,..., the U has ocetral Ch-square dstrbuto b t wth degrees of freedom ad ocetralty parameter = λ. = = = μ Σ. Let X ( X, X,..., X )' has a multvarate dstrbuto wth mea vector ad postve defte covarace matrx The X ' AX s dstrbuted as ocetral wth degrees of freedom f ad oly f s a dempotet matrx of ra. χ ΣA Let X = ( X, X,..., X ) has a multvarate ormal dstrbuto wth mea vector μ ad postve defte covarace matrx Σ. Let the two quadratc forms- χ X ' AX s dstrbuted b t d as wth degrees of ffreedom ad ocetralty parameter μ ' A μ ad X ' AX s dstrbuted as wth degrees of freedom ad ocetralty parameter χ μ A The X ' AX ad X' AX are depedetly dstrbuted f AΣ A = 0. ' μ.

10 0 t- dstrbuto If X has a ormal dstrbuto wth mea 0 ad varace, Y has a χ dstrbuto wth degrees of freedom, ad X ad Y are depedet radom varables, the the dstrbuto of the statstc T = The probablty desty fucto of T s X Y / s called the t-dstrbuto wth degrees of freedom. + + Γ t ft () t = ; - t. + < < Γ π X If the mea of X s o zero the the dstrbuto of Y / s called the ocetral t - dstrbuto wth degrees of freedom ad ocetralty parameter μ.

11 F- dstrbuto If X ad Y are depedet radom varables wth χ - dstrbuto wth m ad degrees of freedom respectvely, the the dstrbuto of the statstc probablty desty fucto of F s F = X / m Y / s called the F-dstrbuto wth m ad degrees of freedom. The m/ m+ m m+ Γ m m ff ( f) f = f ; 0 f. m + < < Γ Γ If X has a ocetral Ch-square dstrbuto wth m degrees of freedom ad ocetralty parameter λ; Y has a dstrbuto wth degrees of freedom, ad X ad Y are depedet radom varables, the the dstrbuto of X / m F = Y / s the ocetral F dstrbuto wth m ad degrees of freedom ad ocetralty parameter λ. χ