Stochastic Structural Dynamics. Lecture-4

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1 Stochstic Structurl Dnmics Lecture-4 Multi-dimensionl rndom vribles- Dr C S Mnohr Dertment of Civil Engineering Professor of Structurl Engineering Indin Institute of Science Bnglore 56 Indi mnohr@civil.iisc.ernet.in

2 Recll Trnsformtion of rndom vribles Let be RV; define =g(); Given df of, wht is the df of? Mthemticl eecttion oertor g( ) g( ) g( d E ) Men, vrince, stndrd devition COV, sewness, urtosis Chrcteristic function Moment generting function

3 Comlete secifiction of RV Secifiction of the robbilit sce. PDF df Moment generting function Chrcteristic function Moments of ll orders 3

4 4 () ) ( n! ) ( ) - (ut n )! ( ) ( )! ( ) (! ) (! ) ( () ) (! ) (,,,, ;! ) ( P n n e e e e e e e e e e e Poisson rndom vrible

5 5 σ! ) ( ) (! ) (! ) ( e e e Vrince

6 6 e e! e ) e(! ) )e( e( e i i i i ω φ Chrcteristic function

7 Gussin rndom vrible m Nm, e ; Are under the curve (=?) m d e d - - m Substitute u du d d e u - - Let I= - - e u du u v I e dudv Substitute u r cos ; v r sin rdrd dudv r I re drd d du 7

8 Eercise

9 Remr A Gussin rndom vrible is comletel secified in terms of its men nd stndrd devition. 9

10 More on Gussin rndom vrible Let ~ N( m, ). - m U ~ N(,). u U u e du U & U. s PU u e ds u s s e ds e ds.5 erf ( u) erf(u)= u u e s ds du d ~ N( m, ) m U u u m e with U u u m e u m ~ N(,)

11 3 3 U U U u du.6868 u du.9545 u du.9973 P m3 m m 3 ~ etremes

12 Moment Genrting function Let Z ~ N(,). e( s ) ese d s e d s s e s s e e d s e d Moment Genrting function Let ~ N(, ). e( s ) ese d Substitute u & roceed. s Show tht e( s ) es

13 A word of cution. Moments m not lws eist. Emle : Cuch rndom vrible /. / d / lim n d; n,, 3

14 Men nd stndrd devition cn be used to obtin bounds on robbilities. Mrov inequlit Let be rndom vrible such tht it tes non-negtive vlues. Tht is, P. Then, for n, P Proof ( ) d ( ) d ( ) d ( ) d ( ) d P P. 4

15 Chebchev inequlit Let be RV with men nd stndrd devition. Then P Proof :Consider P P This is non - negtive. B Mrov inequlit, 5

16 Remrs () The inequlities re ect in nture nd re vlid for n df. () Let =. P - P lies outside -, +. Consider the limit of. P=. is deterministic. m (3) We hve. If m & m. P. (4) Chebchev's bound need not rovide shr bounds, tht is, the utilit s bound m be limited. Emle: Let ~N(,). Consider P 3. The true vlue is.7. Chebchev bound rovides P 3. 9 This is not good enough bound for lictions. (5)If, the bounds hve no significnce. P - min,. 6

17 Multi - dimensionl rndom vribles Consider two rndom vribles nd.. Define E nd E.,, E E E Definition P P P Note: Comm (,) denotes intersection ( ). P, = Joint robbilit distribution function of nd (JPDF) Definition, (, ) Joint robbilit densit function of nd (jdf). (, ) P 7

18 Remrs () Geometric interrettion : Plce oint rndoml in the - lne. P ( () P (3) P (4) P (5) P, ) P(The oint lies in the region, P P P, P P P., P P., P,.. ). 8

19 Remrs (Continued) (6) P? Define S S S 3 S S S ; S S 3 3 3,,,, rcise) PS ( ) P S P S P P P P P P (7) P? (Ee 9

20 Remrs (Continued) (8),,,,, P, u, v dudv P (9) P, P dp, d d Similrl,, d P P u v dudv P u v dudv Mrginl df of Mrginl df of () Knowing, we cn find the mrginl df-s. The other w is not true. () P, is monotone nondecresing in nd.,. Comlete secifiction of two rndom vribles nd through JPDF or jdf.

21 JCSS () Steel s 5-dimensionl rndom vrible Descrition COV ield strength.7 Ultimte tensile.4 strength oung s modulus.3 Poisson s rtio Ultimte strin.6 Distribution: Multivrite lognorml rndom vrible

22 Indeendence of rndom vribles Recll P A Define P A B ; PB. PB PA B PAPB A nd B P P P, P P, Remr If nd re indeendent, the comlete secifiction of A B P B &. nd will be through

23 Conditionl distributions Let nd be two rndom vribles. Let B be n event. Define P B P B P B P B P B Conditionl PDF of given B. Similrl, we introduce B conditionl B P B df of given B. 3

24 Remrs () Similrl, of () g P B g( ) B Thus df. B hs ll One cn define conditionl B conditionl vrince, etc. the roerties of will hve ll d we could define conditionl the roerties eecttion men, PDF. 4

25 5 dudv v u du u d dp dudv v u dudv v u P P P P P P P P B,,,,,,. Let (3) Remrs (Continued)

26 Remrs (Continued) (4)Let P B B,, P P P P B,,,, B P B P u v dudv u v dudv dp d d u du u du d udu, d Remrs (Continued) ( 5) Let nd d d B d As d, B, Accordingl, one gets,., 6

27 Joint Eecttions : Let nd be two rndom vribles. Consider function g(,). Definition g (), g,, Remrs n Clerl, m m n d.similrl, m. n dd,, dd dd 7

28 () n n. Clerl, &. (3) Definition μ η η is the covrince of RVs nd. (4) Definition r correltion coefficient between nd. 8

29 More on Covrince nd Correltion Coefficient () Let b σ σ r η b η b η η η b η b η η σ () Let σ r σ b η, η η η η, η η σ σ b (sign of ) dd dd σ 9

30 ()More on Covrince nd Correltion Coefficient undson(3) r, σ η η dd,, η η dd o B η, dd η, dd bvitueoftheschwrzinequlit σ r r (4) Definition r nd re uncorrelted. (5) nd re uncorrelted nd re uncorrelted does not men tht 3

31 Summr r r re the limits of liner behvior; nd re uncorrelted. 3

32 - dimensionl Gussin rndom vrible nd re sid to be jointl Gussin if, η η r η η e πσ r σ σ σσ σ r ; Eercise: Prove tht () η ; η ; η σ ; η σ ; η η r σσ η σ rσσ Notes : ~ N η r σ σ σ σ r σ σ rσ σ σ is nown s the covrince mtri., r σ /σ (b) Show tht e ; σ σ r π r 3

33 Remrs () r, Tht is, for Gussin rndom vribles, r (b) Eercise: For r, rove tht η,d e ; πσ σ η,d e ; πσ σ 33

34 N N

35 N.9.9 N

36 Functions of rndom vribles Let nd be two rndom vribles. DefineU g(,) ndv h(,). Question :Given the jdf of nd, wht is the jdf of U ndv? Stes () Consider u g(, ) nd v h(, ). Solve for (, ) from these equitons. n, Let be the roots. Note tht ncould i i i be. () Determine u u - u v J OR J v v J u v 36

37 Stes (continued) (3)Find UV OR u,v UV n i n u,v J, i i, J i i i (4)Emine g nd h nd decide uon limits of u Note : Decide if it is esier to wor with J or J i i i i - nd v. 37

38 Emle U V - ~ N u v uv uv ; J ; J. UV u, v J, uv uv uv uv e e u v ; u 4 4 ; v U ~ N V Chec U V UV mdf - s U V U v (o) (o) U (o) (o) u u,vdv~n, v u,vdu~n, It turns out tht U ndv re indeendent UV UV UV u, v u v U V UV V (o) 38

5 Probability densities

5 Probability densities 5 Probbility densities 5. Continuous rndom vribles 5. The norml distribution 5.3 The norml pproimtion to the binomil distribution 5.5 The uniorm distribution 5. Joint distribution discrete nd continuous