CHAPTR Probablt, Statstcs, ad Relablt or geers ad Scetsts MULTIPL RANDOM VARIABLS Secod dto A. J. Clark School o geerg Departmet o Cvl ad vrometal geerg 6b Probablt ad Statstcs or Cvl geers Departmet o Cvl ad vrometal geerg Uverst o Marlad, College Park CHAPMAN HALL/CRC CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet Computatoal Procedures or Momets Dscrete Radom able: The k th momet about the org s gve b M ' k k all k... k... (,,..., ) Cotuous Radom able: The k th momet about the org s gve b P ' k k k k...... (,,..., ) d d d... M...
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet The prevous momets are cosdered as a specal case o mathematcal epectato. The mathematcal epectato o a arbtrar ucto g(), whch s a ucto o the radom vector, s deed the ollowg vewgraph or dscrete ad cotuous cases. CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet Mathematcal pectato Dscrete Radom able: The mathematcal epectato s gve b [ g( )] g( ) P (, ) all...,..., Cotuous Radom able: The mathematcal epectato s gve b... d d d... [ g( )] g( ) (,,..., )...
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 4 Codtoal Momets, Covarace, ad Correlato Coecet For smplct, the presetato o the materals the remag part o ths secto s lmted to two radom varables. For the two-dmesoal case, ad, the codtoal mea or gve that takes value deoted µ, s deed terms the codtoal mass ad dest uctos or the dscrete ad cotuous radom varables. CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 5 Codtoal Momets, Covarace, ad Correlato Coecet Codtoal Mea Dscrete Radom able µ P ( ) ( ) all Cotuous Radom able µ ( ) ( ) d
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 6 Codtoal Momets, Covarace, ad Correlato Coecet Codtoal Mea For statstcall ucorrelated radom varables ad, the codtoal mea s gve b µ µ ( ) ( ) ( ) ( ) ( µ ) ( ) Also, t ca be show that CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 7 Codtoal Momets, Covarace, ad Correlato Coecet Codtoal ace Dscrete Radom ables µ P ( ) ( ) ( ) all Cotuous Radom ables ( ) ( µ ) ( ) d 4
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 8 Codtoal Momets, Covarace, ad Correlato Coecet The varace o a radom varable ca also be computed usg codtoal varace as ollows: ( ) [ ( )] [ ( )] CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 9 Codtoal Momets, Covarace, ad Correlato Coecet Covarace o Two Radom ables The covarace (Cov) o two radom varables ad s deed terms o mathematcal epectato as Cov, µ µ ( ) [( )( )] the covarace o,, or Cov It s commo to use the ollowg otatos or (, ) ad : 5
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet Covarace o Two Radom ables It ca be show that the Cov ca also be determed usg the ollowg equato: Cov where (, ) ( ) ( ) (, ) µ µ d d CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet Covarace o Two Radom ables I ad are statstcall ucorrelated radom varables, the ad Cov (, ) ( ) µ µ 6
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet Correlato Coecet The correlato coecet o two radom varables ad s deed as a ormalzed covarace wth respect to the stadard devatos o ad ad s gve b Cov(, ) ρ The correlato coecet rages betwee -ad, ρ CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Codtoal Momets, Covarace, ad Correlato Coecet Correlato Coecet I the correlato coecet s zero, the the two radom varables are sad to be ucorrelated. I order or ρ to be zero, the Cov( ) must be zero. Thereore ad are statstcall ucorrelated. However, the coverse o ths dg s ot true. The correlato coecet ca be vewed as a measure o the degree o lear assocato betwee ad. 7
8 CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 4 ample: Two Dscrete RV s Gve the ollowg jot dest ucto o radom varables ad ad assume : (a) Fd the margal dest uctos o ad. (b) Determe the mea or epected values o ad. (c) The covarace ad correlato coecet o ad Codtoal Momets, Covarace, ad Correlato Coecet ( ) otherwse ad or, CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 5 ample (cot d): Two Dscrete RV s a) Margal dest uctos Codtoal Momets, Covarace, ad Correlato Coecet ( ) ( ) or, d d ( ) ( ) or, d d
9 CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 6 ample (cot d): Two Dscrete RV s b) pected values o ad Codtoal Momets, Covarace, ad Correlato Coecet ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 45., For Sce d d d CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 7 ample (cot d): Two Dscrete RV s c) Covarace ad correlato o ad Codtoal Momets, Covarace, ad Correlato Coecet ( ) ( ) ( ) ( ) ( ) ( ) ( ).57 the, I or Sce d d d
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 8 Codtoal Momets, Covarace, ad Correlato Coecet ample (cot d): Two Dscrete RV s c) Covarace ad correlato (cot d) The varaces o ( ) ( ) ( ) ( ) Thereore, Cov ρ ad are computed as ollows : [ ].57 (.45) The epected value o the product s dd.5 (, ) ( ) ( ) ( ).5.45(.45) Cov(, ).475.546.546.87.546.475 CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 9 Fuctos o Radom ables Ma egeerg problems deal wth a depedet varables that s a ucto o oe or more depedet varables w P L/ wl PL M.5w 7. 5P 8 4 L
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables Three cases to be cosdered: Probablt dstrbutos or depedet radom varables, Mathematcal epectatos, ad Appromate methods CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables A radom varable s deed as a mappg rom a sample space o a egeerg sstem or epermet to the real le o umbers. I s deed to be a depedet varable terms o a ucto g( ) the s also a radom varable
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables amples P w P M, A, I, c P Mc A I L/ L wl PL M.5w 7. 5P 8 4 CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables Sgle Radom able The stress () a beam s a ucto o a appled load (). I the load s radom, the stress s also radom Lear Case g( ) a b where a ad b are real umbers g( ) ( ) a( ) b a ( )
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 4 Fuctos o Radom ables Multple Radom ables The stress () a beam s a ucto o a appled load, materal propertes, ad geometr: g(,,..., ) Lear Case g( ) a ( ) a a ( ) a ( )... a ( ) ( ) aa jcov(, j ) j a a where a ad b are real umbers... a CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 5 Fuctos o Radom ables Multple Radom ables It should be oted that ( ) ( ) Cov, The varace o ca be also obtaed rom ( ) a a j ρ j j j I the radom varables o the vector are statstcall ucorrelated, the ( ) a ( )
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 6 Fuctos o Radom ables ample: Mea ad ace o a Lear Fucto Assume ad are ucorrelated. Z 5 µ ad µ 5 ad Thereore, ad µ Z Z Z µ a 4. 5µ () 5() 5 ( ) ( ) 5 ( ) COV 4 4. 4 ( Z ). 5 CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 7 Fuctos o Radom ables Mathematcal pectato Mathematcal epectato or g() Dscrete Case Cotuous Case [ g( )] g( ) P ( ) g [ ( )] g( ) ( ) d 4
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 8 Fuctos o Radom ables ace The varace or g() Dscrete Case ( ) [ g( )] ( g( ) [ g( )]) P ( ) Cotuous Case ( ) [ g( )] ( g( ) [ g( )]) ( )d CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. 9 Fuctos o Radom ables Specal Case I the ucto g() a b, the ( ) a( ) b ( ) a ( ) Where a ad b are real umbers. 5
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables Multple Radom ables I the ucto g() s gve b g a a a... a The ad ( ) ( ) a a ( ) a ( )... a ( ) ( ) aa jcov(, j ) aa jρ j j j I the radom varableso ( ) a ( ) j are ucorrelated, the CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables Multple Radom ables I the ucto g() s gve b ad The ( ) g... ( ) ( ) ( ) ( )...( ) [ ] ( ) ( ) ( )...( ) ( ) ( )...( ) 6
CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables ample: Cost o Precast Cocrete The total cost C to mauacture a cocrete pael a precast plat s C.5 where s the cost o materals, s the cost o labor. I the costs ad are assumed to be ucorrelated wth meas o $/pael ad $5/pael, respectvel, ad wth stadard devatos o $/pael ad $5/pael, respectvel, compute the mea,varace, stadard devato, ad COV o the total cost. CHAPTR 6b. MULTIPL RANDOM VARIABLS Slde No. Fuctos o Radom ables ample (cot d): Cost o Precast Cocrete µ C C C.5µ.5.5 ( ) ( 5).5 $65 / pael µ ( ) ( 5),5( $ / pael),5 $. / pael. COV.556 65 7