Statstcs 1: Probablty Theory II 8 1 JOINT AND MARGINAL DISTRIBUTIONS In Probablty Theory I we formulate the concept of a (real) random varable and descrbe the probablstc behavor of ths random varable by the dstrbutons that t nduces on the real lne In that development attenton s drected to a sngle number assocated wth each outcome of the basc experment That s the problem focuses on a sngle random varable that descrbes the outcome of the experment In the study of many random experments there are or can be more than one random varable of nterest The outcome of an experment mght be best descrbed usng several numbers and thus there may be a need to defne two or more random varables These numbers may be vewed separately as values of the ndvdual random varables but they may be consdered jontly as elements of a random vector As n the case of a sngle random varable we use fundamental mappng deas to arrve at a jont dstrbuton that descrbes the probablstc behavor of the random vector as well as the probablstc relatons among the random varables n the random vector 11 Jont Dstrbutons 111 The Noton of a Random Vector Def n: Let P A be a probablty space A random vector functon wth doman and counterdoman that s ' s a 1 ( 1 )': such that for any set of real numbers say x1 x x : 1 x1 x x A Def n: In the bvarate case e for = for a probablty space A P we defne a bvarate random vector Y ' to be a functon wth doman and counterdoman that s ( Y)': such that for any par of real numbers say u and v : ( ) u Y( ) v A How s t defned on a Borel set B of the plane? A y A (() Y()) x A = {: (() Y()) B}
Statstcs 1: Probablty Theory II 9 As n the one-dmensonal case we may map the probablty from the events on the basc space of the Borel subsets on For example n the dagram above we assgn to regon B (a Borel set) on the plane the probablty mass n event A In a general settng suppose and Y are two real valued functons defned on the probablty space A P For any pont the values of and Y can be represented as an ordered par ( ) Y( ) ' so that ths defnes a functon from to Defnng ths (jont) functon of and Y as Y ' we have a functon ( ( ) ( ) ' For ths functon Y ' be a bvarate random vector t must be that for every Borel set B (a regon Y)': whch maps every nto Y or a pont or an nterval) n the Cartesan plane Y beng a bvarate random vector defned over P Also ' probablty measure Borel sets e : ( ) Y( ) ' B A A nduces a P whch translates probabltes on events to probabltes on : ( ) ( ) ' P Y B P B Remars: 1 The functon Y ' nduces a probablty measure on the Borel subsets B of the Cartesan plane Ths probablty measure denoted by P s a functon wth Y doman B ( )and counterdoman the nterval 01 that s P : B( ) [01] and s defned as P ( B) P : ( Y ' B A smlar probablty measure may be defned for the case of a random vector wth three or more random varables However t wll nvolve more complcated notatons and the geometrc vsualzatons wll be more dffcult 3 For convenence we shall use the notaton Y ' : ( Y ' B Example: u Y v : ( ) u Y( ) v Example: B to represent
Statstcs 1: Probablty Theory II 10 11 Defnton of a Jont Dstrbuton Functon Def n: Let 1 be random varables all defned on the same probablty space A P The jont cumulatve dstrbuton functon (or jont CDF) or smply jont dstrbuton functon of the random varables 1 denoted by F s defned as F 1 or smply 1 F x x x P x x x x x ) 1 x 1 1 1 ( 1 x Remars: 1 Thus a jont CDF s a functon wth doman the -dmensonal Eucldean space counterdoman the nterval 01 FY or For = the jont CDF F has doman and counterdoman and 01 In ths case the jont CDF s smply an extenson of the noton of a dstrbuton functon n the unvarate case Whereas we use the nfnte nterval x to defne the CDF n the unvarate case we shall use the nfnte rectangle where B uv ( x y)' R : x u y v Also ( x y)' R : x u ( x y)' R : y v B uv uv F u v P B Y P ) Y ( ) ' B P ) u Y ( ) v P u Y v uv B uv n the bvarate case 1 Suppose we are to toss a balanced con twce and we defne the followng random varables: = number of heads on the 1 st toss Y = number of heads on the nd toss What s the jont CDF of and Y? Consder the experment of tossng two tetrahedra (regular 4-sded polyhedron) each wth sdes labeled 1 to 4 Defne the followng random varables: = the number on the downturned face of the 1 st tetrahedron Y = the larger of the two downturned numbers What s the jont CDF of and Y?
Statstcs 1: Probablty Theory II 11 113 Propertes of Jont Dstrbuton Functons Let F be the jont CDF of two random varables and Y 1 Boundedness: F v lm F u v 0 Thus Proof: Y Y u F u lm F u v 0 Y Y v F lm F u v 1 Y u Y v ( u v) R FY u v 0 1 Monotoncty: If a b c d are any real numbers a b c d R a b and c d then P a b c Y d F b d F b c F a d F a c 0 Y Y Y Y Note: The result s analogous to P(a< b) = F(b) F(a) n the unvarate case Proof: 3 Contnuty from the Rght: F s contnuous from the rght n each of the varables That s for any fxed x F s contnuous from the rght n Y and for any fxed y the rght n That s lm F u h v F u v F u v Proof: h0 h0 Y Y Y lm F u v h F u v F u v Y Y Y Remars: 1 The boundedness property does not mply that F v lm F u v 1 nor Y Y u F u lm F u v 1 Y Y v Both arguments must be nfnte for F to converge to 1 F s contnuous from The monotoncty property s not a strct monotoncty However F s a monotone non-decreasng functon n each of the varables That s for any fxed x F s non-decreasng n Y and for any fxed y F s non-decreasng n For a fxed x y y F x y F x y Proof: For a fxed 1 Y 1 Y y x x F x y F x y 1 Y 1 Y
Statstcs 1: Probablty Theory II 1 3 The rght contnuty property mples the ff: For any fxed constants a b c d R a b and c d Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y P a bc Y d F bd F bc F a d F a c P a bc Y d F b d F b c F a d F a c P a bc Y d F b d F b c F ad F ac P b Y d F bd F b d F bd F b d Note that F Y ( b d ) lm F Y ( b d h) lm F Y ( b d h) h 0 4 If h0 F s contnuous at the pont (bd) n one of the varables then P b Y d (The proof s easy) Thus f P b Y d 0 h0 0 then F must be dscontnuous at the pont (bd) Hence f we want to locate the ponts (bd) at whch P b Y d 0 then we need only to consder those ponts at whch F s dscontnuous e those ponts at whch the functon F jumps 1 Whch of the followng functons do not represent a genune jont CDF? 1 exp x y x 0 and y 0 a FY x y 0 elsewhere b F x y c F x y 1 exp x y x 0 and y 0 0 elsewhere 1 x y0 0 elsewhere Suppose that the jont CDF of the random varables and Y s gven by: 0 x or y 5 3 x and 5 y 3 8 F 1 Y x y x and 5 y 3 1 x and y 3 1 x and y 3 Determne the ponts probabltes at these ponts bd for whch P b Y d 0 and evaluate the
Statstcs 1: Probablty Theory II 13 114 Classfcaton of Jont (Cumulatve) Dstrbuton Functons The classfcaton of the jont dstrbuton functons s carred out on the bass of the nature of the jont dstrbuton functon dscrete and (absolutely) contnuous However the two do not exhaust all the possble cases: t s possble to have bvarate dstrbutons whch are absolutely contnuous n one varable and dscrete n the other 1141 Dscrete Jont Dstrbutons Def n: Gven a probablty space A P a -dmensonal random vector 1 ' s defned to be a -dmensonal dscrete random vector f and only f t can assume values only at a countable number of ponts x x x n the - dmensonal Eucldean space Def n: If 1 ' s a -dmensonal dscrete random vector then the jont probablty mass functon (or jont PMF) or jont dscrete densty functon of the random varables 1 denoted by p or p s defned 1 as p x p 1 x 1 x x P 1 x1 x x ( )' s a possble value of ' where x x1 x x be 0 otherwse ( ) 0 1 (jont probablty functon n general even for later on contnuous random vectors) 1 ; and s defned to 4 The collecton of all ponts x ( x1 x x )' for whch the jont PMF s strctly greater p ( x) p x x x 0 s called the set of mass ponts than zero e 1 1 of the dscrete random vector 1 ' Remars: 1 The random varables 1 are referred to as jontly dscrete random varables Alternatvely we can defne the jont PMF p ( x) p 1 x1 x x functon from nto the nterval 01 satsfyng the followng: a 1 b p x p x x x as a p ( x) p x x x 0 x ( x x x )' 1 1 ( ) 1 1 1 where the summaton s taen over the set of mass ponts of 3 As n the unvarate case we can obtan the jont CDF of a dscrete random vector from ts jont PMF and vce versa For the bvarate case we have
Statstcs 1: Probablty Theory II 14 FY ( u v) P( u Y v) P( x Y y) xu yv py ( a b) P( a Y b) F ( a b) F ( a b) F ( a b ) F ( a b ) Y Y Y Y Theorem: If and Y are jontly dscrete random varables then nowledge of F Y ( ) s equvalent to nowledge of p Y ( ) Also the statement extends to -dmensonal dscrete random vector Proof: (Assgnment) 1 Suppose that we are to toss a balanced con twce and we defne the followng random varables: = the number of heads on the 1 st toss Y = the number of heads on the nd toss What s the jont PMF of and Y? Consder the experment of tossng two tetrahedra each wth sdes labeled 1 to 4 Defne the followng varables: = the number on the downturned face of the 1 st tetrahedron Y = the larger of the two downturned numbers What s the jont PMF of and Y? 3 Suppose that the jont PMF of the random varables and Y s gven by the table below Fnd the jont CDF of and Y What s the probablty that Y? Y 0 1 3 5-1 1/1 /1 1/1 0 3 /1 1/1 0 1/1 4 0 1/1 /1 1/1 4 Suppose that the jont PMF of the random varables and Y s gven by the formula: P( x Y y) ( x y ) x 1013 and y 13 Fnd the value of the constant What s the probablty that 0? Y
Statstcs 1: Probablty Theory II 15 114 Some Specal (Dscrete) Multvarate Dstrbutons A Multnomal Dstrbuton Def n: A Multnomal experment s one that possesses the followng propertes: a The experment conssts of n repeated trals b Each tral can result n any one of the (+1) dstnct possble outcomes denoted by E1 E E 1 c The probablty of the th possble outcome E s P( E ) p 1 1 d The repeated trals are ndependent Def n: If n a Multnomal experment denotes the number of trals (out of n) that result n the outcome E 1 1 then the random varables 1 (excludng 1 ) are sad to have a Multnomal dstrbuton wth jont PMF: n x1 x x1 p1 ( x1 x x ; 1 ) p p p n p1 p p 1 x1 x x 1 wth 1 1 x n and p 1 1 1 Remars: 1 The Bnomal experment becomes a Multnomal experment f each tral can have more than possble outcomes The PMF of the Multnomal Dstrbuton gves the probablty that the outcome E 1 occurs exactly x 1 tmes the outcome E occurs exactly x tmes and so on up to the (+1) th outcome occurs exactly x 1 tmes n the n ndependent trals 3 The Multnomal dstrbuton has (+1) parameters: n p1 p p The quantty p 1 le q n the Bnomal dstrbuton s exactly determned by p 1 1 p 1 p p 4 If 1 has a Multnomal dstrbuton then each s Bnomally dstrbuted e B( n p ) 1 If a par of dce s tossed sx tmes what s the probablty of obtanng a total of 7 or 11 twce and a matchng par once? A certan devce can fal n any one of three possble mutually exclusve ways The probablty that t wll fal n the 1 st way s p1 03 and the probablty that t wll fal n the nd way s p 05 Ten devces are receved for servce The devces can be
Statstcs 1: Probablty Theory II 16 assumed to have faled ndependently of each other What s the probablty that there wll be 4 falures of the 1 st nd and 3 falures of the nd nd? B Generalzed Hypergeometrc Dstrbuton Def n: A generalzed Hypergeometrc experment s one that possesses the followng propertes: a A sample of sze n s taen randomly wthout replacement from a populaton wth N elements b N 1 of the N populaton elements are of 1 nd N of the N populaton elements are of a nd nd and so on and the N +1 of the N populaton elements are of a (+1) th nd Def n: If n a generalzed Hypergeometrc experment denotes the number of sample th elements that are of the nd 1 1 then the random varables 1 (excludng 1) are sad to have a generalzed Hypergeometrc dstrbuton wth jont PMF: N1 N N 1 x1 x x 1 p1 ( x1 x x ; 1 ) N N N N n N n wth 1 1 x n and N N 1 1 Remars: 1 The Hypergeometrc experment n whch each element of the populaton can be classfed as ether a success or falure becomes a generalzed Hypergeometrc experment f each element of the populaton can be classfed nto more than nds The PMF of the generalzed Hypergeometrc dstrbuton gves the probablty that n the sample there wll be exactly x 1 samples of the 1 st nd exactly x sample elements of the nd nd and so on and exactly x 1 sample elements of the (+1) th nd 3 The generalzed Hypergeometrc dstrbuton has (+) parameters: N N1 N N n The quantty N 1 s exactly determned by N 1 N N 1 N N 4 If 1 has a generalzed Hypergeometrc dstrbuton then each s unvarate Hypergeometrc e Hyp( N n N ) 1 Two reflls of a ballpont pen are selected at random from a box that contans 3 blue red and 3 green reflls If denotes the number of blue reflls selected and Y denotes the
Statstcs 1: Probablty Theory II 17 number of red reflls selected fnd the jont PMF of and Y Fnd the probablty that the total number of blue and red reflls selected s less than Three cards are drawn wthout replacement from the 1 face cards (jacs queens or ngs) of an ordnary dec of playng cards If denotes the number of ngs selected and Y denotes the number of jacs selected fnd the jont PMF of and Y Fnd the probablty that the total number of ngs and jacs selected s at least 1143 (Absolutely) Contnuous Jont Dstrbutons Def n: Gven a probablty space A P a -dmensonal random vector 1 ' a postve nteger s defned to be a (-dmensonal absolutely) contnuous random vector f and only f there exsts a nonnegatve functon denoted f ( ) or f () such that for any x x1 x x ' 1 x x1 x1 ( ) 1 ( 1 ) 1 F x f u u u du du du Def n: If ' s a -dmensonal contnuous random vector then the 1 nonnegatve functon f 1 ( ) s called the jont probablty densty functon (or jont PDF) of the random varables 1 Remars: 1 The random varables 1 are referred to as jontly (absolutely) contnuous random varables Alternatvely we can defne the jont PDF f ( x) f 1 ( x1 x x ) as a functon from nto the postve real lne satsfyng the followng: f ( ) f ( x x x ) 0 x ( x x x ) a x 1 1 1 f ( x x x ) dx dx dx 1 b 1 1 1 3 As n the unvarate case we can obtan the jont CDF of a contnuous random vector from ts jont PDF and vce versa For the bvarate case we have F ( u v ) P ( u v u Y v ) f ( x y ) dxdy f F ( x y) ( x y) xy ponts (xy) where F s dfferentable
Statstcs 1: Probablty Theory II 18 v u 4 The followng results are mmedate from F Y ( u v) f Y ( x y) dxdy a f ( x y) dxdy 1 Proof: Y b P ( a b d b c Y d ) f ( x y ) dxdy c a Proof: : 5 Whle probabltes are represented as areas n the case of a contnuous unvarate random varable for a contnuous bvarate random vector probabltes are represented as volumes That s If s a contnuous random varable wth PDF f then P( a b) can be vewed as the area under the curve f ( x ) above the -axs and between the ponts a and b Smlarly f ( )' s a bvarate contnuous random vector wth jont PDF f then P( a b c Y d) can be vewed as the volume under the surface (or plane) f ( x y ) above the -Y plane and wthn the rectangle wth vertces a b c and d Y 6 The jont PDF represents the lmt of the rato of the amount of probablty n a rectangle to the area of the rectangle as the area goes to zero (e as the sdes of the rectangle shrn to zero) Hence the jont PDF reflects how densely the probablty mass s spread over the plane ( ) x y F x y F( x h y ) F( x y ) F( x h y) F( x y) = lm 0 0 h = h P( x x h y Y y ) lm 0 0 h h 7 The jont PDF does not represent a probablty e f ( x y) P( x Y y) The jont PDF f ( x y ) gves the heght (or smply the value) of the functon f at the pont ( ) xy 8 If h and are small then we have P(x<x+h y<yy+) = h f(x y) Y
Statstcs 1: Probablty Theory II 19 1 Let and Y be jontly contnuous random varables wth jont PDF gven by: f ( x y) exp x y I ( x) I ( y) (0 ) (0 ) a Determne the jont CDF F b Fnd P( Y ) c Fnd P( Y) d Fnd P( 1 Y ) Let and Y be jontly contnuous random varables wth jont PDF gven by: f ( x y) ( x y) I ( x) I ( y) (01) (01) a Fnd the constant b Fnd P 1 Y 1 4 3 Two random varables and Y are sad to be jontly unformly dstrbuted over (01) f and only f ther jont PDF s gven by: f ( x y) I ( x) I ( y) (01) (01) a Fnd P( Y 1) b Fnd P(1 3 Y 3 ) c Fnd P( Y) 4 A candy company dstrbutes boxes of chocolates wth a mxture of creams toffees and cordals Suppose that the weght of each box s 1 logram but the ndvdual weghts of the creams toffees and cordals vary from box to box For a randomly selected box let and Y represent the weghts of the creams and the toffees respectvely and suppose that and Y are jontly contnuous random varables wth the jont PDF gven by: f ( x y) 4 xyi ( x) I ( y) x y 1 [01] [01] Fnd the probablty that for a gven box the cordals account more than half the weght of the box 5 Let and Y be jontly contnuous random varables wth jont PDF gven by: f ( x y) 4 xyi ( x) I ( y) (01) (01) Fnd the jont CDF of and Y
Statstcs 1: Probablty Theory II 0 6 Let and Y be jontly contnuous random varables wth jont PDF gven by: f ( x y) 4 y(1 x) I ( y) I ( x) Y (0 x) (01) a Fnd P( Y) b Fnd P( Y) c Fnd P(1/ 3 1/ Y 1/ ) d Fnd PY ( 1/ ) e Fnd P(1/ 3 1/ ) f Fnd P(1/ 3 1/ ) ( Y 1/ ) 7 Let and Y be jontly contnuous random varables wth jont CDF gven by: 0 x 0 or y 0 1 ( x y xy ) 0 x1 0 y1 1 FY ( x y) ( x x) 0 x 1 y 1 1 x1 0 y1 ( y y) 1 x1 x1 Fnd the jont PDF of and Y 1144 Some Specal (Contnuous) Multvarate Dstrbutons A Bvarate Unform Dstrbuton Def n: A bvarate contnuous random vector ( )' s sad to have a Bvarate Unform dstrbuton over the regon A f and only f the jont PDF of and Y s gven by: f ( x y) ( x y) A 0 otherwse where 1 area of A Remars: 1 The constant s chosen so that the total volume under the surface f and above the Y Cartesan plane s equal to 1 e f ( x y) dxdy 1 The constant gves the heght of the functon f at a partcular pont ( x y) A Geometrcally the Bvarate Unform dstrbuton over the regon A can be represented by the (flat) surface Z f ( x y) whch generates a cylndrcal sold wth the Y regon A as ts base as ts heght and the plane Z= as ts top
Statstcs 1: Probablty Theory II 1 3 The regon A s a subset of or any regon n the -Y plane It may tae any shape but commonly t s a rectangle n the -Y plane 1 Let and Y be two contnuous random varables that are jontly Unformly dstrbuted over the unt square What s the jont PDF of and Y? Fnd P( Y 1/ ) Let and Y be jontly contnuous random varables wth jont PDF gven below Fnd P( 31 Y 3/ ) f 1 ( x y) I(4) ( x) I(1) ( y) 3 Let and Y be jontly contnuous random varables havng a Bvarate Unform dstrbuton over a crcle centered at the orgn and wth radus equal to 1 a Fnd the jont PDF of and Y b Fnd P(0 1/ 0 Y 1/ ) c Fnd P( Y) B Bvarate Normal Dstrbuton Def n: A bvarate contnuous random vector ( )' s sad to have a Bvarate Normal dstrbuton f and only f the jont PDF of and Y s gven by: f 1 1 x x y Y y Y ( x y) exp (1 ) Y Y Y 1 where Y Y and are constants such that: 0 1 1 0 We wrte Y Y Y Y ( Y)' BVN ( ) Remars: 1 The total volume under the surface generated by the Bvarate Normal PDF f and Y above the -Y plane s equal to 1 e f ( x y) dxdy 1 For any pont ( xy ) the computed value of the jont PDF s the heght of the functon at that pont Geometrcally the Bvarate Normal dstrbuton can be represented by a bellshaped surface Z f ( x y) whch generates a bell-shaped sold floatng above the Y
Statstcs 1: Probablty Theory II Y plane A horzontal cross-secton of ths sold s an ellptc curve whle a vertcal cross-secton of the sold s a (unvarate) Normal curve 3 The Bvarate Normal dstrbuton has 5 parameters: Y Y and Example: Let and Y be jontly contnuous random varables havng a Bvarate Normal dstrbuton wth parameters Y 0 Y 1 and 0 Fnd the jont PDF of and Y What s the probablty that and Y are both less than? 1 Margnal Dstrbutons Def n: Gven a probablty space a -dmensonal random vector ( 1 )' wth jont dstrbuton F () or F 1 ( ) For some 1 the margnal (cumulatve) dstrbuton functon (or margnal CDF) of denoted F ( ) s defned as: F ( x ) lm F ( x x x x ) all x 1 1 j j F ( x ) 1 Remars: 1 For the bvarate case f and Y have jont CDF F the margnal dstrbutons (margnal CDFs) of and Y are gven by: F ( x) lm F ( x y) F ( x ) Y Y y F ( y) lm F ( x y) F ( y) Y Y Y x In general the jont dstrbuton of any sub-vector of a random vector 1 ' s obtaned from the dstrbuton of by allowng the arguments NOT correspondng to the sub-vector to tend to nfnty For nstance the jont dstrbuton of ( Z )' s obtaned from the jont dstrbuton of ( Y Z )' as: F ( x z) F ( x z) Z Y Z a Let 1 3 4 ' be a random vector wth jont CDF 1 3 4 F ( x ) F ( x ) 4 4 1 3 4 4 F ( x x ) F ( x x ) 4 4 1 3 4 4 F
Statstcs 1: Probablty Theory II 3 b Let 1 3 4 5 ' be a random vector wth jont CDF 1 3 4 5 F ( x ) F ( x ) 3 3 1 3 4 5 3 F ( x x ) F ( x x ) 1 3 1 3 1 3 4 5 1 3 F ( x x x ) F ( x x x ) 1 3 4 1 3 4 1 3 4 5 1 3 4 1 3 Although nowledge of the dstrbuton of F ' s suffcent to determne the jont dstrbuton of any sub-vector of ncludng the margnal dstrbutons of the s the converse s not true That s the margnal dstrbutons are unquely determned from the jont dstrbuton but nowledge of the margnal dstrbuton s NOT suffcent to determne the jont dstrbuton (See example below) 1 Fnd the margnal dstrbuton of the random varables and Y f ther jont dstrbuton s gven by: 0 x 0 or y 0 y y FY ( x y) 1 e e x y y 0 x ( x y) y 1 e e e x 0 y x Consder the followng jont dstrbutons and show that both wll gve rse to the same set of margnal dstrbutons of and Y 0 x 0 or y 0 6 1 3 xy x y 0 x 1 0 y 1 7 6 (1) 6 1 3 FY ( x y) x x 0 x 1 y 1 7 6 6 1 y y x 1 0 y 1 7 6 1 x1 y1 0 x 0 or y 0 36 1 3 1 x x y Y 0 x 1 0 y 1 49 6 6 () 6 1 3 FY ( x y) x x 0 x 1 y 1 7 6 6 1 y y x 1 0 y 1 7 6 1 x1 y1
Statstcs 1: Probablty Theory II 4 11 Dscrete Case Def n: Gven a -dmensonal dscrete random vector 1 ' wth jont PMF p () or p 1 ( ) For some 1 the margnal probablty mass functon (or margnal PMF) of denoted p () s defned as: p ( x ) p ( x x x x ) 1 1 where the summaton s taen wth respect to all j j Remars: 1 For the bvarate case f and Y have jont PMF p the margnal PMFs of and Y are gven by p ( u) p Y ( u y) all mass pts of Y py ( v) p Y ( x v) all mass pts of Thus for a dscrete random vector ( )' the margnal PMF of each random varable s found by summng the jont probabltes over all mass ponts of the random varable If the jont PMF of and Y s expressed as a table wth the rows and columns representng the mass ponts of and the mass ponts of Y respectvely the totals n the horzontal and vertcal margns wll rse to the margnal PMFs Hence the name margnal dstrbutons Y Y y1 x 1 1 1 x Y y j Y y p ( x ) p ( x y ) py ( x1 y ) p Y ( x1 y j ) p ( x 1) 1 p ( x y ) py ( x y ) p Y ( x y j ) p ( x ) x Y 1 p ( x y ) p Y ( x y ) p Y ( x y j ) p ( x ) py ( y ) py ( y 1) py ( y ) py( y j) 1 In general the jont PMF of any sub-vector of a dscrete random vector s obtaned from the dstrbuton of by summng (over the set of mass ponts) the jont PMF wth respect to the varables NOT correspondng to the sub-vector For nstance the jont PMF of ( Z )' s obtaned from the jont PMF of ( W Y Z )' as: p ( x z) p ( w x y z) Z W Y Z all y all w
Statstcs 1: Probablty Theory II 5 3 If ( 1 )' s a dscrete random vector the jont PMF of any sub-vector of ncludng the margnal PMFs of the ' s can be obtaned from the jont PMF of the entre random vector ( 1 )' but not conversely 1 Fnd the margnal PMFs of and Y f ther jont PMF s gven by: p ( ) ( x y) x y x 14 and y 1013 4 Consder the experment of tossng two tetrahedra (regular four sded polygon) each wth sdes labeled 1 to 4 Defne the followng varables: = the number on the downturned face of the 1 st tetrahedron Y = the larger of the two downturned numbers Recall the jont PMF of and Y s gven below Y Y 1 Y Y 3 Y 4 p ( x ) 1 1/16 1/16 1/16 1/16 0 /16 1/16 1/16 3 0 0 3/16 1/16 4 0 0 0 4/16 p ( y ) Y Fnd the margnal PMFs of and Y 3 Fnd the margnal PMFs of and Y assumng that and Y are jontly Trnomally dstrbuted wth parameters n p1 p 1 (Absolutely) Contnuous Case Def n: Gven a -dmensonal (absolutely) contnuous random vector ( 1 )' wth jont PDF f () or f ( ) 1 For some 1 the margnal probablty densty functon (or margnal PDF) of denoted f () s defned as : f ( x ) f ( x x x ) dx dx dx dx dx 1 1 1 1 1 where the ntegrals are taen over the entre real lne
Statstcs 1: Probablty Theory II 6 Remars: 1 For the bvarate case f and Y have jont PDF f the margnal PDFs of and Y are gven by: f ( x) f Y ( x y) dy fy ( y) f Y ( x y) dx Thus for a contnuous random vector ( )' the margnal PDF of each random varable s found by ntegratng (over ) the jont PDF wth respect to the other random varable In general the jont PDF of any sub-vector of a contnuous random vector s obtaned from the dstrbuton of by ntegratng (over ) the jont PDF wth respect to the varables NOT correspondng to the sub-vector For nstance the jont PDF of ( Z )' s obtaned from the jont PDF of ( W Y Z )' as: f ( x z) f ( w x y z) dwdy Z W Y Z 3 If ( 1 )' s a contnuous random vector the jont PDF of any sub-vector of ncludng the margnal PDFs of the ' s can be obtaned from the jont PDF of the entre random vector ( 1 )' but not conversely 4 If and Y are jontly contnuous random varables then and Y are each contnuous But f and Y are contnuous the vector ( )' may not be jontly contnuous 1 Fnd the margnal PDFs of and Y f ther jont PDF s gven by: f ( x y) ( x y) I ( x) I ( y) (01) (01) Fnd the margnal PDFs of and Y f ther jont PDF s gven by: f ( x y) I ( y) I ( x) Y ( x x) (01) 3 Fnd the margnal PDFs of and Y f ther jont PDF s a Bvarate Normal Dstrbuton wth parameters 0 1 and Y Y 4 Fnd the margnal PDFs of and Y f ther jont PDF s gven by: 5 x y 0 x 1 and x y f ( x y) 6 x 0 x 1 and x y 0 ow 5 Fnd the margnal PDFs of and Y and P ( 1/ ) f ther jont PDF s gven by: f ( x y) 6/ 7 1 x y I ( x) I ( y) (01) (01)
Statstcs 1: Probablty Theory II 7 6 Fnd the margnal PDFs of and Y and PY ( 3/ 4) f ther jont PDF s gven by: f ( x y) 10 xy 0 0 y x1 ow 7 Consder the followng jont PDFs and show that both wll gve rse to the same set of margnal PDFs for and Y (1) f ( x y) 1/ 4 1 xy I ( x) I ( y) [ 11] [ 11] () [ 11] [ 11] f ( x y) 1/ 4 I ( x) I ( y)