A be a probability space. A random vector

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1 Statstcs 1: Probablty Theory II 8 1 JOINT AND MARGINAL DISTRIBUTIONS I Probablty Theory I we formulate the cocept of a (real) radom varable ad descrbe the probablstc behavor of ths radom varable by the dstrbutos that t duces o the real le I that developmet atteto s drected to a sgle umber assocated wth each outcome of the basc expermet That s the problem focuses o a sgle radom varable that descrbes the outcome of the expermet I the study of may radom expermets there are or ca be more tha oe radom varable of terest The outcome of a expermet mght be best descrbed usg several umbers ad thus there may be a eed to defe two or more radom varables These umbers may be vewed separately as values of the dvdual radom varables but they may be cosdered jotly as elemets of a radom vector As the case of a sgle radom varable we use fudametal mappg deas to arrve at a jot dstrbuto that descrbes the probablstc behavor of the radom vector as well as the probablstc relatos amog the radom varables the radom vector 11 Jot Dstrbutos 111 The Noto of a Radom Vector Def : Let P A be a probablty space A radom vector fucto wth doma ad couterdoma that s ' s a 1 ( 1 )': such that for ay set of real umbers say x1 x x : 1 x1 x x A Def : I the bvarate case e for = for a probablty space A P we defe a bvarate radom vector ' to be a fucto wth doma ad couterdoma that s ( )': such that for ay par of real umbers say u ad v : ( ) u ( ) v A How s t defed o a Borel set B of the plae? A y A (() ()) x A = {: (() ()) B}

2 Statstcs 1: Probablty Theory II 9 As the oe-dmesoal case we may map the probablty from the evets o the basc space of the Borel subsets o For example the dagram above we assg to rego B (a Borel set) o the plae the probablty mass evet A I a geeral settg suppose ad are two real valued fuctos defed o the probablty space A P For ay pot the values of ad ca be represeted as a ordered par ( ) ( ) ' so that ths defes a fucto from to Defg ths (jot) fucto of ad as ' we have a fucto ( ( ) ( ) ' For ths fucto ' be a bvarate radom vector t must be that for every Borel set B (a rego )': whch maps every to or a pot or a terval) the Cartesa plae beg a bvarate radom vector defed over P Also ' probablty measure Borel sets e : ( ) ( ) ' B A A duces a P whch traslates probabltes o evets to probabltes o : ( ) ( ) ' P B P B Remars: 1 The fucto ' duces a probablty measure o the Borel subsets B of the Cartesa plae Ths probablty measure deoted by P s a fucto wth doma B ( )ad couterdoma the terval 01 that s P : B( ) [01] ad s defed as P ( B) P : ( ' B A smlar probablty measure may be defed for the case of a radom vector wth three or more radom varables However t wll volve more complcated otatos ad the geometrc vsualzatos wll be more dffcult 3 For coveece we shall use the otato ' : ( ' B Example: u v : ( ) u ( ) v Example: B to represet

3 Statstcs 1: Probablty Theory II Defto of a Jot Dstrbuto Fucto Def : Let 1 be radom varables all defed o the same probablty space A P The jot cumulatve dstrbuto fucto (or jot CDF) or smply jot dstrbuto fucto of the radom varables 1 deoted by F s defed as F 1 or smply 1 F x x x P x x x x x ) 1 x ( 1 x Remars: 1 Thus a jot CDF s a fucto wth doma the -dmesoal Eucldea space couterdoma the terval 01 F or For = the jot CDF F has doma ad couterdoma ad 01 I ths case the jot CDF s smply a exteso of the oto of a dstrbuto fucto the uvarate case Whereas we use the fte terval x to defe the CDF the uvarate case we shall use the fte rectagle 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 P ) ( ) ' B P ) u ( ) v P u v uv B uv the bvarate case 1 Suppose we are to toss a balaced co twce ad we defe the followg radom varables: = umber of heads o the 1 st toss = umber of heads o the d toss What s the jot CDF of ad? Cosder the expermet of tossg two tetrahedra (regular 4-sded polyhedro) each wth sdes labeled 1 to 4 Defe the followg radom varables: = the umber o the dowtured face of the 1 st tetrahedro = the larger of the two dowtured umbers What s the jot CDF of ad?

4 Statstcs 1: Probablty Theory II Propertes of Jot Dstrbuto Fuctos Let F be the jot CDF of two radom varables ad 1 Boudedess: F v lm F u v 0 Thus Proof: u F u lm F u v 0 v F lm F u v 1 u v ( u v) R F u v 0 1 Mootocty: If a b c d are ay real umbers a b c d R a b ad c d the P a b c d F b d F b c F a d F a c 0 Note: The result s aalogous to P(a< b) = F(b) F(a) the uvarate case Proof: 3 Cotuty from the Rght: F s cotuous from the rght each of the varables That s for ay fxed x F s cotuous from the rght ad for ay fxed y the rght That s lm F u h v F u v F u v Proof: h0 h0 lm F u v h F u v F u v Remars: 1 The boudedess property does ot mply that F v lm F u v 1 or u F u lm F u v 1 v Both argumets must be fte for F to coverge to 1 F s cotuous from The mootocty property s ot a strct mootocty However F s a mootoe o-decreasg fucto each of the varables That s for ay fxed x F s o-decreasg ad for ay fxed y F s o-decreasg For a fxed x y y F x y F x y Proof: For a fxed 1 1 y x x F x y F x y 1 1

5 Statstcs 1: Probablty Theory II 1 3 The rght cotuty property mples the ff: For ay fxed costats a b c d R a b ad c d P a bc d F bd F bc F a d F a c P a bc d F b d F b c F a d F a c P a bc d F b d F b c F ad F ac P b d F bd F b d F bd F b d Note that F ( b d ) lm F ( b d h) lm F ( b d h) h 0 4 If h0 F s cotuous at the pot (bd) oe of the varables the P b d (The proof s easy) Thus f P b d 0 h0 0 the F must be dscotuous at the pot (bd) Hece f we wat to locate the pots (bd) at whch P b d 0 the we eed oly to cosder those pots at whch F s dscotuous e those pots at whch the fucto F jumps 1 Whch of the followg fuctos do ot represet a geue jot CDF? 1 exp x y x 0 ad y 0 a F x y 0 elsewhere b F x y c F x y 1 exp x y x 0 ad y 0 0 elsewhere 1 x y0 0 elsewhere Suppose that the jot CDF of the radom varables ad s gve by: 0 x or y 5 3 x ad 5 y 3 8 F 1 x y x ad 5 y 3 1 x ad y 3 1 x ad y 3 Determe the pots probabltes at these pots bd for whch P b d 0 ad evaluate the

6 Statstcs 1: Probablty Theory II Classfcato of Jot (Cumulatve) Dstrbuto Fuctos The classfcato of the jot dstrbuto fuctos s carred out o the bass of the ature of the jot dstrbuto fucto dscrete ad (absolutely) cotuous However the two do ot exhaust all the possble cases: t s possble to have bvarate dstrbutos whch are absolutely cotuous oe varable ad dscrete the other 1141 Dscrete Jot Dstrbutos Def : Gve a probablty space A P a -dmesoal radom vector 1 ' s defed to be a -dmesoal dscrete radom vector f ad oly f t ca assume values oly at a coutable umber of pots x x x the - dmesoal Eucldea space Def : If 1 ' s a -dmesoal dscrete radom vector the the jot probablty mass fucto (or jot PMF) or jot dscrete desty fucto of the radom varables 1 deoted by p or p s defed 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 (jot probablty fucto geeral eve for later o cotuous radom vectors) 1 ; ad s defed to 4 The collecto of all pots x ( x1 x x )' for whch the jot PMF s strctly greater p ( x) p x x x 0 s called the set of mass pots tha zero e 1 1 of the dscrete radom vector 1 ' Remars: 1 The radom varables 1 are referred to as jotly dscrete radom varables Alteratvely we ca defe the jot PMF p ( x) p 1 x1 x x fucto from to the terval 01 satsfyg the followg: a 1 b p x p x x x as a p ( x) p x x x 0 x ( x x x )' 1 1 ( ) where the summato s tae over the set of mass pots of 3 As the uvarate case we ca obta the jot CDF of a dscrete radom vector from ts jot PMF ad vce versa For the bvarate case we have

7 Statstcs 1: Probablty Theory II 14 F ( u v) P( u v) P( x y) xu yv p ( a b) P( a b) F ( a b) F ( a b) F ( a b ) F ( a b ) Theorem: If ad are jotly dscrete radom varables the owledge of F ( ) s equvalet to owledge of p ( ) Also the statemet exteds to -dmesoal dscrete radom vector Proof: (Assgmet) 1 Suppose that we are to toss a balaced co twce ad we defe the followg radom varables: = the umber of heads o the 1 st toss = the umber of heads o the d toss What s the jot PMF of ad? Cosder the expermet of tossg two tetrahedra each wth sdes labeled 1 to 4 Defe the followg varables: = the umber o the dowtured face of the 1 st tetrahedro = the larger of the two dowtured umbers What s the jot PMF of ad? 3 Suppose that the jot PMF of the radom varables ad s gve by the table below Fd the jot CDF of ad What s the probablty that? /1 /1 1/1 0 3 /1 1/1 0 1/ /1 /1 1/1 4 Suppose that the jot PMF of the radom varables ad s gve by the formula: P( x y) ( x y ) x 1013 ad y 13 Fd the value of the costat What s the probablty that 0?

8 Statstcs 1: Probablty Theory II Some Specal (Dscrete) Multvarate Dstrbutos A Multomal Dstrbuto Def : A Multomal expermet s oe that possesses the followg propertes: a The expermet cossts of repeated trals b Each tral ca result ay oe of the (+1) dstct possble outcomes deoted 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 depedet Def : If a Multomal expermet deotes the umber of trals (out of ) that result the outcome E 1 1 the the radom varables 1 (excludg 1 ) are sad to have a Multomal dstrbuto wth jot PMF: x1 x x1 p1 ( x1 x x ; 1 ) p p p p1 p p 1 x1 x x 1 wth 1 1 x ad p Remars: 1 The Bomal expermet becomes a Multomal expermet f each tral ca have more tha possble outcomes The PMF of the Multomal Dstrbuto gves the probablty that the outcome E 1 occurs exactly x 1 tmes the outcome E occurs exactly x tmes ad so o up to the (+1) th outcome occurs exactly x 1 tmes the depedet trals 3 The Multomal dstrbuto has (+1) parameters: p1 p p The quatty p 1 le q the Bomal dstrbuto s exactly determed by p 1 1 p 1 p p 4 If 1 has a Multomal dstrbuto the each s Bomally dstrbuted e B( p ) 1 If a par of dce s tossed sx tmes what s the probablty of obtag a total of 7 or 11 twce ad a matchg par oce? A certa devce ca fal ay oe of three possble mutually exclusve ways The probablty that t wll fal the 1 st way s p1 03 ad the probablty that t wll fal the d way s p 05 Te devces are receved for servce The devces ca be

9 Statstcs 1: Probablty Theory II 16 assumed to have faled depedetly of each other What s the probablty that there wll be 4 falures of the 1 st d ad 3 falures of the d d? B Geeralzed Hypergeometrc Dstrbuto Def : A geeralzed Hypergeometrc expermet s oe that possesses the followg propertes: a A sample of sze s tae radomly wthout replacemet from a populato wth N elemets b N 1 of the N populato elemets are of 1 d N of the N populato elemets are of a d d ad so o ad the N +1 of the N populato elemets are of a (+1) th d Def : If a geeralzed Hypergeometrc expermet deotes the umber of sample th elemets that are of the d 1 1 the the radom varables 1 (excludg 1) are sad to have a geeralzed Hypergeometrc dstrbuto wth jot PMF: N1 N N 1 x1 x x 1 p1 ( x1 x x ; 1 ) N N N N N wth 1 1 x ad N N 1 1 Remars: 1 The Hypergeometrc expermet whch each elemet of the populato ca be classfed as ether a success or falure becomes a geeralzed Hypergeometrc expermet f each elemet of the populato ca be classfed to more tha ds The PMF of the geeralzed Hypergeometrc dstrbuto gves the probablty that the sample there wll be exactly x 1 samples of the 1 st d exactly x sample elemets of the d d ad so o ad exactly x 1 sample elemets of the (+1) th d 3 The geeralzed Hypergeometrc dstrbuto has (+) parameters: N N1 N N The quatty N 1 s exactly determed by N 1 N N 1 N N 4 If 1 has a geeralzed Hypergeometrc dstrbuto the each s uvarate Hypergeometrc e Hyp( N N ) 1 Two reflls of a ballpot pe are selected at radom from a box that cotas 3 blue red ad 3 gree reflls If deotes the umber of blue reflls selected ad deotes the

10 Statstcs 1: Probablty Theory II 17 umber of red reflls selected fd the jot PMF of ad Fd the probablty that the total umber of blue ad red reflls selected s less tha Three cards are draw wthout replacemet from the 1 face cards (jacs quees or gs) of a ordary dec of playg cards If deotes the umber of gs selected ad deotes the umber of jacs selected fd the jot PMF of ad Fd the probablty that the total umber of gs ad jacs selected s at least 1143 (Absolutely) Cotuous Jot Dstrbutos Def : Gve a probablty space A P a -dmesoal radom vector 1 ' a postve teger s defed to be a (-dmesoal absolutely) cotuous radom vector f ad oly f there exsts a oegatve fucto deoted f ( ) or f () such that for ay x x1 x x ' 1 x x1 x1 ( ) 1 ( 1 ) 1 F x f u u u du du du Def : If ' s a -dmesoal cotuous radom vector the the 1 oegatve fucto f 1 ( ) s called the jot probablty desty fucto (or jot PDF) of the radom varables 1 Remars: 1 The radom varables 1 are referred to as jotly (absolutely) cotuous radom varables Alteratvely we ca defe the jot PDF f ( x) f 1 ( x1 x x ) as a fucto from to the postve real le satsfyg the followg: f ( ) f ( x x x ) 0 x ( x x x ) a x f ( x x x ) dx dx dx 1 b As the uvarate case we ca obta the jot CDF of a cotuous radom vector from ts jot PDF ad vce versa For the bvarate case we have F ( u v ) P ( u v u v ) f ( x y ) dxdy f F ( x y) ( x y) xy pots (xy) where F s dfferetable

11 Statstcs 1: Probablty Theory II 18 v u 4 The followg results are mmedate from F ( u v) f ( x y) dxdy a f ( x y) dxdy 1 Proof: b P ( a b d b c d ) f ( x y ) dxdy c a Proof: : 5 Whle probabltes are represeted as areas the case of a cotuous uvarate radom varable for a cotuous bvarate radom vector probabltes are represeted as volumes That s If s a cotuous radom varable wth PDF f the P( a b) ca be vewed as the area uder the curve f ( x ) above the -axs ad betwee the pots a ad b Smlarly f ( )' s a bvarate cotuous radom vector wth jot PDF f the P( a b c d) ca be vewed as the volume uder the surface (or plae) f ( x y ) above the - plae ad wth the rectagle wth vertces a b c ad d 6 The jot PDF represets the lmt of the rato of the amout of probablty a rectagle to the area of the rectagle as the area goes to zero (e as the sdes of the rectagle shr to zero) Hece the jot PDF reflects how desely the probablty mass s spread over the plae ( ) 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 ) lm 0 0 h h 7 The jot PDF does ot represet a probablty e f ( x y) P( x y) The jot PDF f ( x y ) gves the heght (or smply the value) of the fucto f at the pot ( ) xy 8 If h ad are small the we have P(x<x+h y<y+) = h f(x y)

12 Statstcs 1: Probablty Theory II 19 1 Let ad be jotly cotuous radom varables wth jot PDF gve by: f ( x y) exp x y I ( x) I ( y) (0 ) (0 ) a Determe the jot CDF F b Fd P( ) c Fd P( ) d Fd P( 1 ) Let ad be jotly cotuous radom varables wth jot PDF gve by: f ( x y) ( x y) I ( x) I ( y) (01) (01) a Fd the costat b Fd P Two radom varables ad are sad to be jotly uformly dstrbuted over (01) f ad oly f ther jot PDF s gve by: f ( x y) I ( x) I ( y) (01) (01) a Fd P( 1) b Fd P(1 3 3 ) c Fd P( ) 4 A cady compay dstrbutes boxes of chocolates wth a mxture of creams toffees ad cordals Suppose that the weght of each box s 1 logram but the dvdual weghts of the creams toffees ad cordals vary from box to box For a radomly selected box let ad represet the weghts of the creams ad the toffees respectvely ad suppose that ad are jotly cotuous radom varables wth the jot PDF gve by: f ( x y) 4 xyi ( x) I ( y) x y 1 [01] [01] Fd the probablty that for a gve box the cordals accout more tha half the weght of the box 5 Let ad be jotly cotuous radom varables wth jot PDF gve by: f ( x y) 4 xyi ( x) I ( y) (01) (01) Fd the jot CDF of ad

13 Statstcs 1: Probablty Theory II 0 6 Let ad be jotly cotuous radom varables wth jot PDF gve by: f ( x y) 4 y(1 x) I ( y) I ( x) (0 x) (01) a Fd P( ) b Fd P( ) c Fd P(1/ 3 1/ 1/ ) d Fd P ( 1/ ) e Fd P(1/ 3 1/ ) f Fd P(1/ 3 1/ ) ( 1/ ) 7 Let ad be jotly cotuous radom varables wth jot CDF gve by: 0 x 0 or y 0 1 ( x y xy ) 0 x1 0 y1 1 F ( x y) ( x x) 0 x 1 y 1 1 x1 0 y1 ( y y) 1 x1 x1 Fd the jot PDF of ad 1144 Some Specal (Cotuous) Multvarate Dstrbutos A Bvarate Uform Dstrbuto Def : A bvarate cotuous radom vector ( )' s sad to have a Bvarate Uform dstrbuto over the rego A f ad oly f the jot PDF of ad s gve by: f ( x y) ( x y) A 0 otherwse where 1 area of A Remars: 1 The costat s chose so that the total volume uder the surface f ad above the Cartesa plae s equal to 1 e f ( x y) dxdy 1 The costat gves the heght of the fucto f at a partcular pot ( x y) A Geometrcally the Bvarate Uform dstrbuto over the rego A ca be represeted by the (flat) surface Z f ( x y) whch geerates a cyldrcal sold wth the rego A as ts base as ts heght ad the plae Z= as ts top

14 Statstcs 1: Probablty Theory II 1 3 The rego A s a subset of or ay rego the - plae It may tae ay shape but commoly t s a rectagle the - plae 1 Let ad be two cotuous radom varables that are jotly Uformly dstrbuted over the ut square What s the jot PDF of ad? Fd P( 1/ ) Let ad be jotly cotuous radom varables wth jot PDF gve below Fd P( 31 3/ ) f 1 ( x y) I(4) ( x) I(1) ( y) 3 Let ad be jotly cotuous radom varables havg a Bvarate Uform dstrbuto over a crcle cetered at the org ad wth radus equal to 1 a Fd the jot PDF of ad b Fd P(0 1/ 0 1/ ) c Fd P( ) B Bvarate Normal Dstrbuto Def : A bvarate cotuous radom vector ( )' s sad to have a Bvarate Normal dstrbuto f ad oly f the jot PDF of ad s gve by: f 1 1 x x y y ( x y) exp (1 ) 1 where ad are costats such that: We wrte ( )' BVN ( ) Remars: 1 The total volume uder the surface geerated by the Bvarate Normal PDF f ad above the - plae s equal to 1 e f ( x y) dxdy 1 For ay pot ( xy ) the computed value of the jot PDF s the heght of the fucto at that pot Geometrcally the Bvarate Normal dstrbuto ca be represeted by a bellshaped surface Z f ( x y) whch geerates a bell-shaped sold floatg above the

15 Statstcs 1: Probablty Theory II plae A horzotal cross-secto of ths sold s a ellptc curve whle a vertcal cross-secto of the sold s a (uvarate) Normal curve 3 The Bvarate Normal dstrbuto has 5 parameters: ad Example: Let ad be jotly cotuous radom varables havg a Bvarate Normal dstrbuto wth parameters 0 1 ad 0 Fd the jot PDF of ad What s the probablty that ad are both less tha? 1 Margal Dstrbutos Def : Gve a probablty space a -dmesoal radom vector ( 1 )' wth jot dstrbuto F () or F 1 ( ) For some 1 the margal (cumulatve) dstrbuto fucto (or margal CDF) of deoted F ( ) s defed 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 ad have jot CDF F the margal dstrbutos (margal CDFs) of ad are gve by: F ( x) lm F ( x y) F ( x ) y F ( y) lm F ( x y) F ( y) x I geeral the jot dstrbuto of ay sub-vector of a radom vector 1 ' s obtaed from the dstrbuto of by allowg the argumets NOT correspodg to the sub-vector to ted to fty For stace the jot dstrbuto of ( Z )' s obtaed from the jot dstrbuto of ( Z )' as: F ( x z) F ( x z) Z Z a Let ' be a radom vector wth jot CDF F ( x ) F ( x ) F ( x x ) F ( x x ) F

16 Statstcs 1: Probablty Theory II 3 b Let ' be a radom vector wth jot CDF F ( x ) F ( x ) F ( x x ) F ( x x ) F ( x x x ) F ( x x x ) Although owledge of the dstrbuto of F ' s suffcet to determe the jot dstrbuto of ay sub-vector of cludg the margal dstrbutos of the s the coverse s ot true That s the margal dstrbutos are uquely determed from the jot dstrbuto but owledge of the margal dstrbuto s NOT suffcet to determe the jot dstrbuto (See example below) 1 Fd the margal dstrbuto of the radom varables ad f ther jot dstrbuto s gve by: 0 x 0 or y 0 y y F ( x y) 1 e e x y y 0 x ( x y) y 1 e e e x 0 y x Cosder the followg jot dstrbutos ad show that both wll gve rse to the same set of margal dstrbutos of ad 0 x 0 or y xy x y 0 x 1 0 y (1) F ( x y) x x 0 x 1 y y y x 1 0 y x1 y1 0 x 0 or y x x y 0 x 1 0 y () F ( x y) x x 0 x 1 y y y x 1 0 y x1 y1

17 Statstcs 1: Probablty Theory II 4 11 Dscrete Case Def : Gve a -dmesoal dscrete radom vector 1 ' wth jot PMF p () or p 1 ( ) For some 1 the margal probablty mass fucto (or margal PMF) of deoted p () s defed as: p ( x ) p ( x x x x ) 1 1 where the summato s tae wth respect to all j j Remars: 1 For the bvarate case f ad have jot PMF p the margal PMFs of ad are gve by p ( u) p ( u y) all mass pts of p ( v) p ( x v) all mass pts of Thus for a dscrete radom vector ( )' the margal PMF of each radom varable s foud by summg the jot probabltes over all mass pots of the radom varable If the jot PMF of ad s expressed as a table wth the rows ad colums represetg the mass pots of ad the mass pots of respectvely the totals the horzotal ad vertcal margs wll rse to the margal PMFs Hece the ame margal dstrbutos y1 x x y j y p ( x ) p ( x y ) p ( x1 y ) p ( x1 y j ) p ( x 1) 1 p ( x y ) p ( x y ) p ( x y j ) p ( x ) x 1 p ( x y ) p ( x y ) p ( x y j ) p ( x ) p ( y ) p ( y 1) p ( y ) p( y j) 1 I geeral the jot PMF of ay sub-vector of a dscrete radom vector s obtaed from the dstrbuto of by summg (over the set of mass pots) the jot PMF wth respect to the varables NOT correspodg to the sub-vector For stace the jot PMF of ( Z )' s obtaed from the jot PMF of ( W Z )' as: p ( x z) p ( w x y z) Z W Z all y all w

18 Statstcs 1: Probablty Theory II 5 3 If ( 1 )' s a dscrete radom vector the jot PMF of ay sub-vector of cludg the margal PMFs of the ' s ca be obtaed from the jot PMF of the etre radom vector ( 1 )' but ot coversely 1 Fd the margal PMFs of ad f ther jot PMF s gve by: p ( ) ( x y) x y x 14 ad y Cosder the expermet of tossg two tetrahedra (regular four sded polygo) each wth sdes labeled 1 to 4 Defe the followg varables: = the umber o the dowtured face of the 1 st tetrahedro = the larger of the two dowtured umbers Recall the jot PMF of ad s gve below p ( x ) 1 1/16 1/16 1/16 1/16 0 /16 1/16 1/ /16 1/ /16 p ( y ) Fd the margal PMFs of ad 3 Fd the margal PMFs of ad assumg that ad are jotly Tromally dstrbuted wth parameters p1 p 1 (Absolutely) Cotuous Case Def : Gve a -dmesoal (absolutely) cotuous radom vector ( 1 )' wth jot PDF f () or f ( ) 1 For some 1 the margal probablty desty fucto (or margal PDF) of deoted f () s defed as : f ( x ) f ( x x x ) dx dx dx dx dx where the tegrals are tae over the etre real le

19 Statstcs 1: Probablty Theory II 6 Remars: 1 For the bvarate case f ad have jot PDF f the margal PDFs of ad are gve by: f ( x) f ( x y) dy f ( y) f ( x y) dx Thus for a cotuous radom vector ( )' the margal PDF of each radom varable s foud by tegratg (over ) the jot PDF wth respect to the other radom varable I geeral the jot PDF of ay sub-vector of a cotuous radom vector s obtaed from the dstrbuto of by tegratg (over ) the jot PDF wth respect to the varables NOT correspodg to the sub-vector For stace the jot PDF of ( Z )' s obtaed from the jot PDF of ( W Z )' as: f ( x z) f ( w x y z) dwdy Z W Z 3 If ( 1 )' s a cotuous radom vector the jot PDF of ay sub-vector of cludg the margal PDFs of the ' s ca be obtaed from the jot PDF of the etre radom vector ( 1 )' but ot coversely 4 If ad are jotly cotuous radom varables the ad are each cotuous But f ad are cotuous the vector ( )' may ot be jotly cotuous 1 Fd the margal PDFs of ad f ther jot PDF s gve by: f ( x y) ( x y) I ( x) I ( y) (01) (01) Fd the margal PDFs of ad f ther jot PDF s gve by: f ( x y) I ( y) I ( x) ( x x) (01) 3 Fd the margal PDFs of ad f ther jot PDF s a Bvarate Normal Dstrbuto wth parameters 0 1 ad 4 Fd the margal PDFs of ad f ther jot PDF s gve by: 5 x y 0 x 1 ad x y f ( x y) 6 x 0 x 1 ad x y 0 ow 5 Fd the margal PDFs of ad ad P ( 1/ ) f ther jot PDF s gve by: f ( x y) 6/ 7 1 x y I ( x) I ( y) (01) (01)

20 Statstcs 1: Probablty Theory II 7 6 Fd the margal PDFs of ad ad P ( 3/ 4) f ther jot PDF s gve by: f ( x y) 10 xy 0 0 y x1 ow 7 Cosder the followg jot PDFs ad show that both wll gve rse to the same set of margal PDFs for ad (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)

21 Statstcs 1: Probablty Theory II 8 CONDITIONAL DISTRIBUTIONS AND STOCHASTIC INDEPENDENCE I Probablty Theory I a codtoal probablty represets the lelhood of occurrece of a evet gve that aother evet has already occurred I such a case codtoal probabltes provde a way of updatg the formato about a evet gve the occurrece of aother evet Moreover f the lelhood of occurrece of oe evet s ot affected by the occurrece of aother evet the evets are sad to be depedet The cocept of codtoal probabltes ca be easly exteded to the case of radom varables Through ts codtoal dstrbuto the behavor of a radom varable(s) ca be explored gve the value of aother radom varable The codtoal dstrbuto provdes a way of updatg the formato about a radom varable(s) gve the value of aother radom varable(s) I the followg sectos codtoal dstrbutos wll be defed however for smplcty oly the bvarate case wll be cosdered Smlarly the cocept of depedece ca also be exteded to radom varables If the behavor of a radom varable(s) does ot chage gve dfferet values of aother radom varable(s) the the former does ot deped o the latter I such a case the radom varables are sad to be depedet 1 Codtoal Dstrbutos 11 Dscrete Case Def : 1 Let ad be jotly dscrete radom varables e ( )' s a bvarate dscrete radom vector wth jot PMF p ( ) The codtoal probablty mass fucto (or codtoal PMF) of gve =x o deoted p ( y x 0) s defed as: p ( x0 y) p ( y x0) P( y x0) p ( x ) provded p ( x0) P( x0) 0 Otherwse t s udefed Smlarly the codtoal probablty mass fucto (or codtoal PMF) of gve =y o deoted p ( x y 0) s defed as: p ( x y0) p ( x y0) P( x y0) p ( y ) provded p ( y0) P( y0) 0 Otherwse t s udefed Remars: 1 Gve a dscrete radom vector ( 1 )' the codtoal PMF of ay subvector of gve aother sub-vector ca be defed aalogously The codtoal PMF 0 0

22 Statstcs 1: Probablty Theory II 9 of the sub-vector 1 gve aother sub-vector x s obtaed by dvdg the jot 0 PMF of ( 1 )' by the PMF of the codtog radom vector x That s 0 p1 ( x 1 x ) 0 p1 ( x 1 x ) P( 0 1 x1 x ) 0 p ( x ) For stace for the dscrete radom vector ( W Z )' wth jot PMF p W Z the codtoal PMF of the sub-vector ( )' gve ( W Z)' ( w0 z0)' s: pw Z ( w0 x y z0) p W Z ( x y w0 z0) P( x y W w0 Z z0) p ( w z ) 0 WZ 0 0 It ca be show that the codtoal PMF p ( x0 ) {or p ( y0 ) } s a geue PMF e t wll possess all the propertes of a PMF: a 0 p ( y x ) 0 y or p ( x y0) 0 x p( y x ) 1 or p ( x y0) 1 b 0 all y all x 3 Aalogously we ca also defe the codtoal cumulatve dstrbuto fucto (or codtoal CDF) of gve =x 0 (or of gve =y 0 ) as: F ( y x ) P( y x ) p ( a x ) ay F ( x y ) P( x y ) p ( a y ) ax F ( x0 ad F ( y0 ) It ca be show that both ) proper CDF possesses all the propertes of a 4 Ay codtoal PMF (or CDF) wll be udefed f the probablty of the codtog radom varable (vector) the deomator s zero Itutvely t wll be meagless to loo at the behavor of oe radom varable (vector) gve the value of aother radom varable (vector) f the latter has zero chace of occurrg 1 Cosder the expermet of tossg two tetrahedra (regular four-sded polyhedro) each wth sdes labeled 1 to 4 Defe the followg radom varables: = the umber o the dowtured face of the 1 st tetrahedro = the larger of the two dowtured umbers Fd the codtoal PMF of gve = e fd p ( y ) Fd the codtoal CDF of gve = e fd F ( y )

23 Statstcs 1: Probablty Theory II 30 Let ad be jotly dscrete radom varables wth jot PMF gve by: p x y x y x ad y ( ) ( ) / Fd the codtoal PMF of gve e fd p ( y x 0) Fd the codtoal PMF of gve e fd p ( x y 0) Fd the codtoal CDF of gve e fd F ( y x 0) 3 Suppose 1 cards are draw wthout replacemet from a ordary dec of cards Defe the followg radom varables: W = umber of aces draw = umber of s draw = umber of 3 s draw Z = umber of 4 s draw Fd the codtoal PMF of ( W )' gve Z e fd p ( ) W Z w x y z 0 Fd the codtoal PMF of ( Z )' gve ( W )' e fd p Z W ( x z w0 y 0) 4 Suppose the jot PMF of the radom varables ad s gve by: p x y x ad y x1 y ( ) 1/( 3 ) 1 1 Fd the codtoal PMF of gve e fd p ( x y 0) Fd the margal PMF of e fd p ( x ) 1 (Absolutely) Cotuous Case Def : Let ad be jotly (absolutely) cotuous radom varables e ( )' s a bvarate cotuos radom vector wth jot PDF f ( ) The codtoal probablty desty fucto (or codtoal PDF) of gve =x 0 deoted by f ( y x ) s defed as: 0 f ( y x ) 0 f ( x y) 0 f ( x ) 0 provded f ( x0 ) 0 Otherwse t s udefed

24 Statstcs 1: Probablty Theory II 31 Smlarly the codtoal probablty desty fucto (or codtoal PDF) of gve =y 0 deoted by f ( x y 0) s defed as: f ( x y ) 0 f ( x y ) 0 f ( y ) 0 provded f ( y0) 0 Otherwse t s udefed Remars: 1 Gve a cotuous radom vector ( 1 )' the codtoal PDF of ay sub-vector of gve aother sub-vector ca be defed aalogously The codtoal PDF of the sub-vector 1 gve aother sub-vector x s obtaed by dvdg the 0 jot PDF of ( 1 )' by the PDF of the codtog radom vector x That 0 s f 1 ( x 1 x ) 0 f 1 ( x 1 x ) 0 f ( x ) 0 For stace for the cotuous radom vector ( W Z )' wth jot PDF f codtoal PDF of the sub-vector ( )' gve ( W Z)' ( w0 z0)' s: fw Z ( w0 x y z0) f W Z ( x y w0 z0) f ( w z ) WZ 0 0 W Z It ca be show that the codtoal PDF f ( x0 ) {or f ( y0 ) } s a geue PDF e t wll possess all the propertes of a PDF: a f ( y x0) 0 y or f ( x y0) 0 x b f ( y x0) dy 1 or f ( x y0) dx 1 3 Aalogously we ca also defe the codtoal cumulatve dstrbuto fucto (or codtoal CDF) of gve =x 0 (or of gve =y 0 ) as: y F ( y x0 ) f ( y ' x0 ) dy ' x F ( x y0 ) f ( x' y0) dx ' It ca be show that both F ( x0 ) ad F ( y0 ) possess all the propertes of a proper CDF 4 Note also that we gve meag to codtoal CDF oly through: the F ( y x ) P( y x ) lm P( y x x h ) h0 F ( x y ) P( x y ) lm P( x y y h ) h0

25 Statstcs 1: Probablty Theory II 3 It s mportat to ote that the codtoal CDF (ether dscrete or cotuos) s NOT derved as: F ( x0 y) F ( y x0) F ( x ) or F ( x y0) F ( x y0) F ( y ) 0 5 If the codtog evet volves a terval rather tha a sgle specfc value of the codtog radom varable we ca use the defto of the codtoal probabltes to evaluate the codtoal CDF e P( x1 x y) F ( y x1 x) P( y x1 x) P( x x ) Let ad be jotly cotuous radom varables wth jot PDF gve by: 1x f ( x y) ( x y) I(01) ( x) I(01) ( y) 5 Fd the codtoal PDF of gve e fd f ( x y 0) Fd the codtoal PDF of gve e fd f ( y x 0) Let ad be jotly cotuous radom varables wth jot PDF gve by: exp{ y} 0 x y f ( x y) 0 ow Fd the codtoal PDF of gve e fd f ( y x 0) Fd the codtoal CDF of gve e fd F ( y x 0) 3 Let ad be jotly cotuous radom varables wth jot PDF gve by: x 1 y y (0 ) (0 ) f ( x y) e e I ( x) I ( y) y Fd P( 1 y0 ) 4 Let ad be jotly cotuous radom varables wth jot PDF gve by: 1 f ( x y) x(1 3 y ) I(0) ( x) I(01) ( y) 4 Fd the codtoal PDF of gve e fd f ( x y 0) Fd P(1/ 4 1/ 1/3 ) Fd the margal PDF of e fd f ( x )

26 Statstcs 1: Probablty Theory II 33 5 Let ad be jotly cotuous radom varables wth jot PDF gve by: 10 x y 0 y x 1 f ( x y) 0 ow Fd the codtoal CDF of gve 0 1/ e fd F ( x 0 1/ ) Fd P(1/3 /3 0 1/ ) 6 Let ad be jotly cotuous radom varables wth jot CDF gve by: 0 x 0 or y 0 xy( x y xy ) 0 x 1 0 y 1 F ( x y) y(1 y y ) x 1 0 y 1 x 0 x 1 y 1 1 x1 y1 Fd the codtoal PDF of gve e fd f ( y x 0) Fd the codtoal PDF of gve e fd f ( x y 0) Fd P( 1/ 1/3 ) 7 At the begg of a gve day a softdr mache cotas gallos of softdr stoc ad dspeses gallos of softdr durg the day Sce the dspeser s ot restoced durg the day It has bee observed that ad have jot PDF: 1/ 0 x y ad 0 y f ( x y) 0 ow Fd the codtoal PDF of gve e fd f ( x y 0) Evaluate the probablty that less tha half a gallo wll be dspesed f at the begg of the day the mache cotaed 1 gallo of softdr Evaluate the probablty that less tha half a gallo wll be dspesed f at the begg of the day the mache was fully stoced Idepedece of Radom Varables Def : 3 Let 1 be radom varables wth jot CDF F 1 ( ) The radom varables 1 are sad to be (stochastcally mutually) depedet f ad oly f ( x1 x x )' F ( x x x ) F ( x ) F ( x ) F ( x ) F ( x ) j j j1 Otherwse 1 are sad to be depedet

27 Statstcs 1: Probablty Theory II 34 Remars: 1 If the radom varables 1 are jotly dscrete ad the jot ad margal PMFs exst the CDFs above ca be replaced accordgly Smlarly f the radom varables 1 are jotly cotuous ad the jot ad margal PDFs exst the CDFs above ca be replaced accordgly I the dscrete case a ecessary ad suffcet codto for 1 to be depedet s for the followg to hold: p ( x x x ) p ( x ) p ( x ) p ( x ) p ( x ) j j j1 I the cotuous case a ecessary ad suffcet codto for 1 to be depedet s for the followg to hold: f ( x x x ) f ( x ) f ( x ) f ( x ) f ( x ) j j j1 For the bvarate case two radom varables ad are depedet f ad oly f p ( x y) p ( x) p ( y) ( x y) (dscrete case); f x y f x f y x y ( ) ( ) ( ) ( ) (cotuous case) 3 The codtoal PMF or PDF ca also be used to determe f several radom varables are depedet For the bvarate case we have the followg results: Let ad be jotly dscrete radom varables wth codtoal PMF p ( y0 ) [or p ( x0 ) ] If ether p ( x y0 ) p ( x) y0 or p ( y x0 ) p ( y) x0 the ad are depedet Let ad be jotly cotuous radom varables wth codtoal PDF ) f ( y0 [or f ( x0 ) ] If ether f ( x y0 ) f ( x) y0 or f ( y x0 ) f ( y) x0 the ad are depedet These results ca be easly geeralzed to the multvarate case 4 To determe whether or ot the radom varables 1 are depedet we eed ot loo at the margal CDFs/PMFs/PDFs explctly It suffces to chec whether the jot CDF/PMF/PDF ca be factored to fuctos (ot ecessarly CDFs/PMFs/PDFs) each depedg oly o oe For stace f ad have jot CDF F ( ) [or jot PMF p ( ) or jot PDF f ( ) ] whch ca be factored as: F ( x y) g( x) h( y) or { p ( x y) g( x) h( y) or f ( x y) g( x) h( y) } where g ( ) 0 depeds o aloe h ( ) 0 depeds o aloe ad s a postve costat the ad are depedet

28 Statstcs 1: Probablty Theory II 35 5 If 1 are depedet radom varables ad g1( ) g( ) g ( ) are fuctos such that 1 g1( 1) g( ) 1 g1( 1) the 1 are depedet radom varables I other words fuctos of depedet radom varables are depedet as well However some cases eve whe the radom varables are depedet ther fuctos may stll be depedet 6 Note that f ad are depedet the the margal CDFs/PMFs/PDFs of ad uquely determe the jot CDF/PMF/PDF respectvely 7 It ca happe that the radom varables 1 are parwse depedet but ot mutually depedet However t s always the case that mutually depedet radom varables are parwse depedet 8 If 1 are depedet radom varables the ay sub-collecto or these radom varables are depedet as well For stace f 1 10 are depedet the are depedet 1 Let ad be jotly cotuous radom varables wth jot CDF gve by: x y ( x y) F ( x y) [1 e e e ] I ( x) I ( y) (0 ) (0 ) Are ad depedet? Let ad be jotly dscrete radom varables wth jot CDF gve by: 0 x 0 or y 0 y y F ( x y) 1 e e x y y 0 x ( x y) y 1 e e e y x x 0 Are ad depedet? 3 Let ad be jotly dscrete radom varables wth jot PMF gve by: 1 p ( x y) x 1 ad y 1 x1 y 3 Are ad depedet? 4 Cosder a expermet wth four equally lely sample pots e { 1 3 4} P{ } 1/ 4 for all Defe the radom varables ad as follows: Show that ad are ot depedet Determe f ad are depedet

29 Statstcs 1: Probablty Theory II 36 5 Cosder the expermet of tossg a far co twce Defe the followg radom varables: 1 = the umber of heads o the 1 st toss = the umber of heads o the d toss 3 = equals 1 f exactly 1 head ad 1 tal show up ad 0 otherwse Verfy that 1 ad 3 are parwse depedet but are ot (mutually) depedet

30 Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As Probablty Theory I the terest most stuatos les ot o the actual dstrbuto of a radom vector but rather o a umber of summary measures whch covey certa dstrbutoal characterstcs such as cetral locato spread etc I ths chapter we geeralze the oto of expectato to the multvarate case What follows are smple extesos of the deftos your Statstcs 11 course 31 Expectato of Fuctos of Several Radom Varables 311 Defto of Expectato Def : Let ( 1 )' be a -dmesoal dscrete or cotuous radom vector wth jot PMF p ( ) or jot PDF f ( ) respectvely 1 1 a The expectato of the radom vector deoted by E ( ) or the jot expectato of the radom varables 1 s defed as: E E 1 E E ( ) [ ] [ ] [ ] ' where E [ ] s the expectato of 1 b The expectato of a fucto of the radom vector say g ( ) s defed as: E[ g( )] g( x) p ( x) for the dscrete case provded A ths sum s fte where A s the set of mass pots of E [ g( )] g( x) f ( x) d x for the cotuous case provded R g ( ) s tself a radom varable (or vector) ad the tegral exsts Remars: 1 I other words the expectato of the radom vector ( 1 )' s smply the vector cosstg of the margal expectatos of each radom varable the vector The tegral defto (1b) ca be expressed expaded form as: E g( )] g( x x x ) f ( x x x dx dx dx [ R R R 1 1 ) For the bvarate case f ( )' s a dscrete or cotuous radom vector wth jot PMF p ( ) or jot PDF f ( ) respectvely the expectato of the radom vector ( )' s: E[( )'] ( E[ ] E[ ])'

31 Statstcs 1: Probablty Theory II 38 If h ( ) s a real-valued fucto h : R R ad f Z h( ) the E( Z) E[ h( )] h( x y) p ( x y) for the dscrete case x y E( Z) E[ h( )] h( x y) f ( x y) dxdy for the cotuous case Specal Cases: Z h( ) ( or ) Z h E or E ( ) { ( )} ( { ( )} ) 1 A far co s tossed three tmes Defe as the umber of heads o the frst two tosses ad the umber of heads o the last two tosses Obta the jot expectato of ad E() ad E(+) Fd E() E() ad E() f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) xi ( x) I ( y) [01] [01] 3 Fd E(3++6Z) E(Z) ad E() f ad Z are jotly cotuous radom varables wth jot PDF gve by: f ( Z x y z ) 8 xyzi ( ) ( ) ( ) (01) x I(01) y I(01) z 4 Suppose ad are depedet radom varables each havg Uform dstrbuto over the terval [01] If Z = max() fd E(Z) 5 Cosder a bvarate radom vector ( )' havg a Tromal dstrbuto cosstg of trals wth parameters p 1 ad p both betwee 0 ad 1 ad 0 p1 p 1 Fd the jot expectato of ad ad E() 31 Basc Propertes of Expectato Result #1 Expectato of Lear Combatos Let ( 1 )' be a -dmesoal radom vector ad let g 1( ) g( ) gm( ) be m real-valued fuctos of m m E a g( ) ae[ g( )] 1 1 for ay costats 1 Some partcular cases: E( ) E( ) E( ) E( ) E( a b) ae( ) b 3 Bvarate Case: Let ( )' be a bvarate radom vector E( a b ) ae( ) be( ) a a a m

32 Statstcs 1: Probablty Theory II 39 Result # Expectato of Products Let ( 1 )' be a -dmesoal radom vector wth 1 depedet ad let g () be a fucto of aloe 1 E a g a E g 1 1 ( ) ( ) for ay costats 1 a a a Some partcular cases: 1 E ) E( ) E( ) E( ) where 1 depedet ( 1 1 Bvarate Case: Let ( )' be a bvarate radom vector wth ad depedet a E( ) E( ) E( ) b E[ g( ) h( )] E[ g( )] E[ h( )] Remars: 1 Result #1 states that the expectato of a lear combato (of rv s) s the lear combato of the expectatos (of the rv s) That s the expectato of a sum ad/or dfferece (of rv s) s the sum ad/or dfferece of the expectatos (of the rv s) No assumpto regardg the depedece of 1 s eeded Result # states that the expectato of a product (of rv s) s the product of the expectatos (of the rv s) provded the radom varables volved are depedet 3 Further Result # provdes a ecessary (but ot suffcet) codto for the depedece of 1 That s f 1 are depedet the the codto below ecessarly follows But the codto below s ot suffcet to coclude that 1 are depedet That s the codto may be satsfed eve f the radom varables are ot depedet E E( ) A far co s tossed three tmes Defe as the umber of heads o the frst two tosses ad the umber of heads o the last two tosses From prevous secto E() E() ad E(+) were derved Show that E(+) = E() +E() Determe f ad are depedet ad show that E( ) E( ) E( ) f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) xi ( x) I ( y) [01] [01] 3 Usg the propertes evaluate E(3++6Z) E(Z) ad E() f ad Z are jotly cotuous radom varables wth jot PDF gve by: f ( Z x y z ) 8 xyzi ( ) ( ) ( ) (01) x I(01) y I(01) z

33 Statstcs 1: Probablty Theory II 40 4 Let be a dscrete radom varable wth PMF gve below Defe = Verfy that E( ) E( ) E( ) eve though ad are ot depedet p ( x y) 05 I ( x) 05 I ( x) {0} { 11} 313 Some Specal Expectatos Def : Let ( 1 )' be a -dmesoal radom vector The a the covarace betwee ay par of radom varables ad j deoted by Cov( j ) s defed as: Cov( ) E [ E( )][ E( )] j j j provded the expectato exsts where E ( ) ad E ( ) are the margal expectatos of ad j respectvely b the correlato coeffcet betwee ay par of radom varables ad j deoted by ( j ) s defed as: Cov( j) ( j) = V ( )V( ) j provded the covarace exsts where V( ) 0 ad V( ) 0 are the varaces of ad j respectvely Remars: 1 A lttle algebra would yeld the followg computatoal formula: Cov( ) E[ ] [ E( ) E( )] j j j for ay j where j 1 Note that f =j the covarace s smply the varace Also the exstece of the covarace s assured by the exstece of the secod momets The covarace s a measure of how two radom varables vary relatve to each other (or how closely they move together) However t possesses oe ma defect: t s sestve to the uts of measuremet of the two radom varables For stace two radom varables say ad measured ches mght gve a larger covarace tha aother par of radom varables W ad Z whch are the same measuremets coverted to cetmeters I ths case ad oly move more closely together tha do W ad Z because the latter are measured smaller uts 3 To address the problem Remar the correlato coeffcet a rescaled verso of the covarace ca be used stead The correlato coeffcet s a ut-free measure of the ter-relatoshp betwee two radom varables It taes values betwee 1 ad +1 A zero correlato coeffcet mples that the radom varables are ucorrelated j j

34 Statstcs 1: Probablty Theory II 41 4 Zero covarace (or correlato) s a ecessary (but ot suffcet) codto for two radom varables to be depedet That s f ad are depedet the t ecessarly follows that Cov() = 0 However Cov() = 0 s ot suffcet to show that ad are depedet That s ad may ot be depedet eve f Cov() = 0 5 The correlato coeffcet may also be vewed as a measure of lear depedece Ths s so because ( ) 1 f ad oly f oe radom varable s a lear fucto of the other wth probablty 1 The sg of ( ) s a dcato of the drecto of the lear depedece whle the magtude of ( ) s a measure of the stregth of the lear depedece 6 The property of "ucorrelatedess" s ot trastve For stace f the radom varable s ucorrelated wth the radom varables ad Z t does ot follow that ad Z are ucorrelated as well 1 Fd the covarace ad the correlato coeffcet betwee ad f ad are jotly dscrete radom varables wth jot PMF gve by: 0 1 p ( x ) 0 3/8 6/8 1/8 1 9/8 6/8 0 3/8 0 0 p ( y ) Obta Cov() ad ( ) f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) 10 x yi ( x) I ( y) [01] [0 x] 3 Cosder a bvarate radom vector ( )' havg a tromal dstrbuto cosstg of trals wth parameters p 1 ad p both betwee 0 ad 1 ad 0 p1 p 1 Fd Cov() ad ( ) 4 A far co s tossed two tmes Defe as the sum of the umber of heads o the tosses ad the dfferece of the umber of heads o the tosses Verfy that the varables ad are ucorrelated but ot depedet 5 Obta Cov() ad ( ) f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) 8 xyi ( x) I ( y) [01] [0 x]

35 Statstcs 1: Probablty Theory II Some Geeral Results of Lear Combatos Result #1: If 1 ad are two radom varables the a1 a R V( a a ) a V( ) a V( ) a a Cov( ) Result #: Let 1 be radom varables The a a a R V a a V ( ) aa jcov( j ) 1 1 j 1 Result #3: Let 1 ad 1 be sets of radom varables ad let a a a 1 R ad b b b 1 R be sets of real umbers The Cov a a jj ab jcov( j ) 1 j1 1 j1 Some Partcular Cases: 1 V( ) V( ) V( ) Cov( ) V( ) V( ) V( ) Cov( ) 3 V( a b) a V( ) b V( ) abcov( ) 4 V( ) V( ) V( ) f ad are depedet 5 V( ) V( ) V( ) f ad are depedet 6 V( a b ) a V( ) b V( ) f ad are depedet 7 If 1 are ucorrelated radom varables the Result # reduces to: V a a V ( ) Let 1 3 be radom varables each havg a mea ad varace Further let Cov( 1 ) Cov( 1 3) 3 ad Cov( 3) 1 Defe U 1 43 Fd the mea ad the varace of U Let 1 be depedet radom varables each havg mea ad varace Let deote the sample mea of 1 Fd the mea ad the varace of

36 Statstcs 1: Probablty Theory II The Cauchy-Schwartz Iequalty Theorem: Let ad be two radom varables wth fte secod (raw) momets The the followg equalty holds: E ( ) E( ) E( ) wth equalty f ad oly f for some costats a ad b P( a b ) 1 e s a lear fucto of wth probablty 1 Remars: 1 The Cauchy-Schwartz Iequalty ca be used to establsh bouds o the value of ( ) That s usg the theorem we ca show that ( ) 1 Recall that ( ) 1 has a specal sgfcace If ad are radom varables such that = a + b where a ad b ( b 0 ) are costats the ( ) 1 More precsely f b > 0 the ( ) 1 ad f b < 0 ( ) 1 What do these mply regardg the behavor of ad relatve to each other? 3 The coverse of remar also holds wth probablty 1 If ( ) 1 the oe radom varable s a fucto of the other wth probablty 1 4 If o the other had ( ) 1 the there s always a ozero probablty that ot all the pots ( )' wll le o a straght le o matter what straght le we draw o the - plae ( R ) However ths does ot preclude the possblty that ad may be related a dfferet fasho perhaps olearly Hece eve f ( ) 0 t s stll possble for ad to be depedet 1 Suppose s a Stadard Normal radom varable Defe 1 1 ad Verfy that ( 1) 1 ad ( ) 1 Let ad be stadardzed radom varables (e radom varables wth zero mea ad ut varace) wth Cov() = Obta ( ) E( ) V( ) ad Cov( ) 3 Let 1 r 1 s Z1 Z Z t be ucorrelated radom varables each havg ut varace Derve ( UV ) where U s the sum of all s ad s ad V s the sum of all s ad Z s

37 Statstcs 1: Probablty Theory II 44 3 Codtoal Expectato 31 Defto of Codtoal Expectato A codtoal expectato s smply the expectato of a codtoal dstrbuto Or more descrptvely t s the expectato of a radom varable/vector gve the value of aother For smplcty we cosder oly the bvarate case Def : Let ( )' be a bvarate dscrete or cotuous radom vector ad let g( ) be a fucto of ad The codtoal expectato of g( ) gve = x 0 deoted as E [ g() = x 0 ] s defed as: E [ g( ) x ] g( x y) p ( y x ) for the dscrete case y E [ g( ) x0 ] g( x0 y) f ( y x0 ) dy for the cotuous case R Smlarly f ( )' s a bvarate dscrete or cotuous radom vector ad g( ) s a fucto of ad the the codtoal expectato of g( ) gve = y 0 deoted as E [ g() = y 0 ] s defed as: E [ g( ) y ] g( x y ) p ( x y ) for the dscrete case x E [ g( ) y0 ] g( x y0 ) f ( x y0 ) dx for the cotuous case R Specal Cases: 1 g() = E [ g( ) x0 ] E [ x0 ] g() = E [ g( ) y0 ] E [ y0 ] 3 g( ) E( x ) 0 E g x0 V x0 4 g( ) E( y ) [ ( ) ] [ ] 0 E g y0 V y0 [ ( ) ] [ ] Remars: 1 If g () s a fucto of aloe the E [ g( ) x0 ] wll be a fucto of x0 ad wll ot volve the radom varable I the same cotext f g () s a fucto of aloe the E [ g( ) y0 ] wll be a fucto of y0 ad wll ot volve the radom varable Gve a radom vector ( 1 )' the codtoal expectato of ay subvector of gve aother sub-vector ca be defed aalogously The codtoal expectato of a fucto of the sub-vectors 1 (wth dmeso m) ad say

38 Statstcs 1: Probablty Theory II 45 g( 1 ) gve x s obtaed by summg over all the mass pots of 1 or tegratg wth respect to 1 the product of g( 1 ) ad the codtoal PMF/PDF of 1 gve x That s E [ g( ) x ] g( x x ) p ( x x ) for the dscrete case x1 E[ g( 1 ) x ] g( x1 x) f ( x1 x) d x1 for the cotuous case R m 1 For stace f ( 1 )' s a radom vector 1 ( 1 3)' ad ( 4 6)' are sub-vectors of the codtoal expectato of g( 1 ) gve x s: E [ g( 1 ) x ] g( x1 x3 x4 x6 ) p1 3 4 ( x 6 1 x3 x4 x6) x1 x3 E g( 1 ) x ] g( x1 x3 x4 x6) f ( x1 x3 x4 x6) dx1dx3 [ R R 3 Just as codtoal dstrbutos satsfy all the propertes of ordary dstrbutos codtoal expectatos ad codtoal varaces also satsfy all the propertes of ordary expectatos ad ordary varaces respectvely For stace f ( )' s a bvarate radom vector ad a ad b are real umbers we have the followg results: E( a b y ) a E( y ) b a 0 0 b 0 0 c d e f E( a b x ) a E( x ) b V( y ) E( y ) [ E( y ) ] V( x ) E( x ) [ E( x ) ] V( a b y ) a V ( y ) 0 0 V( a b x ) a V ( x ) 0 0 Also f g 1( ) ad g ( ) are real-valued fuctos of oly oe of the radom varables we have the followg results: E[ g ( ) x ] g ( x ) a E[ g ( ) y ] g ( y ) b 0 0 E[ g ( ) g ( ) x ] E[ g ( ) x ] E[ g ( ) x ] c E[ g ( ) g ( ) y ] E[ g ( ) y ] E[ g ( ) y ] d E[ g ( ) g ( ) x ] g ( x ) E[ g ( ) x ] e E[ g ( ) g ( ) y ] g ( y ) E[ g ( ) y ] f Note that f ad are depedet the E( y0) E( ) for all mass pots y 0 of or E( x0) E( ) for all mass pots x 0 of Thus the codtoal mea s equal to the ucodtoal mea wth probablty 1 But the coverse s ot true That s

39 Statstcs 1: Probablty Theory II 46 eve f the codtoal mea s equal to the ucodtoal mea wth probablty 1 the varables may ot be depedet 1 Cosder the expermet of tossg two tetrahedra each wth sdes labeled 1 to 4 Let represet the umber o the dowtured face of the 1 st tetrahedro ad the larger of the two dowtured umbers Fd the codtoal mea of gve = ad that of (+) gve = e fd E( ) ad E( ) Fd the codtoal expectato of gve =x 0 e fd E( x0 ) f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) ( x y) I ( x) I ( y) (01) (01) 3 Fd the codtoal mea of 1 gve = 1 e fd E( 1 1) f 1 ad are jotly cotuous radom varables wth jot PDF gve by: f ( y y ) (1/ ) I ( y ) I ( y ) 1 1 (0 y ) 1 (0) 4 Let ad be jotly cotuous radom varables wth jot PDF gve by: f ( x y) 8 xyi ( x) I ( y) (0 y) (01) Fd the codtoal mea of gve = x 0 e fd E( x0 ) Fd the codtoal mea of gve = x 0 e fd E( x ) Fd the codtoal mea of gve = x 0 e fd E( x0 ) 5 Fd the codtoal mea ad varace of gve =y 0 e fd E( y0) ad V( y0) f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) yexp{ xy} I ( x) I ( y) (0 ) (01) 6 Show that the codtoal mea of gve = x 0 s equal to ts ucodtoal mea wth probablty 1 f ( )' s Bvarate Uform over the ut square 7 Suppose ( )' has three mass pots (00) (0) ad (11) each wth probablty 1/3 Verfy that although ad are ot depedet E( x ) E( ) for every mass pot x 0 of 0 0

40 Statstcs 1: Probablty Theory II 47 3 Expectato by Codtog The codtoal expectato E( ) s a fucto of the radom varable ad the specfc value of E( ) at x0 s equal to E( x0 ) If s ot fxed to tae ay specfc value the E( ) s tself a radom varable Theorem: Let ( )' be a bvarate radom vector ad let g () be a real-valued fucto of (or ) The E[ g( )] E { E [ g( ) ]} or E[ g( )] E { E [ g( ) ]} These results smplfy to: Eg( ) x0 p ( x0) for the dscrete case x0 E[ g( )] Eg( ) x f ( x) dx for the cotuous case E[ g( )] y0 E g( ) y p ( y ) for the dscrete case 0 0 E g( ) y f ( y) dy for the cotuous case Specal Cases: E( ) E E ( ) 1 E( ) EE ( ) 3 V( ) E V ( ) V E ( ) 4 V( ) E V ( ) V E ( ) terated varace formulas Remars: 1 Specal cases #1 ad # mply that the mea of oe radom varable s the mea (or expectato) of ts codtoal meas (gve the value of aother radom varable) where the expectato s tae over all the values of the codtog radom varable Specal cases #3 ad #4 mply that the varace of oe radom varable ca be expressed as the sum of: a the mea (or expectato) of ts codtoal varaces (gve the value of aother radom varable) where the expectato s tae over all the dfferet values of the codtog radom varable ad b the varace of ts codtoal meas across the dfferet values of the codtog radom varable

41 Statstcs 1: Probablty Theory II 48 1 Let be a dscrete radom varable wth PMF p ( x) ( x /3) I{1} ( x) Suppose that the codtoal PMF of aother radom varable gve = x s Bomal wth parameters = x ad p = ½ Fd the jot PMF of ad Fd the mea ad varace of ad Fd the codtoal varace of gve = x 0 e fd V ( x0) ad the ucodtoal varace of e V ( ) f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) 8 xyi ( x) I ( y) (0 y) (01) 3 Suppose t represetg the umber of phoe calls arrvg at a exchage durg a perod of tme of legth t s a Posso radom varable wth parameter t The probablty that a operator wll aswer ay gve phoe call s equal to p Fd the expected umber of phoe calls that wll be aswered a perod of tme of legth t 33 Jot Momet Geeratg Fuctos ad Momets 331 Defto of Jot Momet Geeratg Fucto ad Jot Momets The momet geeratg fucto (MGF) of a radom varable provdes a compact way of represetg ts momets I addto t ca ofte be used to determe the dstrbuto of a fucto of a radom varable The cocept of the MGF ca be exteded for the multvarate case (e radom varables) through the so-called jot momet geeratg fucto Def : Let ( 1 )' be a -dmesoal radom vector The jot momet geeratg fucto (or jot MGF) of the radom varables 1 deoted as m ( ) s defed as: 1 1 m 1 ( t1 t t ) exp{ 1 1 } exp{ } E t t t E t 1 provded the expectato exsts for all values t 1 t t such that h t h for some h 0 1 The jot raw momets of the radom varables 1 s deoted as: E r1 r r 1 where r s ether zero or a postve teger 1 The jot cetral momets of the radom varables 1 (about ther respectve meas) s deoted as: r1 r r E ( 1 ) ( 1 ) ( ) where r s ether zero or a postve teger 1

42 Statstcs 1: Probablty Theory II 49 Specal Cases: 1 If r rj 1 ad all other r m s are equal to zero the the jot cetral momets of the radom varables 1 becomes the covarace betwee ad If r E ( 1 ) ( 1 ) ( ) ( ) ( ) j j E ( )( ) ( ) j Cov j j j e ad all other r m s are equal to zero the the jot cetral momets of the becomes the margal varace of e radom varables E ( 1 ) ( 1 ) ( ) ( ) E ( ) Var( ) ( ) V Def : For the bvarate case f ( )' s a bvarate radom vector the jot momet geeratg fucto (or jot MGF) of the radom varables ad deoted as m ( ) s defed as: m ( t t ) E exp{ t t } 1 1 provded the expectato exsts for all values t 1 ad t such that for ay real umbers a ad b a t1 a ad b t b The jot raw momets of the radom varables ad s deoted as: r s E[ ] where r ad s are ether zeros or postve tegers The jot cetral momets of the radom varables ad (about ther respectve meas) s deoted as: r s E[( ) ( ) ] where r ad s are ether zeros or postve tegers Specal Cases: 1 If r = s = 1 the the jot cetral momets of the radom varables ad becomes the covarace betwee ad e 1 1 E[( ) ( ) ] E[( )( )] Cov( ) If r = ad s = 0 the the jot cetral momets of the radom varables ad becomes the margal varace of e 0 E[( ) ( ) ] E[( ) ] Var( ) V( )

43 Statstcs 1: Probablty Theory II 50 Remars: 1 The uqueess of the MGF carres o the bvarate ad multvarate cases That s the jot MGF of 1 uquely determes ther jot dstrbuto; ad coversely f the jot MGF of 1 exsts t s uque By remar 1 the jot MGF of 1 uquely determes the margal MGF of ay of the radom varables 1 ad thus the margal dstrbuto of ay of the radom varables 1 Why? m ( t ) m (00 t 0) E exp{ t } 1 3 A ecessary ad suffcet codto for 1 to be depedet s: m ( t t t ) m ( t ) m ( t ) m ( t ) m ( t ) Geerato of Momets Result #1 Jot Raw Momets Let ( 1 )' be a -dmesoal radom vector The jot raw momets of r r r 1 1 deoted E( 1 ) ca be obtaed from the jot MGF of 1 by dfferetatg the jot MGF r 1 tmes wth respect to t 1 r tmes wth respect to t ad so o ad r tmes wth respect to t The lmt of the resultg dervatve s the taes as all t s go to zero Ths may be exteded to the case of the jot raw momets of ay sub-vector of Bvarate Case: Let ( )' be a bvarate radom vector The jot (rs) th raw momets of ad deoted r s E( ) ca be obtaed from the jot MGF of ad by dfferetatg the jot MGF r tmes wth respect to t 1 ad s tmes wth respect to t ad the tag the lmt of the resultg dervatve as both t 1 ad t go to zero Result # Margal Raw Momets Let ( 1 )' be a -dmesoal radom vector The r th margal raw momet r of deoted E ( ) ca be obtaed from the jot MGF of 1 by dfferetatg the jot MGF r tmes wth respect to t The lmt of the resultg dervatve s the tae as all t s go to zero

44 Statstcs 1: Probablty Theory II 51 Bvarate Case: Let ( )' be a bvarate radom vector The r th margal raw momet of deoted r E ( ) ca be obtaed from the jot MGF of ad by dfferetatg the jot MGF r tmes wth respect to t 1 ad the tag the lmt of the resultg dervatve as both t 1 ad t go to zero Remars: 1 The jot cetral momets may be obtaed drectly from the jot raw momets sce the former wll always be a fucto of the latter For stace the jot (11) th cetral momets of ad about ther respectve meas 1 1 deoted E[( ) ( ) ] s just Cov( ) whch ca be expressed as a fucto of the jot raw momet E( ) ad the margal raw momets E ( ) ad E ( ) The margal cetral momets may be obtaed drectly from the margal raw momets sce the former wll always be a fucto of the latter For stace the jot (0) th cetral momets of ad about ther respectve meas 0 deoted E[( ) ( ) ] E[( ) ] s just V( ) whch ca be expressed ( ) as a fucto of the margal raw momets E ad E ( ) 1 Fd the jot MGF of ad f ad are jotly cotuous radom varables wth jot PDF gve by: f ( x y) exp{ y} I ( x) I ( y) (0 y) (0 ) Fd the margal MGFs the margal meas ad varaces of ad Fd the covarace betwee ad Fd the jot MGF of U ad V f U ad V are jotly cotuous radom varables wth jot PDF gve by: f ( u v) exp{ ( u v)} I ( u) I ( v) UV (0 ) (0 ) Fd the covarace betwee U ad V Are U ad V depedet? 3 Let Z be 3 radom varables (ether dscrete or cotuous) wth jot MGF deoted m ( t t t ) Show that m ( t t t) gves the MGF of (++Z) ad that by Z 1 3 Z m ( Z t t 0) gves the MGF of (+)

45 Statstcs 1: Probablty Theory II 5 4 Let ( )' be a bvarate radom vector havg a tromal dstrbuto cosstg of trals wth parameters p 1 ad p both betwee 0 ad 1 ad 0 p1 p 1 Fd the jot MGF of ad Fd the covarace betwee ad Usg the jot MGF of ad show that the margal dstrbuto of s B(p 1 ) 34 Bvarate Normal Dstrbuto 341 Desty Fucto Def : A bvarate cotuous radom vector ( )' s sad to have a Bvarate Normal Dstrbuto f ad oly f the jot PDF of ad s gve by: 1 1 x x y y f ( x y) exp I x I y 1 (1 ) where ad are costats such that: ( )'~ BVN ( 0 We wrte ) 34 (Jot) Momet Geeratg Fuctos ad Momets y Def : Let ( )' be a bvarate cotuous radom vector havg a Bvarate Normal dstrbuto e ( )'~ BVN ( ) The jot momet geeratg fucto (or jot MGF) of the radom varables ad s defed y t 1 R ad t R m ( t t ) exp t t (1/ )( t t t t ) Remars: 1 If ( )'~ BVN ( ) wth 0 the ad are depedet Normal y radom varables Ths result mples that for the Bvarate Normal case ucorrelatedess wll mply depedece Thus ucorrelatedess s both a ecessary ad suffcet codto for depedece the Bvarate Normal case However f ad are two (uvarate) Normal radom varables ot jotly (bvarate) Normal the ucorrelatedess wll ot ecessarly mply depedece as:

46 Statstcs 1: Probablty Theory II Margal ad Codtoal Destes Theorem: Let ( )' be a bvarate cotuous radom vector havg a Bvarate Normal dstrbuto e ( )'~ BVN ( ) The the margal dstrbutos of y ad are each (uvarate) Normal dstrbutos e ~ N( ) ad ~ N( ) Remars: 1 The theorem states that the margal dstrbutos of a Bvarate Normal dstrbuto are each uvarate Normal Ths result ca also be geeralzed to the Multvarate Normal case It s mportat to ote however that t s qute possble for a bvarate dstrbuto whch s NOT Bvarate Normal to have uvarate Normal margal dstrbutos That s eve f the margal dstrbutos of two radom varables are each uvarate Normal ther jot dstrbuto wll ot ecessarly be Bvarate Normal Theorem: Let ( )' be a bvarate cotuous radom vector havg a Bvarate Normal dstrbuto e ( )'~ BVN ( ) The the codtoal dstrbuto of y gve = x 0 s uvarate Normal ad smlarly the codtoal dstrbuto of gve = y 0 s uvarate Normal e ( x0 ) x0 ~ N (1 ) ( y0 ) y0 ~ N (1 ) Remars: 1 The theorem states that for a Bvarate Normal radom vector the codtoal dstrbuto of oe radom varable gve a specfc value of the other s uvarate Normal From the theorem above the codtoal meas ad varaces of gve = x 0 or that of gve = y 0 ca be easly obtaed 1 Let ( )'~ BVN ( ) y Fd the margal MGFs the margal meas ad margal varaces ad Fd the covarace ad correlato betwee ad Usg codtog fd the margal meas ad margal varaces of ad Show that the margal PDFs of ad are each Stadard Normal f ad are jotly cotuous radom varables wth jot PDF gve by:

47 Statstcs 1: Probablty Theory II 54 1 ( x y ) (0 ) (0 ) ( 0) ( 0) f ( x y) (1/ )e I ( x) I ( y) I ( x) I ( y) 3 Suppose ~N(01) ad cosder tossg a far co oce Defe the radom varable f a head shows up ad otherwse Note that also has a Stadard Normal Dstrbuto Why? Verfy that ad are ucorrelated but are ot depedet 4 Let ( )'~ BVN ( ) If 0 fd such that P( ) 0454 y

48 Statstcs 1: Probablty Theory II 55 4 DISTRIBUTIONS OF FUNCTIONS OF RANDOM VARIABLES It s ofte the case that we ow the dstrbuto of a certa radom varable ad are terested determg the dstrbuto of some fucto of say g() It may also happe that we ow the jot dstrbuto of the radom varables 1 ad are terested fdg the dstrbuto of a fucto of these radom varables say maybe the dstrbuto of ther sum 41 Dstrbuto of a Fucto of a Sgle Radom Varable 411 Dscrete Case Let be a dscrete radom varable wth a coutably fte (or fte) set of mass pots { x1 x } ad wth PMF gve by p () Let = g() be a real-valued fucto wth doma The followg are true about = g(): 1 The radom varable = g() s also a dscrete radom varable wth mass pots y1 g( x1) y g( x) The fucto g () eed ot be oe-to-oe If g () s oe-to-oe the for every mass pot x of there correspods oe ad oly oe mass pot y g( x ) of Otherwse t s possble for at least two dstct mass pots x ad x j of to yeld the same mass pot for e g( x ) g( x ) 3 Sce s dscrete ts dstrbuto ca be expressed terms of ts PMF To fd the PMF of t suffces to specfy the mass pots of ad the probabltes for each of these mass pots The latter s foud by: p ( y ) P( y ) p ( x ) j where the summato s tae over the set { x : g( x ) y } e the set of all mass j j j pots of whch correspod to the specfc mass pot y of 1 Fd the PMF of the radom varable = f s a dscrete radom varable wth PMF gve by p ( x) (1/ 6) I{01} ( x) (1/3) I{ 1} ( x) Let ~ Po( ) 0 Determe the PMF of the rv = + 3 ad fd P ( 7) Is there aother way of evaluatg ths problem wthout recourse to the PMF of? 3 Fd the PMF of the radom varable = f s a dscrete radom varable wth PMF gve by: p ( x ) 1/5 1/6 1/5 1/15 11/30

49 Statstcs 1: Probablty Theory II (Absolutely) Cotuous Case I the prevous secto we have see how the dstrbuto of a fucto = g() of a dscrete radom varable s derved The case whch s absolutely cotuous s ot as smple Wheever s cotuous the dstrbuto of = g() may be foud through several dfferet methods We outle three approaches ths secto Method 1: CDF Techque Let be a cotuous radom varable wth CDF F () The the CDF of the radom varable = g() deoted by F () ca be determed by: ( y) P( y) P( A) y R F where A { x : g( x) y} e { A} ff { y} Cosequetly the CDF of ca be foud sce t ca be represeted terms of the CDF of whch s ow Method : Trasformato Techque Let be a cotuous radom varable wth PDF f () Let g () be a strctly mootoe (creasg or decreasg) dfferetable (ad thus cotuous) fucto The the PDF of the radom varable = g() deoted by f () ca be determed by: 1 1 ( ) [ ( )] [ ( )] ( ) s defed f y f g y D g y x y y g x Otherwse the PDF goes to zero I the above descrpto g 1 ( ) s the verse fucto of g () e we defe g 1 ( y) as that value of x for whch g( x) y Method 3: MGF Techque Let be a cotuous radom varable wth PDF f () Assumg t exsts the MGF of the radom varable = g() ca be derved usg the PDF of as: m ( t) E[exp{ t}] E{exp{ tg( )}] exp{ tg( x)} f ( x) dx If upo smplfcato the resultg fucto of t ca be recogzed as the MGF of some ow (usually specal) dstrbuto the f follows that = g() has that ow dstrbuto Remars: 1 Ule the dscrete case where = g() wll always be dscrete whe s dscrete t s possble for a fucto = g() NOT to be cotuous eve whe s cotuous Certa codtos have to be mposed o the fucto g () to esure that = g() s also of the cotuous type wheever s cotuous Methods 1 ad 3 apply eve f = g() s ot cotuous However method s specfcally desged to fd the dstrbuto ( terms of the PDF) of = g() wheever s cotuous

50 Statstcs 1: Probablty Theory II 57 3 Method ca be geeralzed to the case whe g () s ot cotuous If ths s the case ad the rage of ca be decomposed to sub-rages such that g () becomes oe-tooe each sub-rage the the PDF of = g() ca be determed by: 1 1 g ( x) f ( y) f [ g ( y)] D [ g ( y)] where the summato s tae over all the sub-rages to whch the rage of was decomposed ad g 1 ( ) s the verse fucto of g () for the th sub-rage 1 Let ~ Exp( 1) Use method 1 to fd the dstrbutos of each of the followg: a = exp{} b = l() Let be a cotuous radom varable wth PDF gve below Use method 1 to fd the dstrbuto of = x 1 f ( x) I(0) ( x) 6 3 Let ~ U (01) Use method to fd the dstrbutos of each of the followg: a = exp{} b = - l() 4 Let be a radom varable ot ecessarly cotuous Defe = Show that: F( y) F( y) P( y) y 0 F ( y) 0 y 0 5 Let be a cotuous radom varable wth PDF gve below Use the geeralzato of method (Remar #3) to fd the dstrbuto of = f ( x) (1/ )( x 1) I ( x) [ 11] 6 Let Z ~ N (01) Use method 3 to fd the dstrbutos of each of the followg: a Z b = Z

51 Statstcs 1: Probablty Theory II 58 4 Dstrbuto of a Fucto of Several Radom Varables 41 CDF Techque A Descrpto of Techque Let 1 be radom varables wth jot CDF gve by F ( ) 1 Let 1 be real-valued fuctos defed as: 1 g1( 1 ) g( 1 ) g ( 1 ) The fuctos 1 are also radom varables whose jot CDF by defto s: F ( y y y ) P( y y y ) For smplcty let 1 be deoted as The for each pot ( y1 y y ) the -dmesoal hyper-space the evet defed as: { y y y } { g ( ) y g ( ) y g ( ) y } Thus the jot CDF of the radom varables 1 ca be expressed as: F ( y y y ) P( y y y ) P( g ( ) y g ( ) y g ( ) y ) 1 1 a fucto volvg the jot CDF of the radom varables 1 Cosequetly the jot CDF of the radom varables 1 ca be foud through the jot CDF of the radom varables 1 Ths techque for dervg the jot CDF of the radom varables 1 s called the Cumulatve Dstrbuto Fucto (CDF) Techque Note that f we start out wth radom varables 1 we ca derve the jot CDF of the radom varables 1 r where r That s the umber of radom varables eed ot be equal to the umber of radom varables; we ca defe r fuctos of say g1( ) g( ) gr ( ) 1 Let ~ N (01) Fd the dstrbuto of = g() = (Case for whch =1) Let Z be dscrete radom varables wth jot PMF gve below Fd the dstrbuto of ther sum W = + +Z p ( Z x y z) 1/ 7 x y z 13

52 Statstcs 1: Probablty Theory II 59 B Dstrbuto of the Maxmum ad the Mmum Theorem: CDF of the Mmum ad the Maxmum Let 1 be depedet radom varables wth margal CDFs deoted by F ) F ( ) F ( ) respectvely Let 1 ad be real-valued fuctos defed as: ( 1 m{ } ad max{ } The the margal CDFs of 1 ad are gve by: F ( y) 1 1 F ( y) ad F ( y) F ( y) If the radom varables 1 are depedet ad detcally dstrbuted (d) radom varables wth commo CDF deoted by F () the the margal CDFs of 1 ad reduce to: F ( ) 1 1 ( ) ( ) ( ) y F 1 y ad F y F y Corollary: PDF of the Mmum ad the Maxmum If the radom varables 1 are depedet ad detcally dstrbuted (d) cotuous radom varables wth commo CDF deoted by F () ad commo PDF deoted by f () the the margal PDFs of 1 ad are gve by: 1 1 f ( y) 1 F ( y) f ( y) ad F ( y) F ( y) f ( y) 1 1 Let ad be dscrete radom varables wth jot PMF gve by: x y p ( x ) 1/4 1/1 1/1 1/4 1/ /1 1/4 1/1 5/1 6 1/4 1/6 1/4 1/1 1/3 p ( y ) 1/1 1/3 9/4 5/4 1 Defe U = m() ad V = max() Fd the jot PMF of U ad V ad the margal PMFs of U ad V Let ad be depedet ad detcally dstrbuted U(01) radom varables Defe U = m() ad V = max() Fd the jot CDF ad the jot PDF of U ad V Fd the margal PDFs of U ad V 3 Let Z be dscrete radom varables wth jot PDF gve below Fd the jot PMF of U = m(z) ad V = max(z) p ( Z x y z) 1/ 7 x y z 13

53 Statstcs 1: Probablty Theory II 60 C Dstrbuto of the Sum ad the Dfferece of Two Radom Varables Theorem: Let ad be cotuous radom varables wth jot PDF deoted by f ( ) Let Z ad V be defed as Z = + ad V = - The ad f ( z) f ( x z x) dx f ( z y y) dy Z f ( v) f ( x x v) dx f ( v y y) dy V Corollary: Covoluto Formula If ad are depedet cotuous radom varables ad Z = + the f ( z) f ( z x) f ( x) dx f ( z y) f ( y) dy Z 1 Let ad be d U(01) rv s Fd the PDF of Z = + If we let U = m() ad let V = max() fd the PDF of the rage R = V U Fd the dstrbuto of Z = + f ad are jotly cotuous radom varables wth jot PDF gve by f ( x y) exp{ ( x y)} I[0 x] ( y) I[0 ) ( x) 3 Let ad be d Bomal radom varables wth parameters ad p Fd the PMF of Z = + D Dstrbuto of the Product ad the Quotet of Two Radom Varables Theorem: Let ad be cotuous radom varables wth jot PDF deoted by f ( ) Let Z ad V be defed as Z = ad V = / The ad f ( z) x f ( x z / x) dx y f ( z / y y) dy 1 1 Z f ( v) y f ( vy y) dy V 1 Let ad be d U(01) radom varables Fd the PDF of Z = ad V = / Let ~N(0 ) ad ~N(0 ) ad depedet Fd the PDF of V = / 3 Let ad be cotuous radom varables wth jot PDF gve below Fd the PDF of V = / f ( x y) exp{ ( x y)} I ( y) I ( x) [0 x] [0 )

54 Statstcs 1: Probablty Theory II 61 4 MGF Techque A Descrpto of Techque Let 1 be radom varables wth jot PMF/PDF gve by p ( ) 1 or f ( ) 1 respectvely Let 1 be real-valued fuctos defed as: 1 g1( 1 ) g( 1 ) g ( 1 ) The fuctos 1 are also radom varables For smplcty let 1 be deoted as If the jot MGF of 1 exsts the we ca derve t by defto ad cosequetly we ca express t terms of as: m ( t t t ) E[exp{ t t t }] E[exp{ t g ( ) t g ( ) t g ( )}] 1 1 whch s a expectato of a fucto volvg the radom varables 1 If ths expectato exsts t ca be derved by usg the jot PMF/PDF of the radom varables 1 (Ths ca be doe by summg or tegratg the product of the expoetal fucto above ad the jot PMF/PDF over all the values of the radom varables 1 ) If the resultg fucto ca be recogzed as the jot MGF of some ow dstrbuto the the radom varables 1 wll have that jot dstrbuto Ths techque for dervg the jot PMF/PDF of the radom varables 1 s called the Momet Geeratg Fucto (or MGF) Techque Note that for > 1 the method s of lmted use sce there are very few jot dstrbutos whose MGFs we have defed (eg the Bvarate Normal dstrbuto) For =1 the MGF wll be a fucto of a sgle argumet ad we have a better chace of recogzg the resultg MGF 1 Let 1 ad be depedet Stadard Normal rv s Defe 1 = g 1 ( 1 ) = 1 + ad = g ( 1 ) = 1 - Fd the jot PDF of 1 ad Let ad be d N( ) radom varables Fd the dstrbuto of Z = a + b 3 Let ad be d N( ) radom varables Fd the dstrbuto of Z = + 4 Let ad be d Exp() radom varables Fd the dstrbuto of Z = + 5 Let ad be d Geo(p) radom varables Fd the dstrbuto of Z = +

55 Statstcs 1: Probablty Theory II 6 B Dstrbuto of Sums of Idepedet Radom Varables Theorem: Let 1 be depedet radom varables whose margal MGFs deoted by m () t m () 1 t m () t respectvely all exst for all h < t < h for some h > 0 Let 1 The the MGF of s: m ( t) E[exp{ t }] m ( t) 1 for h < t < h Corollary: Let 1 be depedet ad detcally dstrbuted (d) radom varables wth commo MGF deoted by m () t whch exsts for all h < t < h for some h>0 Let 1 The the MGF of s: m ( ) [exp{ }] ( ) t E t m t for h < t < h 1 Let 1 be depedet Beroull radom varables all wth parameter p Show that 1 ~ B( p) Let 1 be depedet Posso rv s wth havg parameter Show that 1 ~ Po( ) 3 Let 1 be d Expoetal radom varables wth parameter Show that 1 ~ Ga( ) 4 Let 1 be depedet radom varables wth ~ N( ) Show that: a a ~ N( a a ) b a ~ N( a a ) Remars: 1 I Example 4 above we see that ay lear combato of depedet Normal radom varables s tself a Normal radom varable If 1 are depedet ad detcally dstrbuted radom varables wth commo MGF m () t ad S 1 m ( ) ( ) S t m t If the dstrbuto of S belogs to the same parametrc famly as the commo dstrbuto of 1 the we say that the dstrbuto s reproductve the For stace Example above the sum of Posso radom varables s also a Posso radom varable Thus the Posso dstrbuto s reproductve wth respect to It ca be verfed that the Normal Gamma Bomal Negatve Bomal ad Posso dstrbutos are all reproductve

56 Statstcs 1: Probablty Theory II 63 C The Cetral Lmt Theorem I the above dscusso we derved the exact dstrbuto of the sum of certa depedet radom varables I may cases though oe s more terested the average rather tha the sum However f the dstrbuto of the sum s ow the dstrbuto of the average ca be easly derved F ( z) P( z) P z P ( ) z F z 1 m ( t) E[exp{ t }] m ( t / ) If the exact dstrbuto of the average caot be obtaed easly the followg theorem s very useful sce t gves the approxmate (or asymptotc) dstrbuto of Theorem: Cetral Lmt Theorem (CLT) If for each postve 1 are depedet ad detcally dstrbuted radom varables wth commo mea ad commo varace the z FZ () z coverges to () z as E( ) where: Z V( ) 1 Remar: The CLT says that f you have d radom varables wth commo mea ad varace the the stadardzed radom varables Z has a dstrbuto that approaches the Stadard Normal dstrbuto (The radom varable Z s the stadardzed from whch we subtract ts mea ad dvde by ts stadard devato) The mportat thg to ote s that t does ot mae ay dfferece what type of commo dstrbuto 1 have as log as ther commo mea ad commo varace both exst ad the sample sze s very large Corollary: If 1 are depedet ad detcally dstrbuted radom varables wth commo mea ad commo varace the b a P a b Example: Achevemet test scores from all hgh school seors a certa tow have a mea ad a varace of 60 ad 64 respectvely A specfc hgh school class of 100 studets had a mea score of 58 Is there evdece to suggest that ths hgh school s feror to other hgh schools tow?

57 Statstcs 1: Probablty Theory II Method of Trasformato 431 Dscrete Case Let 1 be dscrete radom varables wth jot PMF p ( ) 1 Let H be the set of mass pots of 1 that s: H = ( x1 x x )' : p ( x1 x x ) 0 1 Let 1 be real-valued fuctos defed as: 1 g1( 1 ) g( 1 ) g ( 1 ) The fuctos 1 are also dscrete radom varables wth jot PMF gve by: p ( y y y ) P( y y y ) p ( x x x ) 1 1 where the summato s tae over all mass pots ( x1 x x )' H for whch: ( y y y ) g ( x x x ) g ( x x x ) g ( x x x ) Note that f we start out wth radom varables 1 we ca derve the jot PMF of the radom varables 1 r where r That s the umber of radom varables eed ot be equal to the umber of radom varables; we ca defe r fuctos of 1 say g1( 1 ) g( 1 ) gr( 1 ) Example: ( )' have jot PMF gve by: Let 1 3 ( x1 x x 3)' (000) (001) (011) (101) (110) (111) p ( x x x )' 1/8 3/8 1/8 1/8 1/8 1/ Fd the jot PMF of 1 ad f 1 ad are defed as: g ( ) g ( ) Cotuous Case Let 1 be cotuous radom varables wth jot PDF f ( ) 1 1 be real-valued fuctos defed as: 1 g1( 1 ) g( 1 ) g ( 1 ) Let

58 Statstcs 1: Probablty Theory II 65 The fuctos 1 are also radom varables To fd the jot desty of 1 we assume that the fuctos g 1( ) g ( ) g ( ) satsfy the followg codtos: 1 The fuctos 1 g1( 1 ) g( 1 ) g ( 1 ) are oe-to-oe e the equatos ca be uquely solved for the values of 1 terms of 1 wth solutos gve by: g ( ) 1 1 g ( ) 1 1 g ( ) The fuctos g ( ) g ( ) g ( ) all have cotuous partal 1 dervatves at all pots ( x1 x x ) We defe the Jacoba of the trasformato deoted by J as the determat of the matrx cotag the partal dervatves that s: J 1 1 The the jot PDF of 1 deoted by f 1 ( ) s determed as f ( y y y ) D 1 J f g ( y y y ) g ( y y y ) g ( y y y ) I ( y y y ) where D s the subset of -dmesoal hyper-space cosstg of pots ( y1 y y ) for whch: ( y y y ) g ( x x x ) g ( x x x ) g ( x x x ) Remars: 1 If we wat the jot PDF of the radom varables 1 r where r < we ca troduce addtoal (dummy) radom varables r 1 r usually chose as ay of the s the we ca fd the jot PDF of 1 usg the above method From the jot PDF of 1 we ca fd the jot PDF of 1 r by tegratg out the PDF wth respect to the radom varables r 1 r If ay of the fuctos g ( ) g ( ) g ( ) s ot oe-to-oe the 1 rage of values of ( 1 ) ca be decomposed to several sub-rages so that all

59 Statstcs 1: Probablty Theory II 66 of them are oe-to-oe each sub-rage (See Theorem 15 p 11 of MGB for the complete result) 1 Let 1 ad be d U(01) rvs Fd the jot PDF of 1 1 ad 1 Let ad be rvs wth jot PDF gve below Fd the jot PDF of U = / ad V=+ f ( x y) 8 xyi ( x) I ( y) (0 y) (01) 3 Let ad be rvs wth jot PDF gve below Fd the jot PDF of ad (+) f ( x y) exp{ ( x y)} I ( y) I ( x) [0 x] [0 ) 4 Let ~ Ga(r 1 ) ad ~ Ga(r ) where ad depedet Defe U = (+) ad V=/(+) Fd the jot PDF of U ad V ad the margal PDFs of U ad V 433 Probablty Itegral Trasform Theorem: If s a radom varable wth cotuous CDF F (x) the U = F (x) s uformly dstrbuted over the terval (01) Coversely f U s uformly dstrbuted over the terval (01) the = F -1 (U) has CDF F x () Remar: The trasformato = F () s called the Probablty Itegral Trasform It plays a mportat role the theory of dstrbuto-free statstcs ad goodess-of-ft tests

60 Statstcs 1: Probablty Theory II 67 5 SAMPLING AND SAMPLING DISTRIBUTIONS 51 Itroducto Def : The totalty of elemets whch are uder dscusso ad about whch formato s desred wll be called the target populato Remar: The object of ay vestgato s to fd out somethg about a gve target populato However t s geerally mpossble or mpractcal to exame the etre populato So o the bass of examg a part of t a sample fereces (or geeralzatos) regardg the etre target populato ca be made Iferece employs the process of ductve argumetato Sce we caot be absolutely certa about our geeralzatos ucertaty wll always be preset all ductve fereces we mae Def : Gve a probablty space ad a postve teger a collecto of depedet radom varables 1 all havg commo dstrbuto F s called a radom sample from the populato (wth dstrbuto) F Remars: 1 A radom sample (rs) ca be vewed as a radom vector ( 1 )' defed o the -dmesoal real space Further t ca also be terpreted as the outcome of a seres of depedet trals of a expermet performed uder detcal codtos The commo dstrbuto F s usually called the sampled populato the collecto of all elemets from whch the sample s actually selected (I certa cases F may be replaced wth the correspodg PDF or PMF) 3 Samplg from the dstrbuto F s sometmes referred to as samplg from a fte populato or samplg wth replacemet from a fte populato Implctly samplg wthout replacemet from a fte populato s ruled out the above defto (Why?) 4 Sce the rs ( 1 )' cossts of depedet ad detcally dstrbuted (d) radom varables the the dstrbuto of the rs whch s smply the jot dstrbuto of 1 s gve by: F ( x) F ( x x x ) F ( x ) F ( x ) F ( x ) F ( x ) If the PDF or PMF exsts the F may be replaced accordgly Thus f ( x) f ( x x x ) f ( x ) f ( x ) f ( x ) f ( x ) or p ( x) p ( x x x ) p ( x ) p ( x ) p ( x ) p ( x )

61 Statstcs 1: Probablty Theory II 68 1 I studyg the "relablty" of lght bulbs the lfetme ( hours) of a gve lght bulb s tae to be a Expoetal rv wth parameter A collecto of lght bulbs s put to a "relablty test" ad ther lfetmes are recorded The ( 1 )' ca be cosdered a rs (from a Expoetal populato wth parameter ) What s the PDF of the rs? Let ( 1 )' be a rs from a Beroull populato wth parameter 0 1 What s the probablty mass fucto (PMF) of the rs? Def : Let ( 1 )' be a observable radom vector Ay observable fucto of say T( ) whch s tself a rv (or radom vector) s called a statstc The probablty dstrbuto of a statstc s called a samplg dstrbuto The stadard devato of a statstc s called ts stadard error Remar: A statstc s always: (1) a fucto of observable radom varables () s tself a radom varable ad (3) does ot cota ay uow parameter By "observable" we mea that the value of the statstc T( ) ca be computed drectly from the values of the r v's the rs Some Importat Statstcs Let ( 1 )' be a rs from F Some commo statstcs are: 1 Sample Sum : Sample Mea : 3 Sample Varace : 4 Sample Raw Momets : 5 Sample Cetral Momets : S 1 / 1 1 S ( ) ( 1) r M r ' / r 1 1 r M r ( ) / r 1 6 Sample Mmum : 1 (1) m{ 1 } 7 Sample Maxmum : ( ) max{ 1 } 1

62 Statstcs 1: Probablty Theory II 69 5 The Sample Mea Theorem: Let ( 1 )' be a rs from F wth commo mea E ( ) ad commo varace V( ) The a E( ) E( ) b V( ) V ( ) / Theorem: Wea Law of Large Numbers (WLLN) Let 1 be a rs from the PDF f wth commo mea E ( ) ad commo varace V( ) ( ) Let ad be arbtrary umbers such that 0 ad 0 1 ( ) If V the P E( ) 1 Remars: 1 Explaato of the WLLN: The probablty that wll devate from the true populato mea E ( ) by more tha some arbtrarly small ozero value ca be made arbtrarly small by choosg suffcetly large Thus the sample mea ca be used to estmate E ( ) relably If s suffcetly large the E( ) s lely to be small but ths does ot mply that E( ) s small for all large The result does ot mply that P E( ) 1 Example: Cosder a dstrbuto wth uow mea ad varace 1 How large a sample should be tae so that a probablty of at least 095 s attaed that the sample mea devates from the populato mea by o more tha 04 uts? Theorem: Cetral Lmt Theorem (CLT) Let 1 be a rs from the PDF f wth commo mea E() ad fte commo varace V( ) Let be the sample mea of the rs ad defe the rv Z E( ) E( ) V( ) Sd( ) Z as: The as the dstrbuto of Z approaches the Stadard Normal e Z N(01) Remars: 1 The CLT also mples: N ( / ) as ad S N ( ) as The CLT result holds for all rs's regardless of the form of the paret PMF/PDF for as log as ths dstrbuto has fte varace

63 Statstcs 1: Probablty Theory II 70 3 Importace of the CLT: I mag fereces about populato parameter(s) we eed the dstrbuto of certa statstcs eg the sample mea Dog ths s ofte mathematcally easer f samples are tae from the Normal dstrbuto However f the rs s ot tae from the Normal dstrbuto fdg the samplg dstrbuto of ca become very dffcult The CL T states that for as log as (1) the paret PDF/PMF of the rs has fte varace ad () the sample sze s large ( 30) the approxmate dstrbuto of the sample mea s a Normal dstrbuto 1 Cosder a dstrbuto wth uow mea ad 1 How large a sample should be tae so that a probablty of (exactly) 095 s attaed that the sample mea wll ot devate from the populato mea by more tha 04 uts? A electrcal frm maufactures lght bulbs that have a average legth of lfe equal to 800 hours ad a stadard devato of 40 hours Fd the probablty that a radom sample of 16 bulbs wll have a average lfe of less tha 775 hours Assume ormalty 53 Samplg from the Normal Dstrbuto Def : A cotuous rv s sad to have a Ch-Square dstrbuto wth degrees of freedom (df) f ad oly f the PDF of s: 1 / ( / ) 1 f ( x) x exp{ x / } I(0 ) ( x) ( / ) Notato : Mea : E( ) 1 ~ ( ) MGF : m () t 1 t Varace : V( ) Remar: A Ch-square rv wth degrees of freedom s equvalet to a Gamma rv where r / ad 1/ e ( ) Ga( r / 1/ ) Theorem: If the rv's 1 are Normally ad depedetly dstrbuted wth meas ad varaces U 1 = 1 respectvely the ~ ( ) Corollary: If 1 s a rs from U N ( ) ~ ( ) 1 ad the V ( 1) S ~ ( 1)

64 Statstcs 1: Probablty Theory II 71 Remars: 1 The theorem states that the sum of the squares of depedet Stadard Normal rv's s a Ch-Square rv wth df equal to the umber of rv's (or terms) the sum If ~ N( ) the ~ (1) If ~ N (01) the ~ (1) 3 The Ch-Square famly of destes s reproductve wrt the df a 1 ~ d ( ) S ~ ( ) b ~ S ~ d 1 ( ) ( ) 4 For 0 1 the otato wll be used to deote the value of a Ch-square r v wth df leavg a area of above t That s the probablty above the umber s ad the cumulatve probablty below s equal to (1 ) So f 1 ~ ( ) F ( ) P( ) 1 ad thus F (1 ) Example: If 1 ~ (5) the 0015 F (099) 151 F ( ) P( ) P( 151) Def : A cotuous rv s sad to have a F-dstrbuto wth m (umerator) ad (deomator) degrees of freedom (df) f ad oly f the PDF of s: m m/ ( m ) 1 m x f ( x) I ( )/ (0 )( x) m m m m 1 x Notato : ~ F ( m ) MGF : DNE Mea : E( ) Varace: Theorem: Let U ad V wth U ad V depedet The ~ ( m) U / m ~ F( m ) V / ~ ( ) Corollary: If 1 m s a rs from rs from N ( ) the S S ~ F ( m 1 1) where ( ) ( m ) m( ) ( 4) V ( ) 4 N ad 1 s aother depedet S m ( ) ad S m 1 1 ( ) 1 1

65 Statstcs 1: Probablty Theory II 7 Remars: 1 The theorem states that the rato of two depedet Ch-square rv s over ther respectve df s a F-dstrbuted rv wth umerator df equal to the df of the Chsquare rv the umerator ad deomator df equal to the df of the Ch-square rv the deomator F For 0 1 the otato m wll be used to deote the value of a F-dstrbuted rv wth m umerator ad deomator df leavg a area of above t That s the probablty above the umber F m s ad the cumulatve probablty below F m s equal to (1 ) So f ~ F ( m ) F ( F ) P( F ) 1 ad thus m m Example: If ~ F (3) the 3 Idetty: F F F (099) 308 F 1 m F (1 ) F ( F ) P( F ) P( 308) (1 001) m 1 F m Def : A cotuous rv s sad to have a Studet s t-dstrbuto wth degrees of freedom (df) f ad oly f the PDF of s: 1 ( 1)/ x f ( x) 1 I( ) ( x) Notato : ~ t ( ) MGF : DNE Mea : E ( ) 0 Varace : V( ) /( ) Remars: 1 The t-dstrbuto s symmetrc about ts mea 0 The t-dstrbuto s more varable tha the Stadard Normal dstrbuto 3 For large df the t-dstrbuto reduces to the Stadard Normal dstrbuto Theorem: Let Z ~ N (01) ad T U wth Z ad U depedet The ~ ( ) Z ~ t( ) U Corollary: If 1 s a rs from stadard devato s the T ~ t( 1) s N ( ) wth sample mea ad sample

66 Statstcs 1: Probablty Theory II 73 Corollary: If ~ t ( ) the ~ F (1 ) Remars: 1 The theorem states that the rato of a Stadard Normal rv to the square root of a depedet Ch-square rv over ts df s a t-dstrbuted rv wth df equal to the df of the Ch-square rv the deomator For 0 1 the otato t wll be used to deote the value of a t-dstrbuted rv wth df leavg a area of above t That s the probablty above the umber t s ad the cumulatve probablty below t s equal to (1 ) Thus f ~ t ( ) F t P t ad thus ( ) ( ) 1 1 F t 1 F (1 ) Example: If ~ t (5) the t0015 (099) 3365 F ( t ) P( t ) P( 3365) (1 001) Idetty: t1 t Some Importat Results: Let 1 be a rs from 1 N( ) The 1 1 N( ) ad 1 be aother depedet rs from 1 1 ~ t 1 S1 1 ( 1) ad ~ t S ( ) ( ) ~ N (01) ( ) ( ) ~ t 1 1 S p ( 1) S where S ~ F ( 11 1) S S 5 1 ~ F ( 1 S 1 1) ( 1) S ( 1) S 1 1 p 1 ( 1) assumg assumg 1 1

67 Statstcs 1: Probablty Theory II 74