Chapter 3 Moments of a Distribution

Transcription

1 Chaper 3 Moes of a Disribuio Epecaio We develop he epecaio operaor i ers of he Lebesgue iegral. Recall ha he Lebesgue easure λ(a) for soe se A gives he legh/area/volue of he se A. If A = (3; 7), he λ(a) = 3-7 = 4. The Lebesgue iegral of f o [a,b] is defied i ers of Σ i y i λ(a i ), where = y y... y, A i = { : y i f () < y i+ }, ad λ(a i ) is he Lebesgue easure of he se A i. The value of he Lebesgue iegral is he lii as he y i 's are pushed closer ogeher. Tha is, we brea he y-ais io a grid usig {y } ad brea he -ais io he correspodig grid {A } where A i = { : f () є [y i ; y i+ )}.

2 Taig epecaios: Riea vs Lebesgue Riea s approach Pariio he base. Measure he heigh of he fucio a he ceer of each ierval. Calculae he area of each ierval. Add all iervals. Lebesgue approach Divide he rage of he fucio. Measure he legh of each horizoal ierval. Calculae he area of each ierval. Add all iervals. Taig epecaios A Borel fucio (RV) f is iegrable if ad oly if f is iegrable. For coveiece, we defie he iegral of a easurable fucio f fro (Ω, Σ, μ) o ( R, B), where R = R U {, }, B = σ(b U{{ }, { }}). Eaple: If Ω = R ad μ is he Lebesgue easure, he he Lebesgue iegral of f over a ierval [a, b] is wrie as [a,b] f() d = ab f() d, which agrees wih he Riea iegral whe he laer is well defied. However, here are fucios for which he Lebesgue iegrals are defied bu o he Riea iegrals. If μ=p, i saisics, dp = E = E[] is called he epecaio or epeced value of.

3 Epeced Value Cosider our probabiliy space (Ω, Σ, P). Tae a eve (a se A of ω є Ω) ad, a RV, ha assigs real ubers o each ω єa. If we ae a observaio fro A wihou owig which ω єa will be draw, we ay wa o ow wha value of (ω) we should epec o see. Each of he ω єa has bee assiged a probabiliy easure P[ω], which iduces P[]. The, we use his o weigh he values (ω). P is a probabiliy easure: The weighs su o. The weighed su provides us wih a weighed average of (ω). If P gives he "correc" lielihood of ω beig chose, he weighed average of (ω) --E[] ells us wha values of (ω) are epeced. Epeced Value Now wih he cocep of he Lebesgue iegral, we ae he possible values { i } ad cosruc a grid o he y-ais, which gives a correspodig grid o he -ais i A, where A i = {ωєa : (ω) є [ i ; i+ )}. Le he elees i he -ais grid be A i. The weighed average is i P[ A ] i i i P i [ ] i i i f ( ) As we shri he grid owards, A, becoes ifiiesial. Le dω be he ifiiesial se A. The Lebesgue iegral becoes: i li i i P[ Ai ] P[ d ] P [ i ] f ( ) d i 3

4 The Epecaio of : E() The epecaio operaor defies he ea (or populaio average) of a rado variable or epressio. Defiiio Le deoe a discree RV wih probabiliy fucio p() (probabiliy desiy fucio f() if is coiuous) he he epeced value of, E() is defied o be: E p p ad if is coiuous wih probabiliy desiy fucio f() E f d Soeies we use E[.] as E [.] o idicae ha he epecaio is beig ae over f () d. i i i Ierpreaio of E(). The epeced value of, E(), is he ceer of graviy of he probabiliy disribuio of.. The epeced value of, E(), is he log-ru average value of. (To be discussed laer: Law of Large Nubers) E() 4

5 Eaple: The Bioal disribuio Le be a discree rado variable havig he Bioial disribuio --i.e., = he uber of successes i idepede repeiios of a Beroulli rial. Fid he epeced value of, E(). p p p,,,3,, Ep p p p p! p p!! Eaple: Soluio E p p p p p! p p!!! p p!!!! p p p p!!!!!!!!!! p p p 5

6 Eaple: Soluio!! p p p p p!!!!!! p p p!!!! p p p p p p p p p p p p p Eaple: Epoeial Disribuio Le have a epoeial disribuio wih paraeer. The probabiliy desiy fucio of is: f e The epeced value of is: We will deerie E f d e d e d udv uv vdu 6

7 Eaple: Epoeial Disribuio We will deerie e d I his case u ad dv e d Hece Thus du d ad v e e de e d usig iegraio by pars. e E e d e e e Suary: If has a epoeial disribuio wih paraeer he: E Eaple: The Uifor disribuio Suppose has a uifor disribuio fro a o b. The: ba a b f a, b The epeced value of is: E f d d b ba a b b a a b ba b a a 7

8 Eaple: The Noral disribuio Suppose has a Noral disribuio wih paraeers ad. The: f e The epeced value of is: E f d e d Mae he subsiuio: z dz d ad z Hece Now z E z e dz z z e dz ze dz z z e dz ad ze dz Thus E 8

9 Eaple: The Gaa disribuio Suppose has a Gaa disribuio wih paraeers ad. The: e f Noe: f d e d if, This is a very useful forula whe worig wih he Gaa disribuio. Eaple: The Gaa disribuio The epeced value of is: E f d e d e d e This is ow equal o. d 9

10 Eaple: The Gaa disribuio Thus, if has a Gaa (,) disribuio, he epeced value of is: E() = / Special Cases: (,) disribuio he he epeced value of is:. Epoeial () disribuio: =, arbirary E. Chi-square () disribuio: = /, = ½. E Eaple: The Gaa disribuio E

11 The Epoeial disribuio E The Chi-square ( ) disribuio E

12 Epecaio of a fucio of a RV Le deoe a discree RV wih probabiliy fucio p() (or pdf f() if is coiuous) he he epeced value of g(), E[g()], is defied o be: E g g p g p ad if is coiuous wih probabiliy desiy fucio f() E g g f d Eaples: g() = (-μ) => E[g()] = E[(-μ) ] g() = (-μ) => E[g()] = E[(-μ) ] i i i Eaple: Fucio of a Uiforly disribued RV Suppose has a uifor disribuio fro o b. The: b b f, b Fid he epeced value of A =. If is he legh of a side of a square (chose a rado for o b) he A is he area of he square b ba b b E f d d = /3 he aiu area of he square a 3 b 3 3 b b 3 3 3

13 Media: A aleraive ceral easure A edia is described as he ueric value separaig he higher half of a saple, a populaio, or a probabiliy disribuio, fro he lower half. Defiiio: Media The edia of a rado variable is he uique uber ha saisfies he followig iequaliies: P( ) ½ ad P( ) ½. For a coiuous disribuio, we have ha solves: f ( ) d f ( ) d / Media: A aleraive ceral easure Calculaio of edias is a popular echique i suary saisics ad suarizig saisical daa, sice i is siple o udersad ad easy o calculae, while also givig a easure ha is ore robus i he presece of oulier values ha is he ea. A opialiy propery A edia is also a ceral poi which iiizes he average of he absolue deviaios. Tha is, a value of c ha iiizes E( c ) is he edia of he probabiliy disribuio of he rado variable. 3

14 4 Eaple I: Media of he Epoeial Disribuio Le have a epoeial disribuio wih paraeer. The probabiliy desiy fucio of is: e f The edia solves he followig iegral of : / ) ( d f / e e d e d e Tha is, = l()/λ. Eaple II: Media of he Pareo Disribuio Le follow a Pareo disribuio wih paraeers α (scale) ad s (shape, usually oaed ). The pdf of is: The edia solves he followig iegral of : / ) ( d f / ) ( ) ( / ) ( s s s s s s C C d d Noe: The Pareo disribuio is used o describe he disribuio of wealh. ) ( if if f s s

15 Moes of a Rado Variable The oes of a rado variable are used o describe he behavior of he RV (discree or coiuous). Defiiio: K h Moe Le be a RV (discree or coiuous), he he h oe of is: E - p if f d if is discree is coiuous The firs oe of, = = E() is he ceer of graviy of he disribuio of. The higher oes give differe iforaio regardig he shape of he disribuio of. Moes of a Rado Variable Defiiio: Ceral Moes Le be a RV (discree or coiuous). The, he h ceral oe of is defied o be: E - p if is discree where = = E() = he firs oe of. f d if is coiuous The ceral oes describe how he probabiliy disribuio is disribued abou he cere of graviy,. 5

16 Moes of a Rado Variable s ad d The firs ceral oes is give by: E The secod ceral oe depeds o he spread of he probabiliy disribuio of abou I is called he variace of ad is deoed by he sybol var(). E = d ceral oe. E is called he sadard deviaio of ad is deoed by he sybol. var E Moes of a Rado Variable - Sewess 3 The hird ceral oe 3 E coais iforaio abou he sewess of a disribuio. A popular easure of sewess: Disribuio accordig o sewess: ) Syeric disribuio ,

17 Moes of a Rado Variable - Sewess ) Posiively sewed disribuio , ) Negaively sewed disribuio , Moes of a Rado Variable - Sewess Sewess ad Ecooics - Zero sew eas syerical gais ad losses. - Posiive sew suggess ay sall losses ad few rich reurs. - Negaive sew idicaes los of ior wis offse by rare ajor losses. I fiacial ares, soc reurs a he fir level show posiive sewess, bu a soc reurs a he aggregae (ide) level show egaive sewess. Fro horse race beig ad fro U.S. sae loeries here is evidece supporig he coeio ha gablers are o ecessarily ris-lovers bu sewess-lovers: Log shos are overbe (posive sewess loved!). 7

18 Moes of a Rado Variable - Kurosis The fourh ceral oe 4 E 4 I coais iforaio abou he shape of a disribuio. The propery of shape ha is easured by his oe is called urosis. The easure of (ecess) urosis: Disribuios: ) Mesouric disribuio , oderae i size Moes of a Rado Variable - Kurosis ) Playuric disribuio, sall i size ) Lepouric disribuio, large i size

19 Moes of a Rado Variable Eaple: The uifor disribuio fro o f, Fidig he oes f d d Fidig he ceral oes: f d d Moes of a Rado Variable Fidig he ceral oes (coiuaio): f d d aig he subsiuio w w w dw if eve if odd 9

20 Moes of a Rado Variable Hece, 3, Thus, var The sadard deviaio The easure of sewess The easure of urosis var Aleraive easures of dispersio Whe he edia is used as a ceral easure for a disribuio, here are several choices for a easure of variabiliy: -The rage -he legh of he salles ierval coaiig he daa -The ierquarile rage -he differece bewee he 3 rd ad s quariles. -The ea absolue deviaio (/) Σ i i ceral easure() -The edia absolue deviaio (MAD). MAD= i ( i - () ) These easures are ore robus (o ouliers) esiaors of scale ha he saple variace or sadard deviaio. They especially behave beer wih disribuios wihou a ea or variace, such as he Cauchy disribuio.

21 Rules for Epecaios Rules: E g g p g f d if. E c c where c is a cosa Proof: if if is discree is coiuous he g c E g E c cf d The proof for discree rado variables is siilar. c f d c Rules for Epecaios. E a b ae b where a, b are cosas Proof if he g a b E a b a b f d The proof for discree rado variables is siilar. a f db f d ae b

22 Rules for Epecaios 3. var E Proof: f d The proof for discree rado variables is siilar. E E var E f d f d f d f d E E Rules for Epecaios a ba 4. var var Proof: a b Ea b ae b ab var a b E a b a b E a b a b E a ae a var

23 Moe geeraig fucios The epecaio of a fucio g() is give by: g p if is discree E g g f d if is coiuous Defiiio: Moe Geeraig Fucio (MGF) Le deoe a rado variable. The, he oe geeraig fucio of, (), is defied by: - E e e p if e f d if is discree is coiuous 3

24 MGF: Eaples. The Bioial disribuio (paraeers p, ) p p p,,,, The MGF of, () is: E e e p e p p ep p ab a b e p p MGF: Eaples. The Poisso disribuio (paraeer ) The MGF of, () is:,,, p e! E e e p e e! e e u u e e e usig e!! e e 4

25 MGF: Eaples 3. The Epoeial disribuio (paraeer ) f The MGF of, () is: e E e e f de e d e e d udefied MGF: Eaples 4. The Sadard Noral disribuio ( =, = ) The MGF of, () is: f e E e e f d e e d e d 5

26 MGF: Eaples We will ow use he fac ha e de e d b a e d for all a, b a We have copleed he square e e d e This is MGF: Eaples 4. The Gaa disribuio (paraeers, ) f e The MGF of, () is: E e e f d e e d e d 6

27 MGF: Eaples We use he fac b a a a b e d a b for all, e d e d Equal o The Chi-square disribuio wih degrees of freedo = /, =½): MGF: Properies. () = E e, hece E e E Noe: The MGFs of he followig disribuios saisfy he propery () = ii) Poisso Dis' iii) Epoeial Dis' iv) Sd Noral Dis' v) Gaa Dis' e e i) Bioial Dis' e p p e 7

28 MGF: Properies 3 3.! 3!! We use he epasio of he epoeial fucio: e E e 3 u u u u u! 3!! 3 3 E! 3!! 3 3 E E E E! 3!! 3 3! 3!! MGF: Properies d 3. d Now 3 3! 3!! 3 3! 3!! 3!! ad!! ad 4 3 coiuig we fid 8

29 MGF: Applyig Propery 3 - Bioial Propery 3 is very useful i deeriig he oes of a RV. Eaples: e p p i) Bioial Dis' e p p pe e p p pe p p e p p e p e e p p e '' pe e p p e p e p p pe e p p e p p ( ) p[ p p] p[ p q] p pq MGF: Applyig Propery 3 - Poisso ii) Poisso Dis' e e e e e e e e e e e e e e e e e e e e e e e e 3e 3 3 e 3 e e e e e 9

30 MGF: Applyig Propery 3 - Poisso To fid he oes we se =. e e e e e e e 3 e e 3 MGF: Applyig Propery 3 - Epoeial iii) Epoeial Dis' d d d d ! 5 5! 3

31 MGF: Applyig Propery 3 - Epoeial Thus, 3!! We ca calculae he followig popular descripive saisics: - σ = μ = μ - μ = (/λ ) - (/λ) = (/λ) - γ = μ 3 /σ 3 = (/λ 3 ) / [(/λ) ] 3/ = - γ = μ 4/σ 4-3 = (9/λ 4 ) / [(/λ) 4 ] - 3 = 6 MGF: Applyig Propery 3 - Epoeial Noe: he oes for he epoeial disribuio ca be calculaed i a aleraive way. This is doe by epadig () i powers of ad equaig he coefficies of o he coefficies i: 3 3! 3!! 3 u u u u 3 3 Equaig he coefficies of we ge:!! or 3

32 MGF: Applyig Propery 3 - Noral iv) Sadard oral disribuio ()=ep( /) We use he epasio of e u. e u 3 u u u u u!! 3!! 3 e! 3!! 4 6 3! 3!! We ow equae he coefficies i:!!! MGF: Applyig Propery 3 - Noral If is odd: =. For eve : or!!!!! 4! Thus,, 3, 4 3! 3

33 The log of Moe Geeraig Fucios Le l () = l () = he log of he MGF. The l l l l l l l The Log of Moe Geeraig Fucios Thus l () = l () is very useful for calculaig he ea ad variace of a rado variable. l. l 33

34 Log of MGF: Eaples - Bioial. The Bioial disribuio (paraeers p, ) l l e p p e p q l e p q epq l e p l epq ep ep q ep ep l p p p q p q p p q p p l pq Log of MGF: Eaples - Poisso. The Poisso disribuio (paraeer ) e e l l e l e l e l l 34

35 Log of MGF: Eaples - Epoeial 3. The Epoeial disribuio (paraeer ) udefied l l l l if l l l l Thus ad Log of MGF: Eaples - Noral 4. The Sadard Noral disribuio ( =, = ) e l l l, l l l Thus ad 35

36 Log of MGF: Eaples - Gaa 5. The Gaa disribuio (paraeers, ) l l l l l l Hece l ad l Log of MGF: Eaples Chi-squared 6. The Chi-square disribuio (degrees of freedo ) l l l l l Hece l ad l 36

37 Characerisic fucios Defiiio: Characerisic Fucio Le deoe a rado variable. The, he characerisic fucio of, φ () is defied by: - i ( ) E( e ) Sice e i = cos() + i si() ad e i, he φ () is defied for all. Thus, he characerisic fucio always eiss, bu he MGF eed o eis. Relaio o he MGF: φ () = i () = (i) Calculaio of oes: ( ) i 37