χ be any function of X and Y then

Transcription

1 We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto, ( y). That s f < dχ y Y y dy (, y) ob, Y. f Y, Theoe Let g (, y) χ be ay fucto of ad Y the o [ g(, Y )] g(, y) f (, y) [ g(, Y )] g(, y) f (, y) all χ all y Y Y dχdy aple; We cosde ow, two ado vaables whose jot pdf (o pf) s gve by the at show below Table Table Y /8 / /8 /8 / /8 We defe g (, y) y y g, Y dectly, ad the by usg Theoe. Fd [ ( )] Let Z - Y Y. By specto, Z ca tae o the values,,, ad, wth pobablty,p(z), as show Table (whch ca easly be see to follow fo Table ). Table Table y /8 / /8 /8 / /8 z p Z (z) / / /

2 Fo defto of epectato the, [ g( Y )] [ g (, Y )] ( Z ) z p Z ( z ) all z, s equal to : We get the sae aswe fo [ g( Y )] gve Table : [ Y ] g( y) [ g(, Y )] p 8 y, by applyg Theoe to the jot pdf The advatage, of couse, ejoyed by the latte soluto s that we avod the teedate step of havg to detee p Z (z). Aothe useful popety s the followg: Theoe I If ad Y ae depedet.v. the [ Y ] [ ] [ Y]. e. [ Y ] Yf ( y) f d - coelato ddy Yf dy Y Yf f [ ] [ Y ] Y ddy aple Two fou dce ae tossed, the epected value of the poduct of the two faces s [ Y ] [ ] [ Y ] [ ] ( ) [ ] Y [ Y ]. 5

3 The ea,, of a.v. s aalogous to the cete of gavty of f(). Aothe potat paaete s ts vaace,, o dspeso defed by We ote o [( ) ] ( ) f ( ) [( ) ] ρ[ ] [( ) ] [ ] Stadad devato d [ ] [] [ ] [ ] [] whee [] I, geeal, a oe coplete specfcato of the statstcs of ca be gve f oe ows ts MOMNTS,, defed by Clealy etc. [ ] f ()d [] We also have aothe stadad paaete called CNTRAL MOMNTS,.e., they elate the statstcs to the ea value: we have [( ) ] ( ) f ( ),, d We ca wte [( ) ] ( ) ( ) boal epaso ad due to the addtvty popety,

4 ( ) ( ) [ ] ( ) ( ) I patcula We could also develop tes of Thee ae two potat popetes of vaaces. ) Defe b a Y, the [ ] [ ] [ ] ( ) ( ) [ ] [ ] ( ) [ ] [ ] () Va a a a b a b a Y Va b a b a Y aple A ado vaable s descbed by the pdf () < < f What s the stadad devato Y, whee Y? Fst, we eed to fd the vaace of. But ( ) ( ) ( ) ( ) 8 ad Va d d

5 The, by Theoe I, above, Va 9 8 ( Y ) ( ) Va( ) whch aes the stadad devato Y equal to, o.7. ) If Y whee ae depedet ado vaables. The, Va(Y) Va( ) Va ( ) Va( ) aple: The boal ado vaable beg the su of depedet Beoull s s a obvous caddate fo the above popety. Let deote the ube of successes occug o the th tal. The wth pobablty wth pobablty p - p Wte Y.. We wat to fd the vaace of Y. But ad ( ) p q p ( ) () p () ( p) so Va( ) ( ) [ ( ] p p p( p) pq It follows, the, that the vaace of a boal ado vaable s p(-p): Va) Y) Va( ) p( p)

6 Tchebycheff's Iequalty Suppose s a abtay ado vaable wth desty f() ad a fte vaace ( ) f ( ) d ( ) f ( ) > d f > () d { } { } Tchebcheff's Iequalty Alteately, we ca wte Tchebycheff's Iequalty aothe fo as: { < } > oof: Defe as a sall, postve value ad cosde a abtay pobablty law (depcted hee as a oal oly fo covece) f() { < } We defe the vaace of a ado vaable as follows; Va () ( ) f () d ( ) f ( ) d ( ) f ( ) d ( ) f ( ) If we eglect the secod te, whch s always postve, we have eag d

7 Va () ( ) f () d ( ) f () ( ) f ( ) d f () [ ] [ ] f [ ] d d Hece, Tchebysheff's Theoe ca be suazed as follows: Let be a ado vaable wth a ea of ad a fte vaace of. The fo ay sall value, >, [ ] whee Ths s equvalet to ( < < ) ( < < ) To tepet ths let, the the age fo to ude whateve pobablty law ests, ust cota at least / / / pecee of the pobablty ass fo the ado vaable. aple: Cosde: the daly poducto of electc otos at a ceta factoy aveages, wth a stadad devato of. a) What ca be sad about the facto of days that wll have a poducto level betwee ad. b) Fd the shotest teval ceta to cota at least 9% of the daly poducto levels. Solutos: a) The evet,, s daly poducto level of ay day { } { } { } > we see Hece, fo ( ) ad sce

8 > "At least 75% of days wll have poducto ths teval." So, { } { }. 75 c) Fd shotest teval of poducto values ceta to cota at least 9% of daly poducto. aga epesets daly poducto. We eed to fd. Set.9. {. < <. } {. < <.} O the teval 88. to 5. wll cota at least 9% of daly poducto levels. A oe detaled developet of the soluto s; So!.9 L [ L] [ ]. 9 [ ] > -/.9.. Wth, & - (. ) (. ) The teval s {88 to 5}. Ths eas that 9% of days wll have poducto betwee 88 to 5 otos. aple Suppose that s a epoetal ado vaable wth pdf () f e, >. a) Copute the eact pobablty that taes o a value oe tha two stadad devatos away fo ts ea. b) Use Chebycheff's equalty to get a uppe boud fo the pobablty ased fo pat a). Soluto: Gve () f e, > a) Calculate eact pobablty [ > ]

9 [] Let u dv e du d v e The udv vu [ ] ( e )() e e d vdu l e Let u e () () du v e e udv vu e d vdu e d ( ) dv e e d [ ] [ ] f() { > } [ ] e d. 5

10 b) [ ] [ ]. 5 aple A fa dce s tossed tes. Let deote the outcoe o the th oll. Use Theoe.. to get a lowe boud fo the pobablty that s betwee ad. Dce s tossed tes. Let outcoe o the th oll Fd lowe boud fo { Z betwee ad } { Z } Z But [ Z] [ ] [ ] [ ] [ ] [ ] ( 5 ) [ ] The 9 5 Sce depedet evets: Va So! ( Z ) Z [ Z]

11 [ ] o [ < ] [ < ( ) < ] [ < < ] [ < ( ) < ] Aothe eaple s the desg of a epeet. aple Use Chebysheff's equalty to get a lowe boud fo the ube of tes a fa co ust be tossed ode fo the pobablty to be at least.9 that the ates of the obseved ube of heads to the total ube of tosses be betwee. ad.? Defe whee h h ube of heads tosses head a sgle toss p p tal a sgle toss () ( ) [] h [ ] h h h pected value of Va [] ƒ Va[ h h h ] Va( h ) We wat the ato of to equalty z fo Chebysheff's

12 sce o ad Va. >. ( a) a Va( ).. sce (fa co)..9 <. > > /.9. / (.) ( ) 5 Aothe potat te s soethg called the Moet Geeatg Fucto Moet Geeatg Fucto We see that, eve f we do't ow what the DK o pdf s, we ca estate ceta statstcs ad pobabltes. Befoe we beg, we defe the followg G c,,, If () ( ) the th [ G() ] ( c) oet of.v. aoud costat c th If c, ( ) oet aoud og o just aw oet c If () ea value of the.v. the ( () ) o sply th cetal oet (secod cetal oet s th oet the vaace) aoud the ea Usg Moet Geealty Fucto, we ca evaluate all aw oets, whe they est, of the.v. studed. Defto: The Moet Geealty Fucto of a.v., Ψ ( θ ), ay be defed as the epected value of the fucto e θ, whe θ s sply a abtay eal ube.

13 ( θ ) Ψ θ ( θ ) ( e ) e all θ e f θ () p d () Ψ ay ot always est fo all θ. I othe wods, Ψ ca be ethe fte o fte (does ot est). It follows that a pobablty law o dstbuto ay ot always possess a oet geeatg fucto. Cosequetly, oe ay defe a fucto called a Chaactestc Fucto, ( e θ ), that always ests fo all values of θ. We ca tal late about ths, fo ow we cofe ouselves to Moet Geeatg Fuctos. To splfy, we wll assue M.G.F. ests. (Note: If M.G.F. ests, fo all θ L, L, the all oets est ad the dstbuto uquely deteed by t.) ( ) vaables). Tae epaso of e θ θ θ Cosde [ e ] e f ()d ( Note, Ths apples fo cotuous ad dscete ado θ e θ θ θ!! Hece Ψ Dffeetate wth espect to θ Ψ ' ad lettg θ, we get Ψ ' θ ( θ ) ( e ) ( θ ) f ( ) θ θ! d θ! () ( ) ( ) θ! ( θ ) ( ) θ( ) ( ) () () fst aw oet (the ea value)