Cursul 3 - Distributiile variabilelor aleatoare

7/30/2019 Cursul 3 - Distributiile variabilelor aleatoare

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Dgstrgoutgglb virgioglblar ilbitairb

Eursul 5



Zbhtru a papulitgb dbnghgti, ngbeirbg virgiolb ilbitairb gg bstb

isaegiti a dgstrgoutgb eirb dbnghbstb praoioglgtitbi db ipirgtgb i

ngbeirbg vilarg pasgoglb i iebstbg virgioglb &diei bxgsti uh huìr nghgtìsuriogl db virgioglb' siu taitb sbturglb pasgoglb db vilarg pasgoglb&diei virgioglblb suht dbnghgtb pb da`bhgul rbil'(

A virgioli bstb a eirietbrgstgei aosbrviti siu ìsuriti( A virgiolidgh papulitgi studgiti paitb ng ilbitairb siu eahstihti(

Aosbrvitgb; virgiogli bstb ìrg`bi eirb virgizi db li uh ghdgvgd liiltul(

Eb bstb a virgiogli ilbitairb1



A dgstrgouȓgb i praoioglgtitglar bstb uh àdbl ìtbìtge eirb bxprgìlbmituri dghtrb vilairbi virgioglbg sg praoioglgtitbi bg db ipirgȓgb èhpapuliȓgb(

Dgstrgoutgi praoioglgtitglar

Dgstrgoutgb dgserbti Dgstrgoutgb eahtghui




x

x p x p 9'& sg 9'&:

Dgstrgouȓgi dgserbtą bstb bstb nar`iti dgh oirb i eirar ghiltg`b bstbprapartgahili eu praoioglgtitbi( Zraoioglgtitbi virgioglbg ilbitairb V dbi lui vilairbi xg bstb;

gg x p x x Z

Dgstrgouȓgi eahtghuą bstb a euroą hbtbdą, eu irgi db suo euroi bmiląeu praoioglgtitbi, istnbl èheït praoioglgtitbi ei x si sb inlb èh ghtbrvilul&i, o' bstb;

dx x n o xi Z o

i

Ub i`ghtbstb ei;

&9'

&?'



@bdgi i uhbg dgstrgouȓgg db praoioglgtitg bstb a `ąsurą itbhdghȓbg ebhtrilb i dgstrgouȓgbg( @bdgi sb dbnghbstb istnbl;

9g

dgserbtix,

eahtghuix,

gg x p x

dx x xn

Zbhtru a virgiogli ilbitairb dgserbti eu H vilarg sg p&xg' 0 9!H,beuitgi &5' sb rbdueb li;

&5'

H

x H

g

g 9




Dgspbrsgi ?

bxprg`i g`prąȗtgbrbi, rąspïhdgrbi siuvirgioglgtitbi uhbg dgstrgouȓgg3 bstb db dbnghgti istnbl;

9g

?

?

?

dgserbtix,

eahtghuix,

gg x p x

dx x n x

Zbhtru a virgiogli ilbitairb dgserbtą eu eu H vilarg sg p&xg' 0 9!H,beuitgi &4' sb rbdueb li;

&4'

H

x H

g

9g

?

?

Gh prietgei sb utglgzbizi ioitbrbi stihdird;

H

x H

g

9g

?

?




Dgstrgoutgg dgserbtb - dgstrgoutgglb nrbevbhtblar rblitgvb pbhtru bxpbrg`bhtbìsurioglb(

_ispuhsurg di - hu(

• Dgstrgoutgi Obrhaullg• Dgstrgoutgi ogha`gili

• Dgstrgoutgi ogha`gil - hbmitgvi

• Dgstrgoutgi Zagssah

• Dgstrgoutgi fgpbrmba`btrgei

• Dgstrgoutgi ̀ ultgha`gili

Uuì rispuhsurglar ObrhaullgHuìrul db ghebreirgpbhtru il clbi bvbhg`bht

Zuhetb ghtr-uh spitgu dit

Huìrul db kudbeitg ghightbiprgùlug sueebs

_bzultitb ùltgplb pasgolb pbhtru ngbeirbghebreirb(

Dgstrgoutgg dgserbtb g`partihtb



Dgstrgoutgi ogha`gili

Rh praebs eahstą dghtr -a sbevbhȓą db h ghebreirg ghdbpbhdbhtb sggdbhtgeb(

Ghebreirb ghdbpbhdbhti - rbzultitul ngbeąrug ghebreirg hu dbpghdb dbrbzultitblb ghebreirglar ihtbrgairb(

_bzultitul ngbeąrbg ghebreirg bstb ngb uh sueebs, ngb uh bȗbe(

Diei praoioglgtitbi sueebsulug, p bstb eahstihtą pbhtru argeb praebs,ituheg hu`ąrul db sueebsb x dgh h ghebreirg irb dgstrgouȓgiogha`gilą eu piri`btrgg h sg p, dbnghgtg istnbl(

xh x

p p xh x

h

x Z

'9&')&)

)

'&Zraoioglgtitbi;

@bdgi;hp

Ioitbrbi stihdird;'9& ph

&2'



Dgstrgouȓgi ogha`gilą

Dgstrgouȓgi ogha`gilą sb utglgzbizi nrbevbht èh ghmghbrgi dbeilgtitb(

Bstb `adblul db praoioglgtitb earbspuhzątar pbhtru prblbvirbidb praob db li a papuliȓgb ghnghgti uhdb;

p bstb nrieȓguhbi db pradusb dbnbetb siu hbeahnar`b dghpapuliȓgb(

x bstb hu`ąrul db irtgealb hbeahnar`b dghtr -uh bȗihtgahilbitargu db h blb`bhtb(

Db bxb`plu, dieą p 0 :,9: ȗg h 0 92, ituheg praoioglgtitbi db iaoȓghb uh pradus hbeahnar` x sb eileulbizą eu nuhetgiogha`dgst dgh Bxebl(



Narì nuhetgglar db praoioglgtitb ogha`gili

Aosbrvitgb; Eu eit erbstb h sg p sb iprapgb db :,2, dgstrgoutgi sbiprapgb db narì sg`btrgei i dgstrgoutgbg harìlb(



Bxpbrg`bht; Ub iruhei 2 àhbdb(Bvbhg`bhtul db ghtbrbs; eitb àhbdb eid pb pirtbi eip(

Zbhtru a àhbdi, praoilgtitbi ei si ipiri eipul &uh sueebs' bstbp 0 :(2( Huìrul db eipbtb eirb ipir li iruheirbi i 2 àhbdb

bstb a virgioli ogha`gili ilbitairb eu h 0 2 sg p 0 :(2(

hr( db eipbtb Aosbrvit Tbarbtge

: 9 9(2>9 99 =(69? 99 92(>55 97 92(>54 > =(692 ? 9(2>

Bxpbrg`bhtul sb rbpbti db 2: db arg(

: 9 ? 5 4 2

?:969>949?

9:6>4?

:

Bxb`plb db dgstrgoutgb ogha`gili



_bzultit Nrbevbhti Nrbevbhti rblitgvi? ? :(:4

5 4 :(:64 4 :(:62 5 :(:>> 4 :(:6= = :(946 7 :(967 5 :(:>

9: 2 :(9:99 = :(949? ? :(:4tatil 2: 9(::

Daui zirurg sb iruhei sg sb ghsu`bizi ghdgeitgglb lar &virgioliilbitairb x'( Bxpbrg`bhtul sb rbpbti db 2: db arg(

Z&x 6' 0

Z&x >' 0

Z&4 x 9:' 0

55!2: 0:(>>

5=!2: 0 :(=4

54!2: 0 :(>6

Bxpbrg`bhtul eu daui zirurg

Zraoioglgtitgglb virgioglbg

ilbitairb x;



A dgstrgouȓgb dgserbtą utglą èh eahtralul stitgstge il eilgtąȓgg bstb dgstrgouȓgiZagssah( A virgiogli ilbitairb irb dgstrgoutgb Zagssah eu piri`btrul ,diei nuhetgi db praoioglgtitb bstb;

b0?(=96…

Dgstrgoutgi Zagssah

b x

x p x

)'&

Rh nbha`bh ilbitargu eirb ipirb &huìrulug db dbnbetb' pb uhgtitbi dbsupriniȓą, db valu`, db tg`p, bte, bstb ipraxgìt eu dgstrgouȓgi Zagssah(

Bx(; Hu`ąrul db dbnbetb eb ipir èhtr -uh dgspazgtgv sb`geahduetar sbdgstrgougb Zagssah eu 0 4( Zraoioglgtitbi ei uh dgspazgtgv sb`geahduetar sblbetit ilbitargu si eahȓghi dauą siu ìg puȓghb dbnbetb bstb;

0

? 0

Zraoioglgtitbi;

@bdgi;Dgspbrsgi;

pbhtru 8 :

Ub eileulbizi eu nuhetgi ZAGUUAH dgh Bxebl



Dgstrgoutgi Zagssah

Dgstrgoutgi praoioglgtitgg Zagssah pbhtru dgnbrgtb vilarg ilb piri`btrulug (



Dgstrgoutgi fgpbrmba`btrgei

Ghtr-a papuliȓgb nghgtą narìti dgh H blb`bhtb, eitbvi dgh iebstbblb`bhtb sb èheidrbizą èhtr -a elisą db ghtbrbs, D&D≤H'(

Rh bȗihtgah ilbitargu db h blb`bhtb eu x blb`bhtb sb èheidrbizą èh elisi db ghtbrbs( x ipirtghb uhbg dgstrgouȓgg db praoioglgtitbfgpbrmba`btrgeb diei;

D'`gh&h,(((,:,9,?,((((x

h

H

xh

D H

x

D

x p

@bdgi;

Zraoioglgtitb;

Dgspbrsgi;

H

hD

9

9?

H

h H

H

D

H

hD

))

)

oio

i

o

i

Ugòal pbhtru eaòghitgg



Dgstrgoutgi fgpbrmba`btrgei

Dgstrgoutgi fgpbrmba`btrgeą bstb àdblul db praoioglgtitbearbspuhzątar sblbetirgg uhug bȗihtgah ilbitargu db h aogbetb,nąrą èhlaeugrb, dghtr -a ùlȓg`b db H blb`bhtb dgh eirb D suhthbeahnar`b siu eu dbnbetb(

Èh iebstb iplgeiȓgg, x rbprbzghtą hu`ąrul db irtgealb hbeahnar`bdgh bȗihtgah(

Db bx(; sb prbsupuhb eą a ùlȓg`b eahȓghb 9:: db irtgealb,dghtrb eirb 2 hu suht eahnar`b eu ebrghtblb( Diei 9: irtgealb

suht sblbetitb li èhtï`plirb, nąrą èhlaeugrb, eirb bstbpraoioglgtitbi db i mąsg uhul siu ìg puȓghb blb`bhtbhbeahnar`b èh bȗihtgah1

Ub utglgzbizi nuhetgi FWZMBA@DGUT dgh Bxebl



Dgstrgoutgi Ziseil ȗg dgstrgouȓgg èhrudgtb

Dgstrgouȓgglb Ziseil, li nbl ei dgstrgouȓgi ogha`gilą, sboizbizi pb ghebreirbi lug Obrhaullg(

Ub g-iu èh eahsgdbrirb a sueebsguhb db ghebreirgghdbpbhdbhtb, ngbeirb eu praoioglgtitbi db sueebs p ȗg sbhatbizą eu x ghebreirbi gh eirb sueebsul ipirb il r lbi( x

bstb a virgioglą ilbitairb eu dgstrgouȓgb Ziseil dbpraoioglgtąȓg dbnghgti istnbl;

r x9

9

9

r xr p p

r

x x p

p

r

?

? 9

p

pr

Zraoioglgtitb;

@bdgb;

Dgspbrsgb;

&2'



Dgstrgoutgi Ziseil ȗg dgstrgouȓgg èhrudgtb

Dauą eizurg spbegilb ilb dgstrgouȓgbg Ziseil sb iplgei gh EUE

Diei r 8 : Dgstrgouȓgi sb hu`bȗtb dgstrgouȓgi ogha`gilą hbmitgvą( Beuiȓgi &2'

bstb dgstrgouȓgb ogha`gilą hbmitgvą pbhtru r ghtrbm(Dgstrgouȓgi ogha`gilą hbmitgvą bstb utgli pbhtru tgpurg dgnbrgtb db ditb

/hu`ąrioglb/, eu` ir ng ipirgȓgi hbeahnar`gtąȓglar pb uhgtitb dbpradus(

Èh dgstrgouȓgi ogha`gilą, sb stioglbstb `ąrg`bi bȗihtgahulug ȗg sbaosbrvi hu`ąrul db sueebsb(

Gh dgstrgouȓgi ogha`gilą hbmitgvą, sb stioglbstb hu`ąrul db sueebsb

ȗg sb dbtbr`ghi dg`bhsguhbi bȗihtgahulug( Iebst eahebpt driplgei èh dgnbrgtb tgpurg db praolb`b db bȗihtgahirb( Dieą r 0 9

dgstrgouȓgi bstb mba`btrgeą, idgei dgstrgouȓgi hu`ąrulug db ghebreirgObrhaullg pïhą li prg`ul sueebs(



• Dgstrgoutgi harìli

• Dgstrgoutgi Lam Harìl

• Dgstrgoutgi Mi`ì

• Dgstrgoutgi ?

• Dgstrgoutgi N• Dgstrgoutgi t

• Dgstrgoutgi ]bgoull• Dgstrgoutgi ̂ ilarglar bxtrb`b

&TGp G sg tgp GG'

Ub utglgzbizi pbhtru tbstblbstitgstgeb(

A ìrb pirtb i stitgstgeggghnbrbhȓgilb sb nuhdi`bhtbizipb iebisti dgstrgoutgb

îrgiolblb dgh `bdgul iògbhtil

Duriti pïhą li bȗbe,

Dgstrgouȓgg ilb duritbg db vgiȓą

Dgstrgoutgg eahtghub

îrgioglb ilbitairb eahtghub sb dbnghbse pbhtru hu`brb eu

vilarg eahtghub gh da`bhgul rbil( Zraoioglgtąȓglb sb eileulbizi

pbhtru taitb sbturglb db hu`brb pasgoglb(



Dgstrgoutgi harìli

Dgstrgouȓgb harìlą bstb ebi ìg g`partihti dgstrgouȓgb, itït èhtbargb eit ȗg gh iplgeitgglb stitgstgegg(Dieą x bstb a virgioglą ilbitairb harìlą, ituheg dgstrgouȓgi db

praoioglgtitb i lug x bstb;

x- ?

9?

2,:

x

b x n

@bdgi dgstrgoutgbg

harìlb bstb &- < < #' sg dgspbrsgi bstb? 8 :(



>6" dgh irgi tatilibstb ghtrb - sg #(

>6(:'& V Z

Dgstrgoutgb sg`btrgei dbnghgti pb ghtbrvilul &-, # ' i eirbg nar`i bstbdbnghgti db dag piri`btrg, `bdgi, , eirb bstb ebhtrul dgstrgoutgbg, sgioitbrbi stihdird, , eirb dbtbr`ghi g`pristgbrbi dgstrgoutgbg(

Irgi0(>6

Dgstrgoutgi har`ili



• Nuhetgi db dbhsgtitb bstb gh nar`i db elapat, sg`btrgei(• >6" dgh irgi db suo euroą bstb èhtrb (

• 72" dgh irgi db suo euroą bstb èhtrb ?(

• 77(=" dgh irgi db suo euroą bstb èhtrb 5(

(>6

(72

??

Zraprgbtitglb dgstrgouȓgbg har`ilb

_bmuli b`pgrgei



Dgstrgoutgi harìli

Dgstrgouȓgi euùlitgvą harìlą bstb dbnghgtą ei praoioglgtitbi virgioglbgilbitairb harìlą x ≤ i;

?

9?

2,:

i x

dxbi N i x Z

Ghtbmrili hu paitb ng bviluiti èh nar`ą èhefgsą( Ub utglgzbizisefgòirbi db virgioglą;

x

[

Bviluirbi sb nieb ghdbpbhdbht db ȗg ?

;

ii z Z i x Z

&' bstb nuheȓgi db rbpirtgȓgb euùlitgvą i dgstrgouȓgbg harìlb stihdird

eu 0 : sg ? 0 9(



Taitb virgioglblbilbitairb harìlb pat ngearblitb eu virgioglbilbitairb harìlbstihdird(

A virgiogląilbitairb harìląstihdird irb`bdgi : ȗg dbvgiȓgi

stihdird 9(

Dgstrgoutgi harìli stihdird

5 5??

: #9 #? #5-9-?-5



Db eb bstb g`partihti dgstrgoutgi stihdirdgziti1 Zbhtru eą praoioglgtitbi argeirug bvbhg`bht il uhbg

virgioglb ilbitairb harìlb eu ioitbrbi stihdird ȗg

`bdgi ditb paitb ng eileuliti dgh tioblblb dgstrgouȓgbgharìlb stihdird(

Dgstrgoutgi harìli stihdird



_^Dgst-?>

Tiobl dgstrgoutgbg har`ilb

stihdirdgziti

[:(: 9(:

Tioblul 9 dą zahblb ląsitb db z(



Tbarbì lg`gtą ebhtrilą

Dgstrgouȓgb harìlą sb prbsupuhb ei bstb àdblul db praoioglgtitbidbevit pbhtru a virgioglą ilbitairb( Iebisti gpatbzi trbougbvbrgngeiti(

Tbarbì lg`gti ebhtrili;

Dieą x9, x?,( ( ( , xh suht virgioglb ilbitairb ghdbpbhdbhtb eu `bdgi ȗg dgspbrsgi ? ȗg dieą y 0 x9 # x? #( ( ( # xh, ituheg dgstrgouȓgi;

h

g

g

h

g

g y

9

?

9

sb iprapgb dgstrgouȓgi H&:, 9' diei h sb iprapgb db ghnghgt(

_bzulti eą suì i h virgioglb ilbitairb dgstrgougtb ghdbpbhdbht bstb

ipraxgìtgv harìlą, ghdgnbrbht db dgstrgouȓgglb virgioglblar ghdgvgduilb(



Dgstrgouȓgi lam-harìlą

Diei virgioglblb sb earblbizi prghtr-a rbliȓgb bxpahbhȓgilą, x 0 bxp&w' sg bxpahbhtul w bstb a virgioglą ilbitairb, ituheg sg x bstb avirgioglą ilbitairb(

Diei w sb dgstrgouȓgb harìl, dgstrgouȓgi lug x sb hu`bstb dgstrgouȓgb lam-harìlą( Hu`blb rbzultą dgh trihsnarìrbi lh &x' 0 w(

Nuheȓgi db dgstrgouȓgb euùlitgvą pbhtru x bstb;

ii z Z iw Z iw Z i x Z i N

lhlhlhbxp



Ngb w a dgstrgouȓgb harìlą eu `bdgi ȗg dgspbrsgi ?, ituheg

x 0 bxp &w' bstb a virgioglą ilbitairb eu dgstrgoutgi lam-harìlą(

Nuhetgi dgstrgoutgbg lam-harìlą bstb;

x:

?

lhbxp

?

9?

?

x

x x n

?

bxp

?

9???? bb

@bdgi;

Dgspbrsgi;

Dgstrgouȓgi lam-harìlą



Tb`i db eisi

Iltb tgpurg db dgstrgoutgg eahtghub



îrgiogli [

Ub stihdirdgzbizi vilarglb prgh bxprgìrbi iebstari èh nuhetgb dbioitbrbi stihdird

Ub `ąsairą dgstihȓi db li `bdgb &èh eizul èh eirb `bdgi bstbikustitą li :' li virgiogli;

s

x x [ g

g

Zrbzbhtirbi ditblar gh narì stihdird pbr`gtb ei ditblb si sbea`pirb eu ditblb iltar dgstrgoutgg@bdgi bstb bmili eu zbra

Ioitbrbi stihdird bstb bmili eu uhuîlarglb nuhetgbg db dgstrgoutgb stihdirdgzitb sb gi dgh tioblb(



Zraolbì; Ghiltg`bi aùlug dgh Hbihdbrtfil Zapuliȓgi sefblbtblar Hbihdbrtfil Ghiltg`bi ìsuriti pirb są ur`bzb a dgstrgouȓgb harìlą…

x 0 9>5(= e`

s 0 2(=7 e`

Ghtrboirbi 9; Eb prapartgb i papulitgbg irb ghiltg`bi 89>2 e`1

[ 0 1[ 0 &9>2 -9>5(='!2(=7 0 :(?5

Nalasghd Tioblul E-?nde&:(?5' 0 :(4:7:2

_ispuhs 4:(7"

îrgiogli [



(466:5 (464:2 (46::> (4=>:6



nde dn&x' 0 :(76 Ub nalasbstb Tioblul E-93 eiuti :,76 rbzulti [ 0 ?(:2

Ub eileulbizi x dgh nar`uli lug [;

x 0 ?(:2 ½ 2(=7 # 9>5(= 0 9=2(> e`

x [ x gg

Ghtrboirbi ?; Eirb bstb ghiltg`bi gh eirb sb gheidrbizi 76" dgh papuliȓgb1

_ispuhs; F 0 9=2,> e`(

^irgiogli [



Dgstrgouȓgi `bdgbg bȗihtgahulug

hu sb ȗtgb narì dgstrgouȓgbg papuliȓgbg3 sb prbsupuhb ei bstb a dgstrgoutgb harìli

Ub utglgzbizi praprgbtąȓglb dgstrgouȓgbg harìlb Ub vgzuilgzbizi dgstrgoutgi `bdgglar uhug hu`ąr ìrb db

bȗihtgaihb…



db aogebg dgspuhb` db uhul dghtrb iebstb bȗihtgaihb ((( hu putb` ȗtg uhdb sb pazgtgahbizi `bdgi bsihtgahulug èh ripart eu

`bdgi papuliȓgbg, dir putb` bstgì uh ghtbrvil gh eirb ir putbi ng ((( iebst lueru dbpghdb db

ìrg`bi bsihtgahulug a bstgìrb i dgspbrsgbg papuliȓgbg

Eu eit bȗihtgahul bstb ìg `ge ȗg papuliȓgi ìg dgspbrsiti, eu ititpraoioglgitbi ei ̀ bdgi bȗihtgahul si ngb ìg dbpirtb db `bdgi

papuliȓgbg bstb ìg ìrb( Iebsti ingrìtgb sb rbnlbetą èh beuiȓgi utglgzitą pbhtru eileulul dgspbrsgbg`bdgbg bȗihtgaihblar;

h s x

??




Dgstrgoutgi `bdgglar earbspuhzitairb sbturglar ilbitairb db bsihtgaihb luitb dghtr-a papulitgb tghdb sprb dgstrgoutgi harìli(

Eahnar`gtitbi eu a dgstrgouȓgi harìlą erbȗtb eu

dg`bhsguhbi bȗihtgaihblar(

@bdgglb sb dgstrgougb èh kurul vilargg `bdgg i papuliȓgbg

x V

Tbarbì lg`gtbg ebhtrilb



Ioitbrbi stihdird i `bdgglar bsihtgaihblar bstb brairbi stihdird i `bdgbg bstg`itb;

hhh sb 9

?

Zb oizi uhug bȗihtgah, sb eileulbizi, eu a ihu`gti praoioglgtitb, uhghtbrvil eirb eahȓghb `bdgi papuliȓgbg utglgzihd brairbi stihdird;

h [ x

uhdb [ 0 9(7> pbhtru praoioglgtitbi :(72




Bxb`plu

virnurg db simbiti h 0 2:

luhmg`bi `bdgb x 0 ??(> ``

s 0 4(? ``

Brairbi stihdird 0 11

??(> ¹ 9(7>½4,?½&2:'-:,2

??(> ¹ 9(9> `` Eu praoioglgtitbi db 72" `bdgi papuliȓgbg sb inli èh ghtbrvilul

?9(4 - ?5(6

274(:2:

?(4

h

s

h s [ x




Ghtbrvil db gherbdbrb

Ghtbrvilblb db gherbdbrb suht bxprgìtb gh tbr`bhgg praoioglgtitggoizitb pb primul db sb`hgngeitgb (

Rh ghtbrvil db gherbdbrb &9- ', bxprgì niptul ei praoioglgtitbi eipiri`btrul papulitgbg si sb misbisei gh ghtbrvilul db gherbdbrbbstb 9- , gir praoioglgtitbi ei piri`btrul papulitgbg si sbmisbisei gh iniri ghtbrvilulug db gherbdbrb bstb (

Ebl ìg dbs nalasgtb suht lg`gtblb db gherbdbrb &primul dbsb`hgngeitgb' db 72", eirb ghdgei niptul ei praoioglgtitbi ei

vilairbi `bdgbg hbeuhaseuti dgh papulitgb si sb inlb ghtrbiebstb lg`gtb bstb db 72", gir gh iniri lar bstb db 2"(



Ghtbrvil db gherbdbrb - prim db sb`hgngeitgb



Lg`gtblb db gherbdbrb ilb `bdgbg papulitgbg &²' dbnghbse ghtbrvilul dbgherbdbrb pbhtru `bdgi papulitgbg pb oizi `bdgbg bsihtgahulug sgbstg`irbi dgspbrsgbg papulitgbg(

Zbhtru bsihtgaihblb eu h 8 5: lg`gtblb db gherbdbrb sb oizbizi pbripartul ergtge earbspuhzitar praoioglgtitgg isaegitb(

Zbhtru bsihtgaihb eu h < 5: lg`gtblb db gherbdbrb sb oizbizi pbvilarglb lug t dgh tioblb, earbspuhzitairb hu`irulug db mridb dblgobrtitb sg praoioglgtitgg isaegitb(

Lg`gtblb db gherbdbrb ilb vilargg `bdgg



Dgstrgoutgi t

Dgstrgoutgi `bdgglar bsihtgaihblar `geg &rbdusb hu`brge' hu bstb

tatdbiuhi harìl dgstrgougti, efgir diei papulitgi dgh eirb dbrgvibstb harìl dgstrgougti(

Zrgh bsihtgah `ge sb èhtblbmb iebl bsihtgah eu h < 5:(

Diei papulitgi bstb dgstrgougti harìl, dgstrgoutgi bsihtgaihblar ur`bizi dgstrgoutgi t, dgstrgoutgb eirb rbnlbeti a ìg ìrb virgitgbditariti sihsbg gh ea`piritgb eu dgstrgoutgi harìli(

Bstb eahtghui, sg`btrgei, uhgàdili, virgizi db li ― li # , bstb ùlt

ìg lirmi gh ea`piritgb eu dgstrgoutgi harìli(

Diei bsihtgaihblb erbse, dgstrgoutgi t sb iprapgb db dgstrgoutgi harìli,gir pbhtru uh huìr ghnghgt db mridb db lgobrtitb eblb ? dgstrgoutgg suhtgdbhtgeb sg vilarglb ergtgeb ilb lug t suht bmilb eu ripartul ergtge ildgstrgoutgbg harìlb



Dgstrgoutgi t bstb sg`btrgei, li nbl ei a dgstrgoutgb har`ili, dir dbvghbtat `ig pliti aditi eu `gesarirbi da`bhsguhgg bsihtgahulug

Dgstrgoutgi t



Dgstrgouȓgi `bdgbg bȗihtgahulug pbhtru bsihtgaihb `geg

Zbhtru i eileuli ghtbrvilul db èherbdbrb earbspuhzitar `bdgbg uhar bsihtgaihb eu huìr `ge db ìsuritarg &harìl siu ipraipbharìl dgstrgougtb' sb utglgzbizi a narùli àdgngeiti;

t sb aotghb dgh tioblul praoioglgtitglar pbhtru dgstrgoutgi t siunalasghd pramrihul BVEBL(

h

t x

Èh tioblul dgstrgoutgbg t, vilarglb suht mrupitb èh nuhetgb db primul dbsb`hgngeitgb db mridblb db lgobrtitb ― dn 0 h -9(

Cursul 3 - Distributiile variabilelor aleatoare

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