roslina-chebyshev

DISTRIBUTIONS DISTRIBU TIONS OF RANDOM VARIABLE DISTRIBUSI VARIABEL RANDOM 1.11 Chebyshev’s Inequal!y "#e!a$sa%aan Chebyshev& selen'$a(nyaa $e!a$sa%a selen'$a(ny $e!a$sa%aan an )hebshev 1. *en *en+ah +ahulu uluan #,nse( a!au -u%us yan' be-hubun'an +en'an #e!a$sa%aan Chebyshev 

E$s(e$!as yan' be-$a!an +en'an sua!u va-abel -an+,%

 bla  $,n!nu  bla  +s$-! / Va-ans +a- 0 a$an +la%ban'$an +en'an  23 +an 4$a  2 a+a3 $!a %en+e5ns$annya %en+e5ns$annya +en'an 2 2   / E 6"0 7 8& 93 un!u$ 0 a+alah va-abel -an+,% 4ens +s$-! a!au $,n!nu. Un!u$ %en'h!un' va-ans  2  la%ban' +a- s%(an'an ba$u 

Fun's *e%ban'$! M,%en

 fungsi pembangkit momen +a- sua!u va-abel va-abel -an+,% 0. Msal$an Msal$an a+a blan'an (,s!5 (,s!5 h tx sehn''a un!u$ -h < t < h  e$s(e$!as %a!e%a!$anya3  E (e  ) a+a. :a+

B. T,($  #e!a$sa%aan Chebyshev Dala% ba'an n $!a a$an %e%bu$!$an !e,-e%a yan' %e%un'$n$an $!a un!u$ %ene%u$an ba!as a!as "a!au ba;ah& un!u$ (-,babl!as "(eluan'& !e-!en!u. Ba!as n3  ba'a%ana(un3  ba'a%ana(un3 !+a$ (e-lu +e$a! +e$a! un!u$ (-,babl!as "(eluan'& "(eluan'& yan' !e(a!3 !e(a!3 +an %a$a3 $!a  basanya !+a$ %en''una$an %en''una$an !e,-e%a un!u$ %e%(e-$-a$an %e%(e-$-a$an (-,babl!as. (-,babl!as. *-ns( (en''unaan (en''unaan !e,-e%a +an $asus $husus !u a+alah +ala% +s$us !e,-!s. Teorema 6. Msal$an u"0& a+alah 5un's n,n ne'a!5 +a- va-abel -an+,% 0. 4$a E6u "0&9

a+a3 %a$a3 un!u$ se!a( ) $,ns!an!a (,s!53

*- 6 u"0& < ) 9 =

Bu$!. Bu$!nya +be-$an $e!$a va-abel -an+,% 0 a+alah !(e $,n!nyu3 !e!a( bu$! +a(a! +sesua$an +en'an $asus +s$-! 4$a $!a %en''an! n!e'-al +en'an 4u%lah. Msal$an A/ >  ? u"& < )@ +an %sal$an 5"& %enan+a$an (.+.5 +a- 0. Ma$a E6u "0&9 / $a-ena se!a( n!e'-al + an'',!a an'',!a e$s!-% -uas $anan +a- (e-sa%aan (e-sa%aan sebelu%nya a+alah a+alah n,n ne'a!53 an'',!a -uas $- lebh besa- +a- a!au sa%a +en'an salah sa!u +a- %e-e$a. Se)a-a $husus3

 Na%un3 4$a   A3 $e%u+an u"& < )? )? %a$a3 an'',!a an'',!a -uas $anan +a- +a- $e!a$sa%aan $e!a$sa%aan sebelu%nya !+a$ %enn'$a! 4$a $!a %en''an! u"& +en'an ). sehn''a Se4a$ 

!u %en'$u! E6u "0&9 < ) *- 6 u"0& < ) 9 yan' %e-u(a$an hasl yan' +n'n$an. Te,-e%a sebelu%nya a+alah 'ene-alsas +a- $e!a$sa%aan yan' se-n' +sebu! $e!a$sa%aan Chebyshev. #e!a$sa%aan n se$a-an' a$an +ben!u$. Teorema 7. #e!a$sa%aan Chebyshev Chebyshev.. Msal$an

va-abel -an+,% 0 %e%l$ +s!-bus  (-,babl!as !en!an' a(a a(a yan' $!a asu%s$an asu%s$an bah;a hanya a+a va-ans va-ans yan' !e-ba!as !e-ba!as  2. n3 !en!u sa4a3 %eny-a!$an bah;a a+a %ean "-a!a-a!a& 8. Ma$a un!u$ se!a( $   *- "0 8 < $& = A!au equvalen +en'an *- "0 8 < $& < 1 Bukti. Dala% !e,-e%a  a%bl u"0& / "0 7 μ&2 +an c = k 2σ 2 . $e%u+an $!a %e%(unya

*- 6"0 8&2 < $ 229 = / $a-ena (e%blan' +a- an'',!a -uas $anan +a- $e!a$sa%aan sebelu%nya a+alah  23 +ala%  (e-sa%aan +a(a! +a(a! +!uls

*- "0 8 < $& = yan' %e-u(a$an hasl yan' +n'n$an. Ten!u3 $a% a$an %en'a%bl 4u%lah $ (,s!5 lebh  besa- +a- 1 un!u$ %e%l$ $e!a$sa%aan $e!a$sa%aan yan' + )a-. Gal n !e-lha! bah;a blan'an 1H$ 2 a+alah ba!as a!as un!u$ (-,babl!as (-,babl!as *- "0 8 < $&. Dala% ),n!,h be-$u! n ba!as a!as +an nla yan' !e(a! +a- (-,babl!as +ban+n'$an +ala% $e4a+an $husus. Contoh 1. %sal 0 %e%(unya (.+.5  F"& / 3  . /  yan' lannya. Dsn 8 /  +an  2 / 1. I5 $ / JH23 $!a %e%(unya (-,babl!as e$sa$  *- "0 8 < $& / *- "0 & / 1 7  Den'an $e!a$sa%aan $e!a$sa%aan )hebysev3 )hebysev3 (-,babl!as sebelu%nya %e%(unya ba!as a!as 1H$ 2 / KH. Se4a$ 1 / 31JK3 +en'an (e-$-aan3 (e-$-aan3 (-,babl!as (-,babl!as e$sa$ +ala% $asus $asus n a+alah a+alah 4auh $u-an' $u-an' +a- ba!as a!as KH. :$a $!a %en'a%bl $/23 $!a %e%(unya $e%un'$nan *-"0 < 2& / . In $e%bal +en'an san'a! $u-an' +a- ba!as a!as 1H$ 2 /  %enya4$an +ala% $e!a$sa%aan )hebyshev. Dala% se!a( $e4a+an (a+a ),n!,h sebelu%nya3 sebelu%nya3 (-,babl!as *- "0 8 < $& +an ba!as a!as 2 1H$  !e-$a! be-be+a 4auh. Gal n %enun4u$$an bah;a $e!a$sa%aan n %un'$n +bua! lebh !a4a%. Na%un3 4$a $!a %en'n'n$an $e!a$sa%aan $e!a$sa%aan yan' be-la$u un!u$ se!a( $   +an be-la$u un!u$ se%ua va-abel -an+,% yan' %e%l$ va-ans yan' !e-ba!as3 se(e-!  (enn'$a!an a+alah a+alah !+a$ %un'$n3 %un'$n3 se(e-! yan' +!un4u$$an +!un4u$$an ,leh ),n!,h ),n!,h be-$u!. Contoh 2. Msal 0 va-abel -an+,% !(e +s$-! %e%l$

(-,babl!as 1H3 H3 1H + !!$  / 13313 -esve)!vely. -esve)!vely. Dsn 8 /  +an   / . :$a $ /23 $e%u+an 1H$ 2 /  +an 2

*- " 0 < 1& / . Bah;a 3 (-,babl!as *- " 0 8& < $& be-$u! %en)a(a ba!as a!as 1H$ 2 /  Oleh $a-ena !u $e!a$sa%aan !e-sebu! !+a$ +a(a! +!n'$a!$an !an(a asu%s lebh lan4u! !en!an' +s!-bus +a- 0.

Ta%bahan :$a (eubah a)a$ 0 %e%l$ -e-a!a 8 +an va-an  2 !+a$ n,l3 %a$a $!a +a(a! %en'hubun'$an %en'hubun'$an an!a-a $e+uanya3 +ala% (e-nya!aan (eluan' +an hu$u% n +!e%u$an ,leh se,-an' ahl %a!e%a!$a (a+a aba+ 13 ya!u *. I Chebyshev. Gu$u%nya a!au -u%usnya n +na%a$an $e!a$sa%aan Chebyshev. Bu$u (+5 *en'an!a- S!a!s!$a S!a!s!$a Ma!e%a!$a ,leh G. G. Ma%an Suhe-%an3D-s.3M.S Suhe-%an3D-s.3M.S

1. La!han

1.104 Msal$an 0 "a+alah& va-abel -an+,% +en'an %ean +an %sal$an E6"0 7 8 & 2$ 9 a+a.

Tun4u$$an3 +en'an +3 bah;a *- "0 8 < +& = E6"0 7 8 & 2$ 9H+2$ 

1.105 Msal$an 0 "a+alah& sua!u va-abel -an+,% bah;a

*- "0 = & /  +an %sal$an

8 / E "0&. Tun4u$$an bah;a *- "0 < 2 8 & = 1H2.

1.106 4$a 0 a+alah sua!u va-abel -an+,% ya!u E"0&/J +an E"0 2&/J 'una$an

$e!a$sa%aan )hebyshev’s un!u$ %enen!u$an ba!as ba;ah un!u$ (-,babl!as *-"2  0  &.

1.107 %sal$an 0 "a+alah& sua!u

va-abel -an+,% +en'an 5un's (e%ban'$! %,%en M"!&3

 7 h!h. Bu$!$anlah bah;a *- "0 < a& = e  7a! M "!&3   !  h3 +an bah;a *- "0 = a& = e  7a! M "!&3 h  !  .

Isya-a!. Msal u"& / e ! +an ) / e !a + !e,-e%a . Ca!a!an. Gasl n %eny-a!$an bah;a *- "0< a& +an *- "0=a& $u-an' +a- ba!as ba;ah yan' (aln' -en+ah %asn'%asn' e  7a! M "!& $e!$a   !  h +an $e!$a h  !  

1.108.5un's (e%ban'$! %,%en 0 a+a un!u$ se%ua nlanla -l ! +an +be-$an

M "!& / 3 !  &3 M"& / 1.

Men''una$an hasl +a- la!han sebelu%nya un!u$ %enun4u$$an bah;a *-" 0 < 1&/  +an *-" 0 = 7 1& / . Ca!a!an bah;a + sn h !an(a ba!as.

*enyelesaan 1.1K. D$e!ahu P 0 "a+alah& va-abel -an+,% +en'an %ean +an %sal$an E6"0 7 8 & 2$ 9 a+a D!anya P +en'an +  3 !un4u$$an bah;a *- "0 8 < +& = E6"0 7 8 & 2$9H+2$  :a;ab

P

Den'an %en''una$an bu$! !e,-e%a  *- 6"0 8&2 < $ 229 = /

A%bl *- "0 8 < +& / *- "0 8 2$  < +2$ & $e+ua -uas +$al 2$3 +en'an $/ *- "0 8 < +& / *- "0 8 2$  < +2$ & = Sehnn'a *- "0 8 < +& = !e-bu$! 1.1Q. D$e!ahu P 0 sua!u va-abel -an+,% bah;a *- "0 = & /  8 / E "0& D!anya P Tun4u$$an *- "0 < 2 8 & =  :a;ab

P

A+! "a$an +!un4u$$an& *- "0 < 2 8 & =  Msal

)/28 Den'an %en''una$an !e,-e%a  *- 6 u"0& < ) 9 = 3 %a$a *- 60 < ) 9 = *- 60 < 2 8 9 = *- 60 < 2 8 9 = *- 60 < 2 8 9 =

!e-bu$!

:a+3 !e-bu$! bah;a *- 60 < 2 8 9 =

1.1. D$e!ahu P  E(X) /  / J  E(X 2 ) / 1J

D!anya P ba!as ba;ah *-"2  0  & :a;ab P 2 / E(X 2 ) – 2 / 1J 7 J 2 / 1J 7  2/ K /2 Den'an $e!a$sa%aan Chebyshev3

Ba!as ba;ah *- "0 8 < $& < 1 *- "0 J < 2$& < 1 *- "0 J < 2$& < 1 *- 6 2$  "0 J&  2$9 < 1 *- 6 2$ J  0  2$  J9 < 1 Ba!as un!u$ *-"2  0  &9

2$  J / 2

2$  J / 

2 $ / Q

2$/Q

$/

$/

 ba!as ba;ahnya a+alah 1 / 1 / / 3K :a+3 ba!as ba;ah un!u$ ba!as ba;ah *-"2  0  & a+alah / 3K

1.107.

D$e!ahu : 5un's (e%ban'$! %,%en M"!&3 7 h!h.

D!anya P Bu$!$anlah bah;a *- "0 < a& = e  7a! M "!&3   !  h +an *- "0 = a& = e  7a! M "!&3 h  !  . :a;ab

P

A+b *- "0 < a& = e  7a! M "!&3   !  h +an *- "0 = a& = e  7a! M "!&3 h  !  . Msal u"& / e ! +an ) / e !a Den'an %en''una$an !e,-e%a  *- 6 u"0& < ) 9 = 3 %a$a

0= a 0
 ! h h  !   h

h  *- 6e ! < e !a 9 = *- 6e ! < e !a 9 = e !a M"!&

Gasl n %eny-a!$an bah;a *- "0< a& +an *- "0=a& $u-an' +a- ba!as ba;ah yan' (aln' -en+ah %asn'%asn' e  7a! M "!& $e!$a   !  h +an $e!$a h  !  . Sehn''a ++a(a!$an *- "0 < a& = e  7a! M "!&3   !  h 3 +an *- "0 = a& = e  7a! M "!&3 h  !   :a+3 !e-bu$! bah;a *- "0 < a& = e  7a! M "!&3   !  h 3 +an *- "0 = a& = e  7a! M "!&3 h  !   1. #es%(ulan #e!a$sa%aan Chebshev :$a (eubah a)a$ 0 %e%l$ -e-a!a 8 +an va-an  2 !+a$ n,l3 %a$a $!a +a(a! %en'hubun'$an an!a-a $e+uanya3 -u%us n +na%a$an $e!a$sa%aan )hebyshev. Teorema 6. Msal$an u"0& a+alah 5un's !a$ ne'a!5 +a- va-abel -an+,% 0. 4$a E6u "0&9

a+a3 %a$a3 un!u$ se!a( ) $,ns!an!a (,s!53 *- 6 u"0& < ) 9 =

Teorema 7. #e!a$sa%aan Chebyshev. Msal$an

va-abel -an+,% 0 %e%l$ +s!-bus  (-,babl!as !en!an' a(a yan' $!a asu%s$an bah;a hanya a+a va-ans yan' !e-ba!as  2. n3 !en!u sa4a3 %eny-a!$an bah;a a+a %ean "-a!a-a!a& 8. Ma$a un!u$ se!a( $   *- "0 8 < $& = "ba!as a!as& A!au equvalen +en'an *- "0 8 < $& < 1

"ba!as ba;ah&

 Teorema Chebyshev

BAB 1 *ENDAGULUAN

A. La!a- Bela$an' Va-an sua!u (eubah a)a$ %e%be- 'a%ba-an %en'ena (enyeba-an (en'a%a!an +se$!a- nla -a!aan. Bla va-ans a!au s%(an'an ba$u sua!u (eubah a)a$ $e)l %a$a +a(a! +ha-a($an bah;a u%u%nya (en'a%a!an %en'el,%(,$an +e$a! +se$!a- nla -a!aan. #a-ena !u3 (eluan' sua!u (eubah a)a$ %en+a(a! nla +ala% sua!u selan' !e-!en!u +se$!a-  nla -a!aan a$an lebh besa- +a-(a+a (eubah a)a$ se-u(a yan' lebh besa- s%(an'an  ba$unya. Bla (eluan' +nya!a$an +en'an luas %a$a +a(a! +ha-a($an bah;a sua!u +s!-bus $,n!nu +en'an s%(an'an ba$u yan' $e)l %e%(unya seba'an besa- luasnya +e$a! +en'an . A$an !e!a(3 nla yan' besa- %enya!a$an (enyeba-an yan' lebh besa- sehn''a +a(a! +ha-a($an luas !a+ lebh %enyeba-. B.

Man5aa! Da(a! %enen!u$an (eluan' (eubah a)a$ yan' %en+a(a! nla +ala% 4a-a$ $ s%(an'an  ba$u +a- ha-'a -a!aannya (aln' se+$! "1 7 1H$ 2& +en'an $ blan'an -eal.

BAB 2 ISI

A. Te,-e%a Cheby)hev3 se,-an' %a!e%a!$a;an Rusa3 %ene%u$an bah;a ba'an luas an!a-a +ua nla yan' s%e!- !e-ha+a( nla -a!aan be-$a!an +en'an s%(an'an ba$u. #a-ena luas +  ba;ah $u-va +s!-bus (eluan' a!au +ala% hs!,'-a% (eluan' be-4u%lah 13 %a$a luas an!a-a +ua blan'an se%ba-an' %enya!a$an (eluan' (eubah a)a$ yan' be-san'$u!an %en+a(a! nla an!a-a $e+ua blan'an !e-sebu!. Te,-e%a be-$u! +$e%u$a$an ,leh Cheby)hev3 %e%be-$an !a$s-an yan' $,l,! "$,nse-va!5& !en!an' (eluan' bah;a se!a( (eubah a)a$  X  %en+a(a! nla +ala% k   s%(an'an  ba$u +a- nla -a!aannya un!u$ se!a( blan'an $ -eal a+alah (aln' se+$! 3 ya!uP BU#TI Menu-u! +e5ns !e-+ahulu %en'ena va-ans  x %a$a +a(a! +!uls 2

 /

/ /  $a-ena yan' $e+ua +a- $e!'a n!e'-al !a$ne'a!5. Se$a-an'3 $a-ena +ala% $e+ua n!e'-al lannya. Ma$a +an bah;a Sehn''a Dan !e,-e%a !elah !e-bu$!. Un!u$  k  / 2 !e,-e%a %enya!a$an bah;a (eubah a)a$  X   %e%(unya (eluan' (aln' se+$! 1  7 "1H2&2 /  %en+a(a!nla +ala% 4a-a$ +ua s%(an'an ba$u +a- nla -a!aan. a!u !'a  (e-e%(a! a!au lebh (en'a%a!an se!a( +s!-bus !e-le!a$ +ala% selan' .be'!u (ula !e,-e%a

!e-sebu! %enya!a$an bah;a (aln' se+$! +ela(an (e-se%blan (en'a%a!an se!a( +s!-bus !e-le!a$ +ala% selan'

B. 1.

*e-%asalahan Sua!u (eubah a)a$  X   %e%(unya -a!aan

/ 3 va-ans

2

/ 3 se+an'$an (eluan'

+s!-busnya !+a$ +$e!ahu. G!un'lah P a. b.

2.

P (- < x < 2!) P ("x – #" $ %)

Sua!u (eubah a)a$  X   %e%(unya -a!aan / 123 va-ans 2  / 3 se+an'$an (eluan'

+s!-busnya !+a$ +$e!ahu. Den'an %en''una$an !e,-e%a Chebyshev h!un'lahP a. b.

J.

P (% < x < "#) P (& < x < 2")

Sua!u (eubah a)a$  X   %e%(unya -a!aan / 13 va-ans

2

/ K. Den'an %en''una$an

!e,-e%a Chebyshev h!un'lahP a. b. c.

K.

P ( ' x – "! ' $ &) P ( ' x – "! ' < &) P (  < x < ")

Sua!u (eubah a)a$  X   %e%(unya -a!aan /3 va-ans se+an'$an +s!-busnya !+a$ 

+$e!ahu. G!un'lah a. b.

C. 1.

*e%bahasan D$e!ahu P / 3 2 /  D!anya$anP a. P (- < x < 2!) b.P ("x – #" $ %)

*enyelesaan P a.  b.

 P (- < x < 2!) = P ( - k  < x <  * k  ) $ "  P ('x – #' $ %) = P ( - k  < x <  * k  ) $ " -

a!au 2. D$e!ahu P / 123 2 / 

D!anya$an P a. P (% < x < "#) *enyelesaanP

b. P (& < x < 2")

a. . P (- < x < 2!)= P ( - k  < x <  * k  ) $ " -

J.

b.P (& < x < 2") = P ( - k  < x <  * k  ) $ " –

D$e!ahu P / 13 2 / K.

D!anya$an P a.  P ( ' x – "! ' $ &) b. P ( ' x – "! ' < &) c. P (  < x < ")

*enyelesaan P a. b. c.

P ( ' x – "! ' $ &)= " – P ( ' x – "!'< P ( ' x – "! ' < &) = P (-& < x – "! < &) $ " P (  < x < ") =

K.

D$e!ahu P /3 D!anya$an P a.  b. *enyelesaan P a.  P ( - k  < x <  * k  ) $ " –  P ( - k  < x <  * k  ) $ " –

D. #es%(ulan Te,-e%a Cheby)hev *eluan' bah;a se!a( (eubah a)a$  X  %en+a(a! nla +ala% k  s%(an'an  ba$u +a- nla -a!aan a+alah (aln' se+$! 3 ya!uP

TEOREMA CGEBSGEV Bla $!a %en'ala% $esul!an un!u$ MENDEFINISI#AN +s!-bus (eluan' +a- sebuah va-abel a)a$ 3 +a(a! +(a$a sua!u !a$  s-an ya!u TEOREMA CGEBSGEV P

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