3691 bab 5 distribusi diskrit

7/25/2019 3691 Bab 5 Distribusi Diskrit

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Dra. Noeryanti, M.Si

BEBERAPA DISTRIBUSI

PROBABLITAS DISKRET(SSTS 2305 3 !"!#

1



Pengantar:

Penya$ian %i!tri&'!i ro&a&i)ita! %a)a* &ent'" +rai!, ta&e) ata'

*e)a)'i r'*'!an ti%a" *a!a)a-, yan+ in+in %i)'"i!"an a%a)a- eri)a"'

("e)a"'an# er'&a- aa" ter!e&'t. Serin+ "ita *en$'*ai,

en+a*atan yan+ %i-a!i)"an *e)a)'i ero&aan !tati!ti" yan+

&er&e%a *e*'nyai &ent'" "e)a"'an '*'* yan+ !a*a.

O)e- "arena it' er'&a- aa" %i!"ret yan+ &er"enaan %en+an

ero&aan ter!e&'t %aat %i)'"i!"an %en+an %i!tri&'!i ro&a&i)ita!

yan+ !a*a, %an %aat %inyata"an %en+an r'*'! yan+ !a*a.

Da)a* &anya" ra"te" yan+ !erin+ "ita $'*ai, -anya

*e*er)'"an &e&eraa %i!tri&'!i ro&ai)ita! yan+ entin+ 'nt'"

*enyata"an &anya" er'&a- aa" %i!"ret.

2



Kompetensi:

Sete)a- *e*e)a$ari *ateri o"o" &a-a!an %i!ini, *a-a!i!/a

%i-ara"an

1. Ma*' *en++'na"an "on!e"on!e %a!ar teori %i!tri&'!i

ro&a&i)ita! %i!"et !eara &enar.

2. Ma*' %an tera*i) %a)a* *e)a"'"an -it'n+an-it'n+an yan+

&er"aitan %en+an %i!tri&'!i Sera+a* Di!"ret, %i!tri&'!i Bino*ia)

%an M')tino*ia), %i!tri&'!i -ier+eo*etri", %an %i!tri&'!i

Poi!!on

3. Tera*i) %a)a* *en+er$a"an !oa)!oa) t'+a! %an )ati-an.

3



4

Datar I!i Materi4 Di!tri&'!i Sera+a* Di!"ret

4 Di!tri&'!i Bino*ia) %an M')tino*ia)

4 Di!tri&'!i ier+eo*etri"

4 Di!tri&'!i Poi!!on



5.1. Distribusi Seragam Diskret

Di!tri&'!i ro&a&i)ita! yan+ a)in+ !e%er-ana a%a)a- yan+

!e*'a er'&a- aa"nya *e*'nyai ro&a&i)ita! yan+ !a*a.

Di!tri&'!i ini %i!e&'t %i!tri&'!i ro&a&i)ita! !era+a* %i!"ret.

Definisi (5.1)

i"a er'&a- aa" 6 *en%aat ni)ai %en+an

ro&a&i)ita! yan+ !a*a , *a"a %i!tri&'!i ro&a&i)ita! %i!"ret %i&eri"an

o)e-

La*&an+ (78"# !e&a+ai en++anti (7#, yan+ *en'n$'"an &a-/a

%i!tri&'!i !era+a* ter!e&'t &er+ant'n+ a%a ara*eter 7 Ta&e) 5.1. Di!tri&'!i roa&i)ita! 6

1 2 "7 ,7 ,.....,7

1 2

1

"(78"# 8 'nt'" 7 7 ,7 ,.....,7"

= =

x 1 2 3 4 5 6

F(x;k)=f(

x)

1

6 1

6

1

6

1

6

1

6

1

6

5



Contoh (5.1)

Se&'a- %a%' !ei*&an+ %i)e*ar"an !at' "a)i, *a"a tia 'n!'r

%a)a* r'an+ !a*e) S9:1, 2,3 ;, 5, <=. M'n') %en+an ro&a&i)ita!

1<. a%i $i"a 6 *enyata"an *ata %a%' yan+ *'n'), *a"a 6

ter%i!tri&'!i e)'an+ !era+a* ('nior*# ya"ni (78<#91<, 'nt'" 7 9 1,

2, 3, ;, 5, <Teorema (5.1)

Ni)ai ratarata (*ean# %an >arian!i %i!tri&'!i !era+a* %i!"ret

(78"# a%a)a-

Mean(6#,

?arian(6#8 ata'

B'"ti !&&

1

1

"

i"i

7

== ∑ µ

2 21

1

"i"

i

(7 #

== −∑σ µ 2 2 2E(6 #= −σ µ

6



Men'r't %eini!i,

%an

1 1

1 1 1

" " "

i i i" "i i i

E(6# 7 (78"# 7 ( # 7

= = == = = =∑ ∑ ∑ µ

2 2 2 2 1

1 1

21

1

" "i i "

i i

"

i"i

E(6 # (7 # (78"# (7 # ( #

(7 #

= =

=

= − = − = −

= −

∑ ∑

∑

σ µ µ µ

µ

Contoh (5.2) @ari *ean %an >arian!i %ari onto- (5.1#

a/a&

1 2 3 4 5 63 5

6

.+ + + + +

= = µ

2 2 2 2 2 22

35

12

1 3 5 2 3 5 3 3 5 4 3 5 5 3 5 6 3 5

6

( . # ( . # ( . # ( . # ( . # ( . #− + − + − + − + − + −=

=

σ

7



5.2. Distribusi Binomial dan ultinomial

S'at' ero&aan yan+ ter%iri ata! &e&eraa '!a-a, tiatia

'!a-a, *e*&eri"an -a!i) yan+ %aat %i"e)o*o"an *en$a%i 2

"ate+ori yait' !'"!e! ata' +a+a), %an tiatia ')an+an ero&aan

&e&a! !at' !a*a )ainnya. Pro&a&i)ita! "e!'"!e!an ti%a" &er'&a- %ari

ero&aan !at' "e ero&aan )ainnya. Pro!e! ini %i!e&'t ro!e!

Berno'))i. a%i ro!e! Berno'))i -ar'! *e*en'-i er!yaratan &eri"'t1. Pero&aan ter%iri ata! n'!a-a yan+ &er')an+

2. Tiatia '!a-a *e*&eri"an -a!i) yan+ %aat %i"e)o*o"an

*en$a%i 2"ate+ori, !'"!e! ata' +a+a)

3. Pe)'an+ "e!'"!e!an %inyata"an %en+an , ti%a" &er'&a- %ari

!at' '!a-a "e '!a-a &eri"'tnya.

;. Tia '!a-a &e&a! %en+an '!a-a )ainnya.

8



Contoh (5.!)

Ti+a &aran+ %ia*&i) !eara aa" %ari -a!i) ro%'"!i a&ri",

%ieri"!a, %an yan+ aat %ii!a-"an %ari yan+ ti%a" aat.

Mi!a)"an yan+ aat %i!e&'t aat. Ma"a &anya"nya "e!'"!e!an

*er 'a"an er'&a- aa" 6 %en+an ni)ai no) !a*ai 3.

Ta&e) 5.2.

@9aat 8 T9ti%a" aat (&ai"#

Karena &aran+ %ia*&i) !eara aa", %an

*i!a)"an %ian++a *en+-a!i)"an 25

&aran+ aat, *a"a

Pro&a&i)ita! 'nt'" -a!i) "e*'n"inan yan+ )ain %i)a"'"an %en+an

$a)an an+ !a*a.

a!i) 6

TTTT@TTT@@TTT@@@T@@@T@@@

01112223

3 3 91

4 4 4 64P(T@T# P(T#P(@#P(T# ( #( #( #= = =

9



ta&e) 5.3 Di!tri&'!i ro&a&i)ita! 6

Per"obaan Binomial

Banya"nya 6 yan+ !'"!e! %a)a* n'!a-a Berno'))i %i!e&'t er'&a-

aa" &ino*ia)C, %an %i!tri&'!i %ari er'&a- aa" ini %i!e&'t %i!tri&'!i

Bino*ia)C. i"a *enyata"an ro&a&i)ita! "e!'"!e!an %a)a* !'at'

'!a-a, *a"a %i!tri&'!i er'&a- aa" 6 ini %inyata"an %en+an &(78n,#.

Karena ni)ainya &er+ant'n+ a%a &anya"nya '!a-a (n#

Mi!a)nya 69 &anya"nya &aran+ yan+ aat.

Se)an$'tnya *enent'"an r'*'! yan+ *e*&eri"an roai)ita! 7 !'"!e!

%a)a* n'!a-a !'at' eo&aan &ino*ia) &(78n,#

91

4 642 2 2 3P(6 # ( # &( 8 , #= = = =

7 0 1 2 3

(7# 27

64

9

64

27

64

1

64

10



Pro&a&i)ita! 7 "e!'"!e!an %an n7 "e+a+a)an %a)a* 'r'tan tertent'.

Tia "e!'"!e!an %en+an ro&a&i)ita! %an tia "e+a+a)an %en+an

ro&a&i)ita! 91 . Banya"nya ara 'nt'" *e*i!a-"an n-a!i)

*en$a%i %'a "e)o*o", !e-in++a 7 -a!i) a%a a%a "e)o*o"

erta*a %an !i!anya n7 a%a "e)o*o" "e%'a, $'*)a- ini

%inyata"an !e&a+ai Karena e*&a+ian ter!e&'t !a)in+ teri!a-

(&e&a!# *a"a ro&a&i)ita!nya a%a)a-

n

7

÷

7 n 7n

7

−

÷S'at' '!a-a &erno'))i %aat *en+-a!i)"an "e!'"!e!an %en+an

ro&a&i)ita! %an "e+a+a)an %en+an ro&a&i)ita! 91, *a"a

%i!tri&'!i ro&a&i)ita! er'&a- aa" &ino*ia) 6 yait' &anya"nya

"e!'"!e!an %a)a* n'!a-a &e&a! a%a)a-

0 1 27 n 7n

&(78n,# 87 , , ,....,n7

− = = ÷

11



S'at' ara enya$ian yan+ )ain %ari ta&e) 5.2 n93 %an 1

4 =

31

4

33 0 1 2 3

7 7&(78 , # 87 , , ,7

− = = ÷

Contoh (5.#)S'at' !'"' a%an+ %aat *ena-an '$i +'nan+an tertent'

%en+an ro&a&i)ita! 0.5. it'n+ ro&a&i)ita! &a-/a teat 2 %ari ;

!'"' a%an+ yan+ %i'$i ti%a" a"an r'!a".

a/a&

Mi!a) tia en+'$ian !a)in+ &e&a!

12

2

2

23 3 3 271 4

4 4 4 2 2 1284

42 4

2

F

F F&( 8 , # ( # ( #

= = = ÷

Catatan:

0

1

n

7

&(78n,#

==∑



13

Contoh (5.5) Pro&a&i)ita! !e!eoran+ !e*&'- %ari enya"it $ant'n+ !ete)a-

oera!i a%a)a- 0.;. Bi)a %i"eta-'i 15 oran+ *en%erita enya"it ini,

&eraa e)'an+

a#. !e"'ran+"'ran+nya 10 oran+ %t !e*&'-&#. a%a 3 !a*ai G oran+ y+ !e*&'-

#. teat 5 oran+ y+ !e*&'-

a/a&

Mi! 6 9 *enyata"an &anya"nya oran+ y+ !e*&'-

Di"et 9 0.; n 9 15

a#.

a%i ro&a&i)ita! !e"'ran+"'ran+nya 10 oran+ !e*&'- 9 0.033G

[ ]

9

0

10 1 10 1 0 1 9

1 15 0 4

1 0 9662

0 0338

7

P(6 # P(6 # P(6 # P(6 # P(6 #

&(78 8 . # )i-at ta&e)

.

.

=

≥ = − < = − = + = + =

= − ¬

= −=

∑



14

&#

a%i ro&a&i)ita! ter%aat 3 !a*ai G oran+ y+ !e*&'- 9 0.GH

8 2

0 0

3 8 8 2

15 0 4 15 0 4

0 9050 0 0271

0 8779

7 7

P( 6 # P(6 # P(6 #

&(78 , . # &(78 , . # )i-at ta&e)

. .

.

= =

≤ ≤ = ≤ − ≤

= − ¬

= −=

∑ ∑

#

a%i ro&a&itita! teat 5 oran+ yan+ !e*&'- 9 0.1G5H

5 4

0 0

5 5 15 0 4 5 4

15 0 4 15 0 4

7 7

P(6 # &( 8 8 . # P(6 # P(6 #

&(78 , . # &(78 , . # )i-at ta&e)

0.;032 0.213

0.1G5H

= =

= = = ≤ − ≤

= − ¬

==

∑ ∑



15

Ta&e) 5.; @ara *en++'na"an ta&e) &ino*ia)

n r p

0.01

. . . . . . . 0.4 . . . . . . . . .

15 1

2 0.0271

:::

8 0.9050

9 0.9662

::

15

9

0

15 0 4 0 9662

7

&(78 8 . # .

=→ =∑Unt'" n915, 90.; 8

2

0

15 0 4 0 0271

7

&(78 8 . # .

=

=∑

8

0

15 0 4 0 9050

7

&(78 8 . # .

==∑



16

@ara )ain *enari ni)ai %i!tri&'!i Bino*ia)

'na"an !ot/are R , )an+"a-nya !&&

J &ino*(H,15,0.;#

1 0.H<<1<<

J &ino*(G,15,0.;#

1 0.H0;H52<

J &ino*(2,15,0.;#

1 0.0211;

J &ino*(5,15,0.;#

1 0.;03215<

J &ino*(;,15,0.;#

1 0.212

Teorema(5.2) Di!tri&'!i Bino*ia) &(78n,# *e*'nyai ratarata %an >arian!i

!&&

%ann µ = 2 nσ =



17

Contoh (5.$)

Tent'"an *ean %an >arian!i %ari onto- (5.5# "e*'%i"an

+'na"an teore*a -e&y!-e> 'nt'" *ena!ir"an!e)an+ 2 µ σ ±

%a&ab:

Dari onto- 5.< %i"eta-'i n915 %an 90.;

Diero)e-

%an

Men++'na"an teore*a @-e&y!-e> a%a)a-

a%i, !e)an+ yan+ %itana"an a%a)a- %ari 2.20< !a*ai H.H;

15 0 4 6( #( . # µ = =

1 897.σ =

2 15 0 4 0 6 3 6( #( . #( . # .σ = =

2 µ σ ±

2 9 794 2 2 206. %an . µ σ µ σ + = − =



18

Per"obaan ultinomial

Pero&aan &ino*ia) a"an *en$a%i ero&aan *')tino*ia) $i"a

tia '!a-a %aat *e*&eri"an )e&i- %ari 2 -a!i) yan+ *'n+"in.

Mi!a)nya -a!i) ro%'"!i a&ri" %aat %i"e)o*o"an *en$a%i &aran+

&ai", aat, %an *a!i- &i!a %ier&ai"i.

Bi)a !'at' '!a-a %aat *en+-a!i)"an " *aa* -a!i)

Den+an ro&a&i)ita!nya *a"a %i!tri&'!i er'&a- aa"

yan+ *enyata"an &anya"nya "e$a%ian

Da)a* n'!a-a &e&a! a%a)a-

Den+an %an

1 2 "E ,E ,....,E

1 2 " , ,....,

1 2 "6 ,6 ,....,6 1 2 "E ,E ,....,E

1 2

1 2 1 2 1 21 2

"

"

7 7 7

"8 " "

n

(7 ,7 ,...,7 , ,..., ,n# ...7 , 7 ,..., 7

= ÷

1

"

ii

7 n

==∑

1

1

"

ii

==∑



19

Contoh(5.')

D'a &'a- %a%' %i)ant'n"an < "a)i, &eraa ro&a&i)ita! a"an

*en%aat"an $'*)a- ata' 11 *'n') %'a "a)i, !ea!an+ &i)an+an

yan+ !a*a !at' "a)i, %an "o*ina!i )ainnya 3 "a)i

%a&ab:

Mi!a) E19 *'n') $'*)a- ata' 11 (E1#92H

E29 *'n') a!an+an &i)an+an yan+ !a*a (E2#91< E39 *'n') !e)ain E1 *a''n E2 (E3#9111G

Ni)ai initi%a" &er'&a- %ari "e<'!a-a. Men++'na"an %i!tri&'!i

*')tino*ia) %en+an 7192, 7291 %an 7393 %iero)e-

→

→→

( ) ( ) ( )2 1 3

2 1 11 2 1 11

9 6 18 9 6 18

36 4 1 11

2 1 3 81 6 318

62 1 3 6

2 1 3

0 1127F

F F F

( , , 8 , , , #, ,

.

= ÷

= =

5 ! Di t ib i i t ik



20

5.!. Distribusi iergeometrik

Per&e%aan %i!tri&'!i &ino*ia) %en+an %i!tri&'!i *')tino*ia)

ter)eta" a%a ara en+a*&i)an !a*e)nya. Pen++'naan %i!tri&'!i ini

-a*ir !a*a %en+an %i!tri&'!i &ino*ia). Mi!a)nya %i!tri&'!i &ino*ia)

%itera"an a%a !a*)in+ %ari !e$'*)a- &aran+ (!e"ota" "art',

!e$'*)a- -a!i) ro%'"!i# !a*)in+ -ar'! %i"er$a"an %en+an

en+e*&a)ian !etia &aran+ !ete)a- %ia*ati. Se&a)i"nya %i!tri&'!i

-ier+eo*etri" ti%a" *e*er)'"an "e&e&a!an %an %i%a!ar"an a%a

!a*)in+ tana en+e*&a)ian.

Di!tri&'!i -ier+eo*etri" *e*'nyai !iat

1. Sa*e) aa" &er'"'ran n yan+ %ia*&i) tana en+e*&a)ian %ari N

&en%a.

2. Se&anya" "&en%a %aat %i&eri na*a !'"!e! %an !i!anya N"

%i&eri na*a +a+a).



21

Distribusi iergeometrik

Di!tri&'!i ro&a&i)ita! er'&a- aa" -ier+eo*etri" 6 yan+

*enyata"an &anya"nya "e!'"!e!an %a)a* !a*e) aa" %en+an

'"'ran n yan+ %ia*&i) %ari No&ye" yan+ *e*'at " !'"!e! %an N"

+a+a) %inyata"an !e&a+ai

0 1 2

" N "

7 n 7

N

n

-(78N,n,"# 8 7 , , ,......,n

− ÷ ÷−

÷

= =

Contoh (5.*)

S'at' anitia 5 oran+ %ii)i- !eara aa" %ari 3 "i*ia/an %an 5

i!i"a/an. it'n+ %i!tri&'!i ro&a&i)ita! &any"nya "i*ia/an yan+

%'%'" %a)a* anitia.

%a&ab:

Mi )" 6 t " & " "i i % ) iti



22

Mi!a)"an 69 *enyata"an &anya"nya "i*ia/an %a)a* anitia.

69:0,1,2,3=

Di!tri&'!i ro&a&i)ita!nya %inyata"an %en+an r'*'!

8

8

Ta&e) 5.< Di!tri&'!i -ier+eo*etri"

( ) ( )

( )

3 5

0 5 1

5685

0 0 8 5 37 -( 8 , , #= → = =

( ) ( )( )

3 5

1 4 15

568

5

1 1 8 5 37 -( 8 , , #= → = =

( ) ( )

( )

3 5

2 3 30

5685

2 2 8 5 37 -( 8 , , #= → = =

( ) ( )( )

3 5

3 2 10

568

5

3 3 8 5 37 -( 8 , , #= → = =

x 0 1 2 3

h(x;8,5,3)

1

56

15

56

30

56

10

56

( ) ( )( )

3 55

8

5

8 5 3 0 1 2 37 7

-(78 , , # 8 7 , , ,−= =



23

Teorema(5.!) Di!tri&'!i -ier+eo*etri" -(78N,n,"# *e*'nyai ratarata %an

>arian!i !&&

%ann"

N µ =

2

1

1N n " "

N n n(n#( #( #σ −

−= −

Contoh (5.+)

Tent'"an *ean %an >arian!i %ari onto- (5.G# "e*'%i"an

+'na"an teore*a -e&y!-e> 'nt'" *ena!ir"an!e)an+2 µ σ ±

%a&ab:

Dari onto- 5.G %i"eta-'i n915 %an 90.;

Diero)e- %an

Men++'na"an teore*a @-e&y!-e> a%a)a-

a%i, !e)an+ yan+ %itana"an a%a)a- %ari 0,;1 !a*ai 1,;H1

5 3 3

40 8

0 375( #( #

, µ = = =

( ) ( ) ( )

2 40 5 3 3

39 40 40

5 1 0 3113( # ,σ −= − =

2 µ σ ±

2 1 491 2 0 741, %an , µ σ µ σ + = − = −

C t h (5 1,)



2424

Contoh (5.1,)

S'at' a&ri" &an *e*'nyai %ata &a-/a %ari en+iri*an !e&anya"

5000 &an "e !e&'a- to"o tertent' ter%aat 1000 aat. i"a a%a

!e!eoran+ *e*&e)i 10 &an ini !eara aa" %ari to"o ter!e&'t, &eraa

ro&a&i)ita!nya *e*'at teat 3 yan+ aat.

%a&ab:

Karena n910 '"' "ei) %i&an%in+"an N95000, *a"aro&a&i)ita!nya %i-a*iri %en+an &ino*ia) %en+an 9 1050009 0,2

a%a)a- ro&ai)ita! *en%aat !at' &an. a%i ro&a&i)ita! *en%aat

teat 3 &an aat

3 2

0 0

3 5000 10 1000 3 10 0 2

10 0 2 10 0 2

0 8791 0 6778

0 2013

7 7

-( 8 , , # &( 8 , . #

&(78 , . # &(78 , . #

, ,

,

= =

=

= −

= −=

∑ ∑

%ika dihit ng dengan soft are -



25

%ika dihitung dengan soft&are -

J -yer(3,5000,10,1000# ti%a" &i!a *en+-it'n+

1 0

Di-it'n+ %en+an en%e"atan %i!tri&'!i &ino*ia)J &ino*(3,10,0.2#

1 0.GH12<1

J &ino*(2,10,0.2#

1 0.<HH5

a%i ro&a&i)ita! *en%aat teat 3 &an aat a%a)a-

3 5000 10 1000 3 10 0 2

0 8791261 0 6777995

0 2013266

-( 8 , , # &( 8 , . #

, ,

,

== −=

5 # Distribusi Poisson



26

5.#. Distribusi Poisson

Pero&aan yan+ *en+-a!i)"an r'&a- aa" 6 yna+ *enyata"an

&anya"nya -a!i) !e)a*a %a)a* !e)an+ /a"t'%aera- tertent' %i!e&'t

%i!tri&'!i oi!!onC.

Pro!e! oi!!on *e*i)i"i !iat!iat &eri"'t

1. Banya"nya "e!'"!e!an yan+ ter$a%i %a)a* !'at' %aera- (!e)an+#

/a"t' tertent' in%een%en %en+an %aera- )ainya.

2. Pro&a&i)ita! !'"!e! %a)a* %aera-!e)an+ yan+ "ei) ti%a"

ter+ant'n+ &anya"nya !'"!e! yan+ ter$a%i %i)'ar !e)an+.

3. Pe)'an+ ter$a%inya )e&i- %ari !at' !'"!e! %a)a* %aera- yan+

!e*it %ia&ai"an.

i"a 6 er'&a- aa" oi!!on *a"a %i!tri&'!i oi!!on ini %inyata"an

%en+an , %i*ana a%a)a- ratarata -a!i)(7, t#λ tλ

Di i& i & - " P i 6 " & "



27

Di!tri&'!i er'&a- aa" Poi!!on 6 yan+ *enyata"an &anya"nya

"e!'"!e!an yan+ ter$a%i %a)a* !'at' !e)an+ /a"t'%aera- tertent' t,

%inata"an

%i*ana e92,1G2G %an *enyata"an ratarata &anya"nya !'"!e!

yan+ ter$a%i er !at'an /a"t'.

Mi!a)"an , 'nt'" &e&eraa ni)ai tertent' %ari 0,1 !a*ai 1G

%i&eri"an a%a ta&e) Poi!!on. Ata' %en+an &ant'an !ot/are R

0 1 2

t 7e ( t#

7F(7, t# 87 , , ,.....

λ

λ λ

−

= =

tλ

t µ λ =

Contoh (5.11)

Ratarata &anya"nya Tan"er *inya" yan+ ti&a tia -ari %i !'at'

e)a&'-an a%a)a- 10. Pe)a&'-an ter!e&'t -anya *a*' *ena*'n+

a)in+ &anya" 15 Tan"er er-ari. Beraa ro&a&i)ita! a%a !'at' -ari

tertent' Tan"er tera"!a er+i "arena e)a&'-an en'- %an ti%a"

*a*' *e)ayani.

%a&ab:



28

%a&ab:

Mi!a)"an 6 9 &anya"nya Tan"er *inya" y+ ti&a tia -ari

6 9 :1, 2, 3, . . . . . , 15=

Ma"a

a%i e)'an+ % !'at' -ari tertent' Tan"er tera"!a er+i 9 0.0;G

15

0

15 1 15 1 10

1 0 9513 0 0487

7

P(6 # P(6 # (78 # ta&e)

. .

=> = − ≤ = − ¬

= − =

∑

4 @ara )ain, $i"a ti%a" *en++'na"an ta&e), +'na"an ro+ra* R,

)an+"a-nya !&&

J oi!(15,10#

1 0.H5125H<

Artinya 15

0

10

7

(78 # 0.H5125H<

==∑

Teorema(5 #)



29

Teorema(5.#)

Di!tri&'!i oi!!on *e*'nyai ratarata %an >arian!i

!&& %ant µ λ =

Contoh (5.12)

Ratarata &anya"nya arti"e) ra%io ati yan+ *e)eati !'at'

en+-i'n+ !e)a*a 1 *i)i%ti" %a)a* !'at' ero&aan %i )aoratoi'* a%a)a-

;. &eraa ro&a&i)ita! < arti"e) *e)e/ati en-t'n+ it %a)a* 1 *i)i%eti"

tertent'. Ke*'%i"an +'na"an teore*a -e&y!-e> 'nt'"

*ena!ir"an!e)an+

a/a&

%ari ta&e) oi!!on %en+an %iero)e-

%ari %iero)e-

a%i, !e)an+ yan+ %itana"an a%a)a- %ari 0 !a*ai G

2 µ σ ±

6 47 8 t µ λ = = =

2 8 2 0%an µ σ µ σ + = − =

(7, t#λ

2 tσ λ =

6 54 64

60 0

6 4 4 4 0 8893 0 7851 0 1042e ( #

F7 7

( 8 # (78 # (78 # , , ,−

= == = − = − =∑ ∑

24 4t %an µ λ σ = = =

Ta&e) 5 @ara *en++'na"an ta&e) Poi!!on



30

r

0. 1 . . . . . . . 4.0 . . . . . . . . .

0

1

:::

5 0,7851

6 0,8893

::

16

6

0

4 0 8893

7

(78 # .

=→ =∑

Unt'" n915, 90.;, *en++'na"an ta&e) %iero)e-

5

0

4 0 7851

7

(78 # .

==∑

Ta&e) 5.. @ara *en++'na"an ta&e) Poi!!on

µ

Me++'na"an R

J oi!(<,;#

1 0.GGH32<

J oi!(5,;#

1 0.G5130;

Teorema(5 5)



3131

Teorema(5.5)

Mi!a)"an 6 er'&a- aa" &ino*ia) %en+an %i!tri&'!i ro&a&i)ita!

&(7,n,#. i"a n , %an teta !a*a *a"a

Contoh (5.12)

Da)a* !'at' ro!e! ro%'"!i yan+ *en+-a!i)"an &aran+ %ari +e)a!,

ter$a%i +e)e*&'n+(aat# yan+ "a%an+ *enye&a&"an !')it %ia!ar"an.i"a %i"eta-'i ratarata 1 %ari 1000 &aran+ yan+ %i-a!i)"an *e*'nyai

!at' ata' )e&i- +e)e*&'n+. Beraa ro&ai)ia!nya &a-/a %a)a* !a*e)

aa" !e&e!ar G000 &aran+ a"an eri!i "'ran+ %ari yan+ &er+e)e*&'n+

%a&ab:

n9G000, 90.001 %i-a*iri %en+an %i!tri&'!i oi!!on %en+an

%iero)e- *en++'na"an ta&e)

n µ =→ ∞ 0→

&(7,n,# (7, # µ →

8000 0 001 8( #( , # µ = =



32

i"a ti%a" *en++'na"an ta&e), ara )ain, +'na"an R, )an+"a-nya

J &ino*(<,G000,0.001# 1 0.3132521

J oi!(<,G#

1 0.3133;3

Diero)e-

Dan

6 6

0 0

7 8000 0 001 8 0 3134

7 7

P(6 # &(78 , . # (78 # ,

= =< = = =∑ ∑

6

0

8000 0 001

7&(78 , . # 0.3132521

==∑

6

0

8

7

(78 # 0.3133;3

==∑

3691 bab 5 distribusi diskrit

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