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Dra. Noeryanti, M.Si
BEBERAPA DISTRIBUSI
PROBABLITAS DISKRET(SSTS 2305 3 !"!#
1
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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 $'*ai, -anya
*e*er)'"an &e&eraa %i!tri&'!i ro&ai)ita! yan+ entin+ 'nt'"
*enyata"an &anya" er'&a- aa" %i!"ret.
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Kompetensi:
Sete)a- *e*e)a$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$a"an !oa)!oa) t'+a! %an )ati-an.
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4
Datar I!i Materi4 Di!tri&'!i Sera+a* Di!"ret
4 Di!tri&'!i Bino*ia) %an M')tino*ia)
4 Di!tri&'!i ier+eo*etri"
4 Di!tri&'!i Poi!!on
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5.1. Distribusi Seragam Diskret
Di!tri&'!i ro&a&i)ita! yan+ a)in+ !e%er-ana a%a)a- yan+
!e*'a er'&a- aa"nya *e*'nyai ro&a&i)ita! yan+ !a*a.
Di!tri&'!i ini %i!e&'t %i!tri&'!i ro&a&i)ita! !era+a* %i!"ret.
Definisi (5.1)
i"a er'&a- aa" 6 *en%aat ni)ai %en+an
ro&a&i)ita! yan+ !a*a , *a"a %i!tri&'!i ro&a&i)ita! %i!"ret %i&eri"an
o)e-
La*&an+ (78"# !e&a+ai en++anti (7#, yan+ *en'n$'"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
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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 $i"a 6 *enyata"an *ata %a%' yan+ *'n'), *a"a 6
ter%i!tri&'!i e)'an+ !era+a* ('nior*# ya"ni (78<#91<, 'nt'" 7 9 1,
2, 3, ;, 5, <Teorema (5.1)
Ni)ai ratarata (*ean# %an >arian!i %i!tri&'!i !era+a* %i!"ret
(78"# a%a)a-
Mean(6#,
?arian(6#8 ata'
B'"ti !&&
1
1
"
i"i
7
== ∑ µ
2 21
1
"i"
i
(7 #
== −∑σ µ 2 2 2E(6 #= −σ µ
6
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Men'r't %eini!i,
%an
1 1
1 1 1
" " "
i i i" "i i i
E(6# 7 (78"# 7 ( # 7
= = == = = =∑ ∑ ∑ µ
2 2 2 2 1
1 1
21
1
" "i i "
i i
"
i"i
E(6 # (7 # (78"# (7 # ( #
(7 #
= =
=
= − = − = −
= −
∑ ∑
∑
σ µ µ µ
µ
Contoh (5.2) @ari *ean %an >arian!i %ari onto- (5.1#
a/a&
1 2 3 4 5 63 5
6
.+ + + + +
= = µ
2 2 2 2 2 22
35
12
1 3 5 2 3 5 3 3 5 4 3 5 5 3 5 6 3 5
6
( . # ( . # ( . # ( . # ( . # ( . #− + − + − + − + − + −=
=
σ
7
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5.2. Distribusi Binomial dan ultinomial
S'at' ero&aan yan+ ter%iri ata! &e&eraa '!a-a, tiatia
'!a-a, *e*&eri"an -a!i) yan+ %aat %i"e)o*o"an *en$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.
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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
$a)an an+ !a*a.
a!i) 6
TTTT@TTT@@TTT@@@T@@@T@@@
01112223
3 3 91
4 4 4 64P(T@T# P(T#P(@#P(T# ( #( #( #= = =
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ta&e) 5.3 Di!tri&'!i ro&a&i)ita! 6
Per"obaan Binomial
Banya"nya 6 yan+ !'"!e! %a)a* n'!a-a Berno'))i %i!e&'t er'&a-
aa" &ino*ia)C, %an %i!tri&'!i %ari er'&a- aa" ini %i!e&'t %i!tri&'!i
Bino*ia)C. i"a *enyata"an ro&a&i)ita! "e!'"!e!an %a)a* !'at'
'!a-a, *a"a %i!tri&'!i er'&a- aa" 6 ini %inyata"an %en+an &(78n,#.
Karena ni)ainya &er+ant'n+ a%a &anya"nya '!a-a (n#
Mi!a)nya 69 &anya"nya &aran+ yan+ aat.
Se)an$'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
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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, $'*)a- ini
%inyata"an !e&a+ai Karena e*&a+ian ter!e&'t !a)in+ teri!a-
(&e&a!# *a"a ro&a&i)ita!nya a%a)a-
n
7
÷
7 n 7n
7
−
÷S'at' '!a-a &erno'))i %aat *en+-a!i)"an "e!'"!e!an %en+an
ro&a&i)ita! %an "e+a+a)an %en+an ro&a&i)ita! 91, *a"a
%i!tri&'!i ro&a&i)ita! er'&a- aa" &ino*ia) 6 yait' &anya"nya
"e!'"!e!an %a)a* n'!a-a &e&a! a%a)a-
0 1 27 n 7n
&(78n,# 87 , , ,....,n7
− = = ÷
11
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S'at' ara enya$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,#
==∑
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13
Contoh (5.5) Pro&a&i)ita! !e!eoran+ !e*&'- %ari enya"it $ant'n+ !ete)a-
oera!i a%a)a- 0.;. Bi)a %i"eta-'i 15 oran+ *en%erita enya"it ini,
&eraa e)'an+
a#. !e"'ran+"'ran+nya 10 oran+ %t !e*&'-&#. a%a 3 !a*ai G oran+ y+ !e*&'-
#. teat 5 oran+ y+ !e*&'-
a/a&
Mi! 6 9 *enyata"an &anya"nya oran+ y+ !e*&'-
Di"et 9 0.; n 9 15
a#.
a%i ro&a&i)ita! !e"'ran+"'ran+nya 10 oran+ !e*&'- 9 0.033G
[ ]
9
0
10 1 10 1 0 1 9
1 15 0 4
1 0 9662
0 0338
7
P(6 # P(6 # P(6 # P(6 # P(6 #
&(78 8 . # )i-at ta&e)
.
.
=
≥ = − < = − = + = + =
= − ¬
= −=
∑
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14
&#
a%i ro&a&i)ita! ter%aat 3 !a*ai G oran+ y+ !e*&'- 9 0.GH
8 2
0 0
3 8 8 2
15 0 4 15 0 4
0 9050 0 0271
0 8779
7 7
P( 6 # P(6 # P(6 #
&(78 , . # &(78 , . # )i-at ta&e)
. .
.
= =
≤ ≤ = ≤ − ≤
= − ¬
= −=
∑ ∑
#
a%i ro&a&itita! teat 5 oran+ yan+ !e*&'- 9 0.1G5H
5 4
0 0
5 5 15 0 4 5 4
15 0 4 15 0 4
7 7
P(6 # &( 8 8 . # P(6 # P(6 #
&(78 , . # &(78 , . # )i-at ta&e)
0.;032 0.213
0.1G5H
= =
= = = ≤ − ≤
= − ¬
==
∑ ∑
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15
Ta&e) 5.; @ara *en++'na"an ta&e) &ino*ia)
n r p
0.01
. . . . . . . 0.4 . . . . . . . . .
15 1
2 0.0271
:::
8 0.9050
9 0.9662
::
15
9
0
15 0 4 0 9662
7
&(78 8 . # .
=→ =∑Unt'" n915, 90.; 8
2
0
15 0 4 0 0271
7
&(78 8 . # .
=
=∑
8
0
15 0 4 0 9050
7
&(78 8 . # .
==∑
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@ara )ain *enari ni)ai %i!tri&'!i Bino*ia)
'na"an !ot/are R , )an+"a-nya !&&
J &ino*(H,15,0.;#
1 0.H<<1<<
J &ino*(G,15,0.;#
1 0.H0;H52<
J &ino*(2,15,0.;#
1 0.0211;
J &ino*(5,15,0.;#
1 0.;03215<
J &ino*(;,15,0.;#
1 0.212
Teorema(5.2) Di!tri&'!i Bino*ia) &(78n,# *e*'nyai ratarata %an >arian!i
!&&
%ann µ = 2 nσ =
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17
Contoh (5.$)
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 + = − =
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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
==∑
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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') $'*)a- ata' 11 (E1#92H
E29 *'n') a!an+an &i)an+an yan+ !a*a (E2#91< E39 *'n') !e)ain E1 *a''n E2 (E3#9111G
Ni)ai initi%a" &er'&a- %ari "e<'!a-a. Men++'na"an %i!tri&'!i
*')tino*ia) %en+an 7192, 7291 %an 7393 %iero)e-
→
→→
( ) ( ) ( )2 1 3
2 1 11 2 1 11
9 6 18 9 6 18
36 4 1 11
2 1 3 81 6 318
62 1 3 6
2 1 3
0 1127F
F F F
( , , 8 , , , #, ,
.
= ÷
= =
5 ! Di t ib i i t ik
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20
5.!. Distribusi iergeometrik
Per&e%aan %i!tri&'!i &ino*ia) %en+an %i!tri&'!i *')tino*ia)
ter)eta" a%a ara en+a*&i)an !a*e)nya. Pen++'naan %i!tri&'!i ini
-a*ir !a*a %en+an %i!tri&'!i &ino*ia). Mi!a)nya %i!tri&'!i &ino*ia)
%itera"an a%a !a*)in+ %ari !e$'*)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).
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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
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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 , , ,−= =
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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,)
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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 -
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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
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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$a%inya )e&i- %ari !at' !'"!e! %a)a* %aera- yan+
!e*it %ia&ai"an.
i"a 6 er'&a- aa" oi!!on *a"a %i!tri&'!i oi!!on ini %inyata"an
%en+an , %i*ana a%a)a- ratarata -a!i)(7, t#λ tλ
Di i& i & - " P i 6 " & "
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27
Di!tri&'!i er'&a- aa" Poi!!on 6 yan+ *enyata"an &anya"nya
"e!'"!e!an yan+ ter$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:
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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, $i"a ti%a" *en++'na"an ta&e), +'na"an ro+ra* R,
)an+"a-nya !&&
J oi!(15,10#
1 0.H5125H<
Artinya 15
0
10
7
(78 # 0.H5125H<
==∑
Teorema(5 #)
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29
Teorema(5.#)
Di!tri&'!i oi!!on *e*'nyai ratarata %an >arian!i
!&& %ant µ λ =
Contoh (5.12)
Ratarata &anya"nya arti"e) ra%io ati yan+ *e)eati !'at'
en+-i'n+ !e)a*a 1 *i)i%ti" %a)a* !'at' ero&aan %i )aoratoi'* a%a)a-
;. &eraa ro&a&i)ita! < arti"e) *e)e/ati en-t'n+ it %a)a* 1 *i)i%eti"
tertent'. Ke*'%i"an +'na"an teore*a -e&y!-e> 'nt'"
*ena!ir"an!e)an+
a/a&
%ari ta&e) oi!!on %en+an %iero)e-
%ari %iero)e-
a%i, !e)an+ yan+ %itana"an a%a)a- %ari 0 !a*ai G
2 µ σ ±
6 47 8 t µ λ = = =
2 8 2 0%an µ σ µ σ + = − =
(7, t#λ
2 tσ λ =
6 54 64
60 0
6 4 4 4 0 8893 0 7851 0 1042e ( #
F7 7
( 8 # (78 # (78 # , , ,−
= == = − = − =∑ ∑
24 4t %an µ λ σ = = =
Ta&e) 5 @ara *en++'na"an ta&e) Poi!!on
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30
r
0. 1 . . . . . . . 4.0 . . . . . . . . .
0
1
:::
5 0,7851
6 0,8893
::
16
6
0
4 0 8893
7
(78 # .
=→ =∑
Unt'" n915, 90.;, *en++'na"an ta&e) %iero)e-
5
0
4 0 7851
7
(78 # .
==∑
Ta&e) 5.. @ara *en++'na"an ta&e) Poi!!on
µ
Me++'na"an R
J oi!(<,;#
1 0.GGH32<
J oi!(5,;#
1 0.G5130;
Teorema(5 5)
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3131
Teorema(5.5)
Mi!a)"an 6 er'&a- aa" &ino*ia) %en+an %i!tri&'!i ro&a&i)ita!
&(7,n,#. i"a n , %an teta !a*a *a"a
Contoh (5.12)
Da)a* !'at' ro!e! ro%'"!i yan+ *en+-a!i)"an &aran+ %ari +e)a!,
ter$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 = =
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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
==∑
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