00 kuliah 03-02 distribusi probabilitas kontinyu teoritis beta (beta distribution) distribusi...

1DISTRIBUSI PROBABILITAS KONTINYU TEORITISSuprayogi

Distribusi Probabilitas Kontinyu Teoritis

2DISTRIBUSI PROBABILITAS KONTINYU TEORITISSuprayogi, 2006

Dist. Prob. Teoritis Kontinyu (1)

Distribusi seragam kontinyu (continuous uniform distribution)

Distribusi segitiga (triangular distribution)Distribusi segitiga kiri (left triangular distribution)

Distribusi segitiga kanan (right triangular distribution)

Distribusi normal (normal distribution)Distribusi normal baku (standard normal distribution)

Distribusi lognormal (lognormal distribution)

Distribusi gamma (gamma distribution)Distribusi Erlang (Erlang distribution)


Dist. Prob. Teoritis Kontinyu (2)

Distribusi eksponensial (exponential distribution)

Distribusi khi‐kuadrat (chisquare distribution)

Distribusi Weibull (Weibull distribution)

Distribusi student t (Student t distribution)

Distribusi F (F distribution)

Distribusi beta (beta distribution)


Distribusi Seragam Kontinyu (1)

X ∼ seragam kontinyu (a, b)

( )⎪⎩

⎪⎨

⎧ <<−=lainnya ;0

;1

x

bxaabxf

Parameter:

a, b bilangan riil (b > a)

Rataan:

2

baX

+=μ

Variansi:( )12

22 abX

−=σ

a : batas bawah

b : batas atas

Fungsi distribusi probabilitas:


Distribusi Seragam Kontinyu (2)

Fungsi distribusi probabilitas kumulatif:

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

<<−−

<

=

bx

bxaab

ax

ax

xF

;1

;

;0


Contoh Kurva Distribusi Seragam Kontinyu

0.00

0.05

0.10

0.15

0.20

0.25

5.00 10.00 15.00 20.00 25.00

x

f(x)

a = 10, b = 20


Probabilitas dari Variabel Random SeragamKontinyu

( ) ∫ −=<< 2

1

121

x

xdx

abxXxP

( )xf

xa b1x 2x


Contoh Perhitungan

Diameter komponen yang dinyatakan dengan variabel random X diketahui berdistribusi seragam kontinyu dengan batas bawah10 mm dan batas atas = 20 mm. Probabilitas bahwa diameter komponen kurang dari 15 mm?

( )

,50 10

10‐15

1020

115

15

10

=

=

−=< ∫ dxXP


Distribusi Segitiga

X ∼ segitiga (a, b, c)

Parameter:

a, b, c bilangan ril (a < b < c)

Rataan:

3

cba ++=μ

Variansi:

18

2222 bcacabcba −−−++=σ

( )

( )( )( )

( )( )( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≤≤−−

−

≤≤−−

−

=

lainnya ;0

;2

;2

cxbbcac

xc

bxaabac

ax

xf



Distribusi Segitiga (2)

Fungsi probabilitas kumulatif:

( )

( )( )( )

( )( )( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<<<<

>−−

−−

−−−<

=cxb

bxa

cxbcac

xc

abac

axax

xF

;

;

;1

1

;0

2

2


Contoh Kurva Distribusi Segitiga

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 5.00 10.00 15.00 20.00 25.00

x

f(x)

a = 5, b = 10, c = 20


Probabilitas dari Variabel Random Segitiga

x1 x2a b c

f(x)

x

( ) ( )( )( )∫ −−

−=<< 2

1

221

x

xdx

abac

axxXxP



f(x)

x1 x2a b c x

( ) ( )( )( )∫ −−

−=<< 2

1

221

x

xdx

bcac

xcxXxP



x1 x2a b c x

f(x)( ) ( )

( )( )( )

( )( )∫∫ −−−

+−−

−=<< 2

1

2221

x

b

b

xdx

bcac

xcdx

abac

axxXxP


Contoh Perhitungan

Misal variabel random X menyatakan waktu perjalanan yang diketahui berdistribusi segitiga dengan nilai optimistik a = 5 menit, nilai pesimistik c = 20 menit, danmost likely b = 10 menit.

Probabilitas bahwa waktu perjalanan antara 8 menit dan 12 menit?

85 12 1510 x

( ) ( )( )( )

( )( )( )

( ) ( )

4533,0

2400,02133,0 150

202

75

52

1020520

202

510520

52128

12

10

10

8

12

10

10

8

=+=

−+

−=

−−−

+−−

−=<<

∫∫

∫∫

dxx

dxx

dxx

dxx

XP


Distribusi Segitiga Kanan danDistribusi Segitiga Kiri

X ∼ segitiga (a, b, c)

b = a

b = c

X ∼ segitiga kiri (left triangular)

X ∼ segitiga kanan (right triangular)

( )( )( )

⎪⎩

⎪⎨⎧ <<

−−

=lainnya ;0

;2

2cxa

ac

xc

xf

( )( )( )

⎪⎩

⎪⎨⎧ <<

−−

=lainnya ;0

;2

2cxa

ac

ax

xf


Contoh Kurva Distribusi Segitiga Kiri

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 5.00 10.00 15.00 20.00 25.00

x

f(x)

a = 5, b = 5, c = 20


Contoh Kurva Distribusi Segitiga Kanan

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 5.00 10.00 15.00 20.00 25.00

x

f(x)

a = 5, b = 20, c = 20


Distribusi Normal

X ∼ normal (μ, σ2)Parameter:

μ bilangan ril; σ2 > 0

Rataan:

μμ =X

Variansi:22 σσ =X

∞<<∞=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

−

x‐exf

x

;2

1)(

2

2

1

σμ

πσ

μ : rataan

σ2 : variansi



Contoh Kurva Distribusi Normal

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00

x

f(x)

μ = 10, σ2 = 1

μ = 10, σ2 = 4


Distribusi Normal Baku

Rataan:

0=Zμ

Variansi:

12 =Zσ

X ∼ normal (μ, σ2)

σμ−

=X

Z

Z ∼ normal baku (standard normal)

∞<<∞=⎟⎠⎞

⎜⎝⎛ −

z‐ezfz

;2

1)(

2

2

1

π



Kurva Distribusi Normal Baku

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

‐5.00 ‐4.00 ‐3.00 ‐2.00 ‐1.00 0.00 1.00 2.00 3.00 4.00 5.00

z

f(z)

μZ = 0; σZ2 = 1


Probabilitas dari Variabel Random Berdistribusi Normal

( )

⎟⎠⎞

⎜⎝⎛ −

<−⎟⎠⎞

⎜⎝⎛ −

<=

⎟⎠⎞

⎜⎝⎛ −

<<−

=<<

σμ

σμ

σμ

σμ

12

2121

x

ZPx

ZP

xZ

xPxXxP

μ

2σ( )xf

x1x 2x 0

( )zf

z1z 2z


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09-4.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000-3.90 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000-3.80 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001-3.70 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001-3.60 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001-3.50 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002-3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002-3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003-3.20 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005-3.10 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007-3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010-2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014-2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019-2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026-2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036-2.50 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048-2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064-2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084-2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110-2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143-2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183-1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233-1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294-1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367-1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455-1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559-1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681-1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823-1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985-1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170-1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379-0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611-0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867-0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148-0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451-0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776-0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121-0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483-0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859-0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247-0.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

TabelDistribusiNorm

al Baku


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99933.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.99953.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.99973.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.99983.50 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.99983.60 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99993.70 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99993.80 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.99993.90 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.00004.00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

TabelDistribusiNorm

al Baku


Contoh Perhitungan (1)

Tinggi badan yang dinyatakan dengan variabelrandom X diketahui berdistribusi normal denganrataan μ = 160 cm dan variansi σ2 = 16 cm2. Probabilitas bahwa tinggi badan antara 150 cm dan 165 cm?( )

( )( ) ( )

8862,0

0062,08944,0

50,225,1

25,150,2

4

160165

4

160150165150

=−=

<−<=<<−=

⎟⎠⎞

⎜⎝⎛ −

<<−

=<<

ZPZP

ZP

ZPXP

μ X= 160

150 165

σX2 = 16

x



Nilai ujian mahasiswa yang diasumsikan memiliki distribusinormal (rataan μ = 80, variansi σ2 = 100). Jika mahasiswa yang lulus diinginkan sebesar 99 persen, batas nilai kelulusan?

Misal variabel random X nilai ujian mahasiswa

( ) ( )

( )7,56

33,21080

33,210

80

01,010

80

01,099,0

=−=

−=−

=⎟⎠⎞

⎜⎝⎛ −

<

=<⇒=>

x

x

x

xZP

xXPxXP

xα μX = 80

σX2 = 100

x


Hampiran Distribusi Normal terhadapDistribusi Binomial

X ∼ binomial (n, p); n →∞, p → 0,5rataan μ = np; variansi σ2 = np(1− p)

( )pnp

npXZ

−−

=1

Z ∼ normal baku


Contoh Perhitungan

Probabilitas bahwa seseorang pada suatu daerah terinfeksi virus demam berdarah adalah p = 0,4. Jika sebanyak n = 100 orangdipilih secara random dari daerah tersebut, probabilitas bahwaterdapat kurang dari 30 orang terinfeksi?

Misal variabel random X banyaknya orang yang terinfeksi

( )( )( ) ( )( )( )

( ) ( ) 0162,014,224

403030

244,014,01001

404,01002

=−<=⎟⎠⎞

⎜⎝⎛ −

<=<

=−=−=

===

ZPZPXP

pnp

np

X

X

σ

μ


Distribusi Lognormal

X ∼ lognormal (μ, σ2)

Parameter:

μ bilangan ril; σ2 > 0

Rataan:

22σμμ += eX

Variansi:

( )12222 −= + σσμσ eeX

⎪⎪

⎩

⎪⎪

⎨

⎧>

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

−

lainnya ,0

0 ;2

1

)(

2ln

2

1

xexxf

x

σμ

πσ



Contoh Kurva Distribusi Lognormal

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

x

f(x)

1;5,0 2 == σμ

1;1 2 == σμ


Hubungan Distribusi Normal denganLognormal

XeY =

Y ~ lognormal (μ, σ2)

X ~ normal (μ, σ2)

YX ln=


Probabilitas dari Variabel Random Berdistribusi Lognormal

X ∼ Lognormal(μ, σ2)

( ) ( ) ( )( )

( )⎟⎠⎞

⎜⎝⎛ −

<=

<=<

σμx

ZP

xXPxXP

ln

lnln


Contoh Soal

Waktu perbaikan suatu mesin diketahui memiliki distribusilognormal (μ = 2, σ2 = 1). Probabilitas bahwa waktu perbaikanmesin lebih dari 20 menit?

Misal X variabel random waktu perbaikan

( ) ( ) ( )( )( )

( )

( )

0,1587

0,84131

00,11

1

220ln1

20ln1

20lnln20

=−=

<−=

⎟⎠⎞

⎜⎝⎛ −

<−=

⎟⎠⎞

⎜⎝⎛ −

<−=

>=>

ZP

ZP

ZP

XPXP

σμ


Distribusi Gamma

X ∼ gamma (α, β)

Parameter:

α, β > 0

Rataan:

Variansi:

( ) ( )

⎪⎪⎩

⎪⎪⎨

⎧>

Γ=

−−

lainnya ;0

0 ;1 1

x

xex

xf

x

βαα αβ

αβμ =X

22 αβσ =X



Fungsi Gamma

( ) ∫∞ −−=Γ0

1 dxex xαα

( ) ( ) ( )11 −Γ−=Γ ααα

( ) ( )! 1−=Γ nn

π=⎟⎠⎞

⎜⎝⎛Γ2

1


Contoh Kurva Distribusi Gamma

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x

f(x)

α = 0,5; β = 1

α = 1; β = 1

α = 2; β = 1


Distribusi Erlang

X ∼ Erlang (n, β)

Parameter:

n bulat > 0, β > 0

Rataan:

Variansi:

βμ nX =

22 βσ nX =

( ) ( )

⎪⎪⎩

⎪⎪⎨

⎧>

−=

−−

lainnya ;0

0 ;!1

1 1

x

xexn

xf

x

n

n

β

β



Contoh Kurva Distribusi Erlang

‐0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x

f(x)

n = 1; β = 1

n = 2; β = 1

n = 5; β = 1


Hubungan Distribusi Erlang dan Gamma

X ∼ gamma (α, β); α = n bulat



Distribusi Eksponensial (1)

X ∼ eksponensial (β)Parameter:

β > 0

Rataan:

Variansi:

βμ =X

22 βσ =X

( )

⎪⎪⎩

⎪⎪⎨

⎧>

=

−

lainnya ;0

0 ;1

x

xe

xf

x

β

β

β : rata‐rata



Distribusi Eksponensial (2)


( )⎪⎩

⎪⎨

⎧>−=

−

lainnya ;0

0 ;1

x

xexF

x

β


Contoh Kurva Distribusi Eksponensial

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

x

f(x)

β = 1

β = 2β = 4


Probabilitas dari Variabel Random Berdistribusi Eksponensial

X ∼ eksponensial (β)

( ) ∫−

=<< 2

1

121

x

x

x

dxexXxP β

β

x1 x2

β


Contoh Perhitungan

Umur lampu (dinotasikan dengan X) merupakan variabel random yang memiliki distribusi eksponensial dengan rataan β = 6 bulan. Probabilitas bahwa umur lampu X lebih dari 8 bulan?

( )

2636,0

6

11

6

18

8

0

6

8

6

=

−=

=>

∫

∫−

∞ −

dxe

dxeXP

x

x


Hubungan Distribusi Erlang danEksponensial

Xi ∼ eksponensial (β)

∑=

=n

iiXY

1

Y ∼ Erlang (n, β)

Xi saling independen


Hubungan Distribusi Gamma, Erlang danEksponensial


α = 1

n = 1

α = n bulat X∼ eksponensial (β)

X ∼ gamma (α, β)


Hubungan Distribusi Eksponensial danPoisson

X ∼ Poisson (λt)

P(tidak ada kejadian sebelum t) = P(X = 0) =( ) t

t

ete λ

λ λ −−

=!0

0

P(tidak ada kejadian sebelum t) = P(kejadian pertama terjadi padaatau setelah saat t)

= te λ−( )

( )

( ) ( )

λβ

β

λ

β

λ

λ

λ

1;

1)(

1

1

==

==

−=

−=

−

−

−

−

t

t

t

t

etf

edt

tdFtf

etF

tFe

t0

λ

t ∼ eksponensial(β = 1/λ);


Distribusi Khi‐kuadrat

X ∼ khi‐kuadrat (v) Parameter:

v bilangan bulat > 0

Rataan:

Variansi:

vX =μ

vX 22 =σ

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧>

⎟⎠⎞

⎜⎝⎛Γ

=

−−

lainnya ;0

0 ;

22

12

12

2

x

xexv

xf

xv

v

v : derajat kebebasan(degree of freedom)



Contoh Kurva Distribusi Khi‐kuadrat

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.00 5.00 10.00 15.00 20.00 25.00 30.00

v = 5

v = 10

v = 15


Probabilitas dari Variabel Random Berdistribusi Khi‐kuadrat

( ) ( ) αχαχ

α ==>Χ ∫∞

2

22 dxxfP

χ2α

α

Simbol umum untuk variabel random khi‐kuadrat Χ2

Χ2 ∼ khi‐kuadrat (v)dengan fungsi distribusi probabilitas f(x)


v 0.999 0.975 0.995 0.990 0.975 0.950 0.900 0.500 0.100 0.050 0.025 0.010 0.005 0.025 0.0011 0.000 0.001 0.000 0.000 0.001 0.004 0.016 0.455 2.706 3.841 5.024 6.635 7.879 5.024 10.8282 0.002 0.051 0.010 0.020 0.051 0.103 0.211 1.386 4.605 5.991 7.378 9.210 10.597 7.378 13.8163 0.024 0.216 0.072 0.115 0.216 0.352 0.584 2.366 6.251 7.815 9.348 11.345 12.838 9.348 16.2664 0.091 0.484 0.207 0.297 0.484 0.711 1.064 3.357 7.779 9.488 11.143 13.277 14.860 11.143 18.4675 0.210 0.831 0.412 0.554 0.831 1.145 1.610 4.351 9.236 11.070 12.833 15.086 16.750 12.833 20.5156 0.381 1.237 0.676 0.872 1.237 1.635 2.204 5.348 10.645 12.592 14.449 16.812 18.548 14.449 22.4587 0.598 1.690 0.989 1.239 1.690 2.167 2.833 6.346 12.017 14.067 16.013 18.475 20.278 16.013 24.3228 0.857 2.180 1.344 1.646 2.180 2.733 3.490 7.344 13.362 15.507 17.535 20.090 21.955 17.535 26.1249 1.152 2.700 1.735 2.088 2.700 3.325 4.168 8.343 14.684 16.919 19.023 21.666 23.589 19.023 27.877

10 1.479 3.247 2.156 2.558 3.247 3.940 4.865 9.342 15.987 18.307 20.483 23.209 25.188 20.483 29.58811 1.834 3.816 2.603 3.053 3.816 4.575 5.578 10.341 17.275 19.675 21.920 24.725 26.757 21.920 31.26412 2.214 4.404 3.074 3.571 4.404 5.226 6.304 11.340 18.549 21.026 23.337 26.217 28.300 23.337 32.90913 2.617 5.009 3.565 4.107 5.009 5.892 7.042 12.340 19.812 22.362 24.736 27.688 29.819 24.736 34.52814 3.041 5.629 4.075 4.660 5.629 6.571 7.790 13.339 21.064 23.685 26.119 29.141 31.319 26.119 36.12315 3.483 6.262 4.601 5.229 6.262 7.261 8.547 14.339 22.307 24.996 27.488 30.578 32.801 27.488 37.69716 3.942 6.908 5.142 5.812 6.908 7.962 9.312 15.338 23.542 26.296 28.845 32.000 34.267 28.845 39.25217 4.416 7.564 5.697 6.408 7.564 8.672 10.085 16.338 24.769 27.587 30.191 33.409 35.718 30.191 40.79018 4.905 8.231 6.265 7.015 8.231 9.390 10.865 17.338 25.989 28.869 31.526 34.805 37.156 31.526 42.31219 5.407 8.907 6.844 7.633 8.907 10.117 11.651 18.338 27.204 30.144 32.852 36.191 38.582 32.852 43.82020 5.921 9.591 7.434 8.260 9.591 10.851 12.443 19.337 28.412 31.410 34.170 37.566 39.997 34.170 45.31521 6.447 10.283 8.034 8.897 10.283 11.591 13.240 20.337 29.615 32.671 35.479 38.932 41.401 35.479 46.79722 6.983 10.982 8.643 9.542 10.982 12.338 14.041 21.337 30.813 33.924 36.781 40.289 42.796 36.781 48.26823 7.529 11.689 9.260 10.196 11.689 13.091 14.848 22.337 32.007 35.172 38.076 41.638 44.181 38.076 49.72824 8.085 12.401 9.886 10.856 12.401 13.848 15.659 23.337 33.196 36.415 39.364 42.980 45.559 39.364 51.17925 8.649 13.120 10.520 11.524 13.120 14.611 16.473 24.337 34.382 37.652 40.646 44.314 46.928 40.646 52.62026 9.222 13.844 11.160 12.198 13.844 15.379 17.292 25.336 35.563 38.885 41.923 45.642 48.290 41.923 54.05227 9.803 14.573 11.808 12.879 14.573 16.151 18.114 26.336 36.741 40.113 43.195 46.963 49.645 43.195 55.47628 10.391 15.308 12.461 13.565 15.308 16.928 18.939 27.336 37.916 41.337 44.461 48.278 50.993 44.461 56.89229 10.986 16.047 13.121 14.256 16.047 17.708 19.768 28.336 39.087 42.557 45.722 49.588 52.336 45.722 58.30130 11.588 16.791 13.787 14.953 16.791 18.493 20.599 29.336 40.256 43.773 46.979 50.892 53.672 46.979 59.70340 17.916 24.433 20.707 22.164 24.433 26.509 29.051 39.335 51.805 55.758 59.342 63.691 66.766 59.342 73.40250 24.674 32.357 27.991 29.707 32.357 34.764 37.689 49.335 63.167 67.505 71.420 76.154 79.490 71.420 86.66160 31.738 40.482 35.534 37.485 40.482 43.188 46.459 59.335 74.397 79.082 83.298 88.379 91.952 83.298 99.60770 39.036 48.758 43.275 45.442 48.758 51.739 55.329 69.334 85.527 90.531 95.023 100.425 104.215 95.023 112.31780 46.520 57.153 51.172 53.540 57.153 60.391 64.278 79.334 96.578 101.879 106.629 112.329 116.321 106.629 124.83990 54.155 65.647 59.196 61.754 65.647 69.126 73.291 89.334 107.565 113.145 118.136 124.116 128.299 118.136 137.208

100 61.918 74.222 67.328 70.065 74.222 77.929 82.358 99.334 118.498 124.342 129.561 135.807 140.169 129.561 149.449

Παv

α

χ2α

Tabel nilai χ2 untuk derajat kebebasan v dan α


Contoh Perhitungan

( )307,18

05,02

22

=

=>Χ

α

α

χ

χP

α

χ2α

Variabel random Χ2 diketahui berdistribusikhi‐kuadrat dengan derajat kebebasan v = 10.Nilai χ2α agar probabilitas di sebelah kananα = 0,05?

αv


Hubungan Distribusi Gamma danKhi‐kuadrat

X ∼ gamma (α, β); α = v/2; v bulat; β = 2

X ∼ khi‐kuadrat (v)


Hubungan Distribusi Normal Baku, Khi‐Kuadrat dan Gamma

Y ∼ khi‐kuadrat (v); v = n

∑=

=n

iiXY

1

2

Y ∼ gamma (α, β); α = n/2; β = 2

Xi ∼ normal baku


Distribusi Weibull (1)

X ∼Weibull (α, β)

Parameter:

α, β > 0

Rataan:

Variansi:

⎟⎠⎞

⎜⎝⎛Γ=αα

βμ 1X

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛Γ−⎟

⎠⎞

⎜⎝⎛Γ=

222 112

2αααα

βσ X

( )( )

⎪⎩

⎪⎨⎧ >

=−−−

lainnya ;0

0 ;1

x

xexxf

x αβαααβ



Distribusi Weibull (2)


( )( )

⎪⎩

⎪⎨⎧ >−

=−

lainnya ;0

0 ;1

x

xexF

x αβ


Contoh Kurva Distribusi Weibull

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

α = 0,5; β = 1

α = 1; β = 1

α = 2; β = 1


Probabilitas dari Variabel Random Berdistribusi Weibull


( ) ( )∫ −−−=<< 2

1

121

x

x

x dxexxXxPαβαααβ


Hubungan Distribusi Weibull danEksponensial


X ∼ eksponensial (β)

α = 1


Distribusi Student t

X ∼ student t (v) Parameter:

v > 0

Rataan:

Variansi:

0=Xμ

2;2

2 >−

= vv

vXσ

( ) ( )( )( )

( )

∞<<∞

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Γ+Γ

=+−

x

v

x

v

v

vxf

v

‐

;12

211212

π

v : derajat kebebasan(degree of freedom)



Contoh Kurva Distribusi Student t

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

‐5.00 ‐4.00 ‐3.00 ‐2.00 ‐1.00 0.00 1.00 2.00 3.00 4.00 5.00

x

f(x)

v = 2

v = 5

v = 100


Probabilitas dari Variabel Random Berdistribusi Student t

0

α

tα

Simbol umum untuk variabel random student‐t T

( ) ( ) αα

α ==> ∫∞

dttftTPt

T ∼ student t (v)dengan fungsi distribusi probabilitas f(t)

αα tt −=−1

Sifat simetris :


v 0.400 0.300 0.200 0.100 0.050 0.025 0.010 0.005 0.025 0.0011 0.325 0.727 1.376 3.078 6.314 12.706 31.821 63.657 12.706 318.3092 0.289 0.617 1.061 1.886 2.920 4.303 6.965 9.925 4.303 22.3273 0.277 0.584 0.978 1.638 2.353 3.182 4.541 5.841 3.182 10.2154 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.604 2.776 7.1735 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.032 2.571 5.8936 0.265 0.553 0.906 1.440 1.943 2.447 3.143 3.707 2.447 5.2087 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.499 2.365 4.7858 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.355 2.306 4.5019 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.250 2.262 4.297

10 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.169 2.228 4.14411 0.260 0.540 0.876 1.363 1.796 2.201 2.718 3.106 2.201 4.02512 0.259 0.539 0.873 1.356 1.782 2.179 2.681 3.055 2.179 3.93013 0.259 0.538 0.870 1.350 1.771 2.160 2.650 3.012 2.160 3.85214 0.258 0.537 0.868 1.345 1.761 2.145 2.624 2.977 2.145 3.78715 0.258 0.536 0.866 1.341 1.753 2.131 2.602 2.947 2.131 3.73316 0.258 0.535 0.865 1.337 1.746 2.120 2.583 2.921 2.120 3.68617 0.257 0.534 0.863 1.333 1.740 2.110 2.567 2.898 2.110 3.64618 0.257 0.534 0.862 1.330 1.734 2.101 2.552 2.878 2.101 3.61019 0.257 0.533 0.861 1.328 1.729 2.093 2.539 2.861 2.093 3.57920 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.845 2.086 3.55221 0.257 0.532 0.859 1.323 1.721 2.080 2.518 2.831 2.080 3.52722 0.256 0.532 0.858 1.321 1.717 2.074 2.508 2.819 2.074 3.50523 0.256 0.532 0.858 1.319 1.714 2.069 2.500 2.807 2.069 3.48524 0.256 0.531 0.857 1.318 1.711 2.064 2.492 2.797 2.064 3.46725 0.256 0.531 0.856 1.316 1.708 2.060 2.485 2.787 2.060 3.45026 0.256 0.531 0.856 1.315 1.706 2.056 2.479 2.779 2.056 3.43527 0.256 0.531 0.855 1.314 1.703 2.052 2.473 2.771 2.052 3.42128 0.256 0.530 0.855 1.313 1.701 2.048 2.467 2.763 2.048 3.40829 0.256 0.530 0.854 1.311 1.699 2.045 2.462 2.756 2.045 3.39630 0.256 0.530 0.854 1.310 1.697 2.042 2.457 2.750 2.042 3.38540 0.255 0.529 0.851 1.303 1.684 2.021 2.423 2.704 2.021 3.30750 0.255 0.528 0.849 1.299 1.676 2.009 2.403 2.678 2.009 3.26160 0.254 0.527 0.848 1.296 1.671 2.000 2.390 2.660 2.000 3.23270 0.254 0.527 0.847 1.294 1.667 1.994 2.381 2.648 1.994 3.21180 0.254 0.526 0.846 1.292 1.664 1.990 2.374 2.639 1.990 3.19590 0.254 0.526 0.846 1.291 1.662 1.987 2.368 2.632 1.987 3.183

100 0.254 0.526 0.845 1.290 1.660 1.984 2.364 2.626 1.984 3.174∞ 0.253 0.524 0.842 1.282 1.645 1.960 2.326 2.576 1.960 3.090

α

0

α

tα

v

Tabel nilai t untukderajat kebebasan vdan α



( )812,1

05,0

==<

α

α

t

tTP

Variabel random Tdiketahui berdistribusi tdengan derajat kebebasanv = 10. Nilai tα agar probabilitasdi sebelah kananα = 0,05?

αv

0

α

tα



( )812,1

05,0

1

1

−==<

−

−

α

α

t

tTP

Variabel random Tdiketahui berdistribusi tdengan derajat kebebasanv = 10. Nilai tα agar probabilitasdi sebelah kiri 0,05?

αv

0

α

tα

05,005,01 tt −=−


Hubungan Distribusi Normal Baku, Khi‐kuadrat dan Student t

Z ∼ normal bakuY ∼ khi‐kuadrat (v)

vY

ZT =

T ∼ student t (v)


Hubungan Distribusi Student t dan Normal Baku

Z ∼ normal baku

X ∼ student t (v); v→∞


Distribusi F

X ∼ distribusi F (v1, v2)Parameter:

v1, v2 > 0

Rataan:

Variansi:

0;2

2

2

2 >−

= vv

vXμ

( )( )( ) 4;

24

2222

221

21222 >

−−−+

= vvvv

vvvXσ

v1, v2 : derajat kebebasan(degree of freedom)

( )

( )( )( ) ( )

( )

( )( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧>

+⎟⎟⎠

⎞⎜⎜⎝

⎛ΓΓ+Γ

=

+

−

lainnya

0 ;122

22

21

122

2

1

21

21

21

11

x

xvxv

x

v

v

vv

vv

xf

vv

vv



Contoh Kurva Distribusi F

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

x

f(x)

v1 = 10, v2 = 2

v1 = 10, v2 = 5

v1 = 10, v2 = 10


Probabilitas dari Variabel Random Berdistribusi F

Simbol umum untuk variabel random F F

0

α

( ) ( ) αα

α ==> ∫∞

dxxffFPf

F ∼ distribusi F (v1, v2) dengan fungsi distribusi probabilitas f(x)

fα


v2

α = 0,10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 39.863 49.500 53.593 55.833 57.240 58.204 58.906 59.439 59.858 60.195 60.473 60.705 60.903 61.073 61.220 61.350 61.464 61.566 61.658 61.740

2 8.526 9.000 9.162 9.243 9.293 9.326 9.349 9.367 9.381 9.392 9.401 9.408 9.415 9.420 9.425 9.429 9.433 9.436 9.439 9.441

3 5.538 5.462 5.391 5.343 5.309 5.285 5.266 5.252 5.240 5.230 5.222 5.216 5.210 5.205 5.200 5.196 5.193 5.190 5.187 5.184

4 4.545 4.325 4.191 4.107 4.051 4.010 3.979 3.955 3.936 3.920 3.907 3.896 3.886 3.878 3.870 3.864 3.858 3.853 3.849 3.844

5 4.060 3.780 3.619 3.520 3.453 3.405 3.368 3.339 3.316 3.297 3.282 3.268 3.257 3.247 3.238 3.230 3.223 3.217 3.212 3.207

6 3.776 3.463 3.289 3.181 3.108 3.055 3.014 2.983 2.958 2.937 2.920 2.905 2.892 2.881 2.871 2.863 2.855 2.848 2.842 2.836

7 3.589 3.257 3.074 2.961 2.883 2.827 2.785 2.752 2.725 2.703 2.684 2.668 2.654 2.643 2.632 2.623 2.615 2.607 2.601 2.595

8 3.458 3.113 2.924 2.806 2.726 2.668 2.624 2.589 2.561 2.538 2.519 2.502 2.488 2.475 2.464 2.455 2.446 2.438 2.431 2.425

9 3.360 3.006 2.813 2.693 2.611 2.551 2.505 2.469 2.440 2.416 2.396 2.379 2.364 2.351 2.340 2.329 2.320 2.312 2.305 2.298

10 3.285 2.924 2.728 2.605 2.522 2.461 2.414 2.377 2.347 2.323 2.302 2.284 2.269 2.255 2.244 2.233 2.224 2.215 2.208 2.201

11 3.225 2.860 2.660 2.536 2.451 2.389 2.342 2.304 2.274 2.248 2.227 2.209 2.193 2.179 2.167 2.156 2.147 2.138 2.130 2.123

12 3.177 2.807 2.606 2.480 2.394 2.331 2.283 2.245 2.214 2.188 2.166 2.147 2.131 2.117 2.105 2.094 2.084 2.075 2.067 2.060

13 3.136 2.763 2.560 2.434 2.347 2.283 2.234 2.195 2.164 2.138 2.116 2.097 2.080 2.066 2.053 2.042 2.032 2.023 2.014 2.007

14 3.102 2.726 2.522 2.395 2.307 2.243 2.193 2.154 2.122 2.095 2.073 2.054 2.037 2.022 2.010 1.998 1.988 1.978 1.970 1.962

15 3.073 2.695 2.490 2.361 2.273 2.208 2.158 2.119 2.086 2.059 2.037 2.017 2.000 1.985 1.972 1.961 1.950 1.941 1.932 1.924

16 3.048 2.668 2.462 2.333 2.244 2.178 2.128 2.088 2.055 2.028 2.005 1.985 1.968 1.953 1.940 1.928 1.917 1.908 1.899 1.891

17 3.026 2.645 2.437 2.308 2.218 2.152 2.102 2.061 2.028 2.001 1.978 1.958 1.940 1.925 1.912 1.900 1.889 1.879 1.870 1.862

18 3.007 2.624 2.416 2.286 2.196 2.130 2.079 2.038 2.005 1.977 1.954 1.933 1.916 1.900 1.887 1.875 1.864 1.854 1.845 1.837

19 2.990 2.606 2.397 2.266 2.176 2.109 2.058 2.017 1.984 1.956 1.932 1.912 1.894 1.878 1.865 1.852 1.841 1.831 1.822 1.814

20 2.975 2.589 2.380 2.249 2.158 2.091 2.040 1.999 1.965 1.937 1.913 1.892 1.875 1.859 1.845 1.833 1.821 1.811 1.802 1.794

v1

Tabel nilai f untuk derajat kebebasan v dan α


v 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.883 240.543 241.882 242.983 243.906 244.690 245.364 245.950 246.464 246.918 247.323 247.686 248.013

2 18.513 19.000 19.164 19.247 19.296 19.330 19.353 19.371 19.385 19.396 19.405 19.413 19.419 19.424 19.429 19.433 19.437 19.440 19.443 19.446

3 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 8.763 8.745 8.729 8.715 8.703 8.692 8.683 8.675 8.667 8.660

4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5.936 5.912 5.891 5.873 5.858 5.844 5.832 5.821 5.811 5.803

5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735 4.704 4.678 4.655 4.636 4.619 4.604 4.590 4.579 4.568 4.558

6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060 4.027 4.000 3.976 3.956 3.938 3.922 3.908 3.896 3.884 3.874

7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637 3.603 3.575 3.550 3.529 3.511 3.494 3.480 3.467 3.455 3.445

8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347 3.313 3.284 3.259 3.237 3.218 3.202 3.187 3.173 3.161 3.150

9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137 3.102 3.073 3.048 3.025 3.006 2.989 2.974 2.960 2.948 2.936

10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978 2.943 2.913 2.887 2.865 2.845 2.828 2.812 2.798 2.785 2.774

11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854 2.818 2.788 2.761 2.739 2.719 2.701 2.685 2.671 2.658 2.646

12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753 2.717 2.687 2.660 2.637 2.617 2.599 2.583 2.568 2.555 2.544

13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671 2.635 2.604 2.577 2.554 2.533 2.515 2.499 2.484 2.471 2.459

14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602 2.565 2.534 2.507 2.484 2.463 2.445 2.428 2.413 2.400 2.388

15 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544 2.507 2.475 2.448 2.424 2.403 2.385 2.368 2.353 2.340 2.328

16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494 2.456 2.425 2.397 2.373 2.352 2.333 2.317 2.302 2.288 2.276

17 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.494 2.450 2.413 2.381 2.353 2.329 2.308 2.289 2.272 2.257 2.243 2.230

18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412 2.374 2.342 2.314 2.290 2.269 2.250 2.233 2.217 2.203 2.191

19 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378 2.340 2.308 2.280 2.256 2.234 2.215 2.198 2.182 2.168 2.155

20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 2.310 2.278 2.250 2.225 2.203 2.184 2.167 2.151 2.137 2.124

v1

v2

α = 0,05



v1

v2

α = 0,01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 4052.2 4999.5 5403.4 5624.6 5763.6 5859.0 5928.4 5981.1 6022.5 6055.8 6083.3 6106.3 6125.9 6142.7 6157.3 6170.1 6181.4 6191.5 6200.6 6208.7

2 98.503 99.000 99.166 99.249 99.299 99.333 99.356 99.374 99.388 99.399 99.408 99.416 99.422 99.428 99.433 99.437 99.440 99.444 99.447 99.449

3 34.116 30.817 29.457 28.710 28.237 27.911 27.672 27.489 27.345 27.229 27.133 27.052 26.983 26.924 26.872 26.827 26.787 26.751 26.719 26.690

4 21.198 18.000 16.694 15.977 15.522 15.207 14.976 14.799 14.659 14.546 14.452 14.374 14.307 14.249 14.198 14.154 14.115 14.080 14.048 14.020

5 16.258 13.274 12.060 11.392 10.967 10.672 10.456 10.289 10.158 10.051 9.963 9.888 9.825 9.770 9.722 9.680 9.643 9.610 9.580 9.553

6 13.745 10.925 9.780 9.148 8.746 8.466 8.260 8.102 7.976 7.874 7.790 7.718 7.657 7.605 7.559 7.519 7.483 7.451 7.422 7.396

7 12.246 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.719 6.620 6.538 6.469 6.410 6.359 6.314 6.275 6.240 6.209 6.181 6.155

8 11.259 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.911 5.814 5.734 5.667 5.609 5.559 5.515 5.477 5.442 5.412 5.384 5.359

9 10.561 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351 5.257 5.178 5.111 5.055 5.005 4.962 4.924 4.890 4.860 4.833 4.808

10 10.044 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.942 4.849 4.772 4.706 4.650 4.601 4.558 4.520 4.487 4.457 4.430 4.405

11 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632 4.539 4.462 4.397 4.342 4.293 4.251 4.213 4.180 4.150 4.123 4.099

12 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.388 4.296 4.220 4.155 4.100 4.052 4.010 3.972 3.939 3.909 3.883 3.858

13 9.074 6.701 5.739 5.205 4.862 4.620 4.441 4.302 4.191 4.100 4.025 3.960 3.905 3.857 3.815 3.778 3.745 3.716 3.689 3.665

14 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.140 4.030 3.939 3.864 3.800 3.745 3.698 3.656 3.619 3.586 3.556 3.529 3.505

15 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895 3.805 3.730 3.666 3.612 3.564 3.522 3.485 3.452 3.423 3.396 3.372

16 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.890 3.780 3.691 3.616 3.553 3.498 3.451 3.409 3.372 3.339 3.310 3.283 3.259

17 8.400 6.112 5.185 4.669 4.336 4.102 3.927 3.791 3.682 3.593 3.519 3.455 3.401 3.353 3.312 3.275 3.242 3.212 3.186 3.162

18 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.597 3.508 3.434 3.371 3.316 3.269 3.227 3.190 3.158 3.128 3.101 3.077

19 8.185 5.926 5.010 4.500 4.171 3.939 3.765 3.631 3.523 3.434 3.360 3.297 3.242 3.195 3.153 3.116 3.084 3.054 3.027 3.003

20 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457 3.368 3.294 3.231 3.177 3.130 3.088 3.051 3.018 2.989 2.962 2.938

v1



Contoh Perhitungan

Variabel random F memiliki distribusi F (v1 = 10, v2 = 10)

Nilai fα sehingga P(F > fα) = 0,05 ?

978,2=αf

v2

v1Tabel nilai f untuk α = 0,05


Hubungan Distribusi F dengan Khi‐kuadrat

Χ12 ∼ khi‐kuadrat (v1)

Χ22 ∼ khi‐kuadrat (v2)

222

121

v

vF

ΧΧ

=

F ∼ distribusi F (v1, v2)

independen


Distribusi Beta

X ∼ beta (α1, α2)

Parameter:

α1, α2 > 0

Rataan:

Variansi:

21

1

αααμ+

=X

( ) ( )221

2

21

212

1+++=

αααααασ X

( )

( )( )

⎪⎪⎩

⎪⎪⎨

⎧≤≤

Β−

=

−−

lainnya ;0

10 ;,

1

21

11 21

x

xxx

xfαα

αα



Fungsi Beta

( ) ( )∫ −− −=Β1

0

1121

21 1, dttt αααα

( ) ( )1221 ,, αααα Β=Β

( ) ( ) ( )( )21

2121 , αα

αααα+ΓΓΓ

=Β


Contoh Kurva Distribusi Beta (1)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x

f(x)

α1 = 1,5; α2 = 5 α1 = 5; α2 = 1,5

α1 = 1,5; α2 = 3 α1 = 3; α2 = 1,5



0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x

f(x)

α1 = 1; α2 = 1

α1 = 2; α2 = 2

α1 = 5; α2 = 5

α1 = 10; α2 = 10



0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x

f(x)

α1 = 0,8; α2 = 2 α1 = 2; α2 = 0,8

α1 = 1; α2 = 2 α1 = 2; α2 = 1



0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

x

f(x)

α1 = 0,8; α2 = 0,2 α1 = 0,2; α2 = 0,8

α1 = 0,5; α2 = 0.5


Hubungan Gamma dan Beta

X1 ∼ gamma (α1, β)X2 ∼ gamma (α2, β)

21

1

XX

XX

+=



Hubungan Distribusi Beta danSeragam Kontinyu

X ∼ seragam kontinyu (a, b); a = 0, b = 1

X ∼ beta (α1, α2);α1 = 1, α2 = 1


Hubungan Distribusi Beta danSegitiga


X ∼ segitiga kiri (left triangular)

X ∼ segitiga kanan (right triangular)

α1 = 2, α2 = 1

α1 = 1, α2 = 2

00 kuliah 03-02 distribusi probabilitas kontinyu teoritis beta (beta distribution) distribusi...

Documents