Distribusi Probabilitas Kontinyu Teoritis

Distribusi Probabilitas Kontinyu Teoritis

DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi

1

Dist. Prob. Teoritis Kontinyu (1) Distribusi seragam kontinyu (continuous uniform distribution) Distribusi segitiga (triangular distribution) 9 Distribusi segitiga kiri (left triangular distribution) 9 Distribusi segitiga kanan (right triangular distribution)

Distribusi normal (normal distribution) 9 Distribusi normal baku (standard normal distribution)

Distribusi lognormal (lognormal distribution) Distribusi gamma (gamma distribution) 9 Distribusi Erlang (Erlang distribution) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

2

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 PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

3

Distribusi Seragam Kontinyu (1) X ∼ seragam kontinyu (a, b) Fungsi distribusi probabilitas:

⎧ 1 ⎪ b − a ; a < x < b f (x ) = ⎨ ⎪0; x lainnya ⎩

Parameter: a, b bilangan riil (b > a) a : batas bawah b : batas atas Rataan:

μX = Variansi:

σ = 2 X

DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

a+b 2

(b − a )2 12 4

Distribusi Seragam Kontinyu (2) Fungsi distribusi probabilitas kumulatif: ⎧ ⎪0; x < a ⎪⎪ x − a F (x ) = ⎨ ; a < x < b ⎪b − a ⎪1; x > b ⎪⎩


5

Contoh Kurva Distribusi Seragam Kontinyu

0.25

0.20

a = 10, b = 20

f(x)

0.15

0.10

0.05

0.00 5.00

10.00

15.00

20.00

25.00

x


6

Probabilitas dari Variabel Random Seragam Kontinyu P ( x1 < X < x 2 ) = ∫

x2

x1

1 dx b−a

f (x )

a

x1


x2

b

x 7

Contoh Perhitungan Diameter komponen yang dinyatakan dengan variabel random X diketahui berdistribusi seragam kontinyu dengan batas bawah 10 mm dan batas atas = 20 mm. Probabilitas bahwa diameter komponen kurang dari 15 mm? 1 dx 10 20 − 10 15 ‐ 10 = 10 = 0,5

P ( X < 15) = ∫

15


8

Distribusi Segitiga X ∼ segitiga (a, b, c) Fungsi distribusi probabilitas: ⎧ 2(x − a ) ⎪ (c − a )(b − a ) ; a ≤ x ≤ b ⎪ ⎪ 2(c − x ) f (x ) = ⎨ ; b ≤ x ≤ c ( )( ) c − a c − b ⎪ ⎪0; lainnya ⎪⎩ DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

Parameter: a, b, c bilangan ril (a < b < c) Rataan:

a+b+c 3 Variansi: a 2 + b 2 + c 2 − ab − ac − bc 2 σ = 18 9

μ=

Distribusi Segitiga (2) Fungsi probabilitas kumulatif: ⎧0; x < a ⎪ (x − a )2 ⎪ ⎪ (c − a )(b − a ) F (x ) = ⎨ ; a < x < b 2 ( ) c − x ⎪1 − ; b < x < c ⎪ (c − a )(c − b ) ⎪⎩1; x > c


10

Contoh Kurva Distribusi Segitiga 0.14 0.12

a = 5, b = 10, c = 20

0.10

f(x)

0.08 0.06 0.04 0.02 0.00 0.00

5.00

10.00

15.00

20.00

25.00

x


11

Probabilitas dari Variabel Random Segitiga P ( x1 < X < x 2 ) = ∫

f(x)

x2

x1

a x1

x2 b


2(x − a ) dx (c − a )(b − a )

c

x 12

Probabilitas dari Variabel Random Segitiga P ( x1 < X < x 2 ) = ∫

f(x)

x2

x1

a

b x1

x2

c

2(c − x ) dx (c − a )(c − b)

x


13

Probabilitas dari Variabel Random Segitiga f(x)

x2 2(x − a ) 2(c − x ) dx + ∫ dx x1 (c − a )(b − a ) b (c − a )(c − b )

P ( x1 < X < x 2 ) = ∫

a

x1

b

b


x2

c

x 14

Contoh Perhitungan Misal variabel random X menyatakan waktu perjalanan yang diketahui berdistribusi segitiga dengan nilai optimistik a = 5 menit, nilai pesimistik c = 20 menit, dan most likely b = 10 menit. Probabilitas bahwa waktu perjalanan antara 8 menit dan 12 menit? 12 2(x − 5) 2(20 − x ) dx + ∫ dx 8 (20 − 5)(10 − 5) 10 (20 − 5)(20 − 10 ) 10 2( x − 5) 122(20 − x ) = ∫ dx + ∫ dx 8 10 75 150 = 0,2133 + 0,2400

P (8 < X < 12) = ∫

10

= 0,4533

5 8 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

10

12

x

15 15

Distribusi Segitiga Kanan dan Distribusi Segitiga Kiri X ∼ segitiga kiri (left triangular)

⎧ 2(c − x ) ; a < x < c ⎪ f (x ) = ⎨ (c − a )2 ⎪⎩0; lainnya

b = a X ∼ segitiga (a, b, c) b = c

⎧ 2(x − a ) ; a < x < c ⎪ X ∼ segitiga kanan (right triangular) f (x ) = ⎨ (c − a )2 ⎪⎩0; lainnya DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

16

Contoh Kurva Distribusi Segitiga Kiri 0.14 0.12

a = 5, b = 5, c = 20

0.10

f(x)

0.08 0.06 0.04 0.02 0.00 0.00

5.00

10.00

15.00

20.00

25.00

x


17

Contoh Kurva Distribusi Segitiga Kanan 0.14 0.12

a = 5, b = 20, c = 20

0.10

f(x)

0.08 0.06 0.04 0.02 0.00 0.00

5.00

10.00

15.00

20.00

25.00

x


18

Distribusi Normal X ∼ normal (μ, σ2)

Parameter:

μ bilangan ril; σ2 > 0

Fungsi distribusi probabilitas:

f ( x) =

1 e σ 2π

⎛ 1 ⎛ x − μ ⎞2 ⎞ ⎜− ⎜ ⎟ ⎜ 2 ⎝ σ ⎟⎠ ⎟ ⎝ ⎠

; ‐∞ < x < ∞

μ : rataan σ2 : variansi Rataan:

μX = μ Variansi:

σ X2 = σ 2


19

Contoh Kurva Distribusi Normal 0.45 0.40

μ = 10, σ2 = 1

0.35

f(x)

0.30 0.25 0.20

μ = 10, σ2 = 4

0.15 0.10 0.05 0.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

x


20

Distribusi Normal Baku X ∼ normal (μ, σ2)

Z=

X −μ

σ

Z ∼ normal baku (standard normal)

Rataan:

μZ = 0


1 f (z) = e 2π

⎛ 1 2⎞ ⎜− z ⎟ ⎝ 2 ⎠

Variansi:

σ Z2 = 1

; ‐∞ < z < ∞


21

Kurva Distribusi Normal Baku 0.45 0.40

μZ = 0; σZ2 = 1

0.35

f(z)

0.30 0.25 0.20 0.15 0.10 0.05 0.00 ‐5.00

‐4.00

‐3.00

‐2.00

‐1.00

0.00

1.00

2.00

3.00

4.00

5.00

z


22

Probabilitas dari Variabel Random Berdistribusi Normal x −μ ⎞ ⎛x −μ P ( x1 < X < x 2 ) = P ⎜ 1
f (z )

f (x )

σ

x1

μ

2

x2

z1

x

0

z2

z

Tabel Distribusi Normal Baku


0.00 0.01 0.02 -4.00 0.0000 0.0000 0.0000 -3.90 0.0000 0.0000 0.0000 -3.80 0.0001 0.0001 0.0001 -3.70 0.0001 0.0001 0.0001 -3.60 0.0002 0.0002 0.0001 -3.50 0.0002 0.0002 0.0002 -3.40 0.0003 0.0003 0.0003 -3.30 0.0005 0.0005 0.0005 -3.20 0.0007 0.0007 0.0006 -3.10 0.0010 0.0009 0.0009 -3.00 0.0013 0.0013 0.0013 -2.90 0.0019 0.0018 0.0018 -2.80 0.0026 0.0025 0.0024 -2.70 0.0035 0.0034 0.0033 -2.60 0.0047 0.0045 0.0044 -2.50 0.0062 0.0060 0.0059 -2.40 0.0082 0.0080 0.0078 -2.30 0.0107 0.0104 0.0102 -2.20 0.0139 0.0136 0.0132 -2.10 0.0179 0.0174 0.0170 -2.00 0.0228 0.0222 0.0217 -1.90 0.0287 0.0281 0.0274 -1.80 0.0359 0.0351 0.0344 -1.70 0.0446 0.0436 0.0427 -1.60 0.0548 0.0537 0.0526 -1.50 0.0668 0.0655 0.0643 -1.40 0.0808 0.0793 0.0778 -1.30 0.0968 0.0951 0.0934 -1.20 0.1151 0.1131 0.1112 -1.10 0.1357 0.1335 0.1314 -1.00 0.1587 0.1562 0.1539 -0.90 0.1841 0.1814 0.1788 -0.80 0.2119 0.2090 0.2061 -0.70 0.2420 0.2389 0.2358 -0.60 0.2743 0.2709 0.2676 -0.50 0.3085 0.3050 0.3015 -0.40 0.3446 0.3409 0.3372 -0.30 0.3821 0.3783 0.3745 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS -0.20 0.4207 0.4168 0.4129 -0.10 0.4602 0.4562 0.4522 Suprayogi, 2006 -0.00 0.5000 0.4960 0.4920

23

0.03 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0006 0.0009 0.0012 0.0017 0.0023 0.0032 0.0043 0.0057 0.0075 0.0099 0.0129 0.0166 0.0212 0.0268 0.0336 0.0418 0.0516 0.0630 0.0764 0.0918 0.1093 0.1292 0.1515 0.1762 0.2033 0.2327 0.2643 0.2981 0.3336 0.3707 0.4090 0.4483 0.4880

0.04 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0006 0.0008 0.0012 0.0016 0.0023 0.0031 0.0041 0.0055 0.0073 0.0096 0.0125 0.0162 0.0207 0.0262 0.0329 0.0409 0.0505 0.0618 0.0749 0.0901 0.1075 0.1271 0.1492 0.1736 0.2005 0.2296 0.2611 0.2946 0.3300 0.3669 0.4052 0.4443 0.4840

0.05 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0006 0.0008 0.0011 0.0016 0.0022 0.0030 0.0040 0.0054 0.0071 0.0094 0.0122 0.0158 0.0202 0.0256 0.0322 0.0401 0.0495 0.0606 0.0735 0.0885 0.1056 0.1251 0.1469 0.1711 0.1977 0.2266 0.2578 0.2912 0.3264 0.3632 0.4013 0.4404 0.4801

0.06 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0006 0.0008 0.0011 0.0015 0.0021 0.0029 0.0039 0.0052 0.0069 0.0091 0.0119 0.0154 0.0197 0.0250 0.0314 0.0392 0.0485 0.0594 0.0721 0.0869 0.1038 0.1230 0.1446 0.1685 0.1949 0.2236 0.2546 0.2877 0.3228 0.3594 0.3974 0.4364 0.4761

0.07 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0008 0.0011 0.0015 0.0021 0.0028 0.0038 0.0051 0.0068 0.0089 0.0116 0.0150 0.0192 0.0244 0.0307 0.0384 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.1423 0.1660 0.1922 0.2206 0.2514 0.2843 0.3192 0.3557 0.3936 0.4325 0.4721

0.08 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0007 0.0010 0.0014 0.0020 0.0027 0.0037 0.0049 0.0066 0.0087 0.0113 0.0146 0.0188 0.0239 0.0301 0.0375 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.1401 0.1635 0.1894 0.2177 0.2483 0.2810 0.3156 0.3520 0.3897 0.4286 0.4681

0.09 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0002 0.0003 0.0005 0.0007 0.0010 0.0014 0.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 0.0183 0.0233 0.0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 0.1379 0.1611 0.1867 0.2148 0.2451 0.2776 0.3121 0.3483 240.3859 0.4247 0.4641

Tabel Distribusi Normal Baku

0.00 0.01 0.02 0.00 0.5000 0.5040 0.5080 0.10 0.5398 0.5438 0.5478 0.20 0.5793 0.5832 0.5871 0.30 0.6179 0.6217 0.6255 0.40 0.6554 0.6591 0.6628 0.50 0.6915 0.6950 0.6985 0.60 0.7257 0.7291 0.7324 0.70 0.7580 0.7611 0.7642 0.80 0.7881 0.7910 0.7939 0.90 0.8159 0.8186 0.8212 1.00 0.8413 0.8438 0.8461 1.10 0.8643 0.8665 0.8686 1.20 0.8849 0.8869 0.8888 1.30 0.9032 0.9049 0.9066 1.40 0.9192 0.9207 0.9222 1.50 0.9332 0.9345 0.9357 1.60 0.9452 0.9463 0.9474 1.70 0.9554 0.9564 0.9573 1.80 0.9641 0.9649 0.9656 1.90 0.9713 0.9719 0.9726 2.00 0.9772 0.9778 0.9783 2.10 0.9821 0.9826 0.9830 2.20 0.9861 0.9864 0.9868 2.30 0.9893 0.9896 0.9898 2.40 0.9918 0.9920 0.9922 2.50 0.9938 0.9940 0.9941 2.60 0.9953 0.9955 0.9956 2.70 0.9965 0.9966 0.9967 2.80 0.9974 0.9975 0.9976 2.90 0.9981 0.9982 0.9982 3.00 0.9987 0.9987 0.9987 3.10 0.9990 0.9991 0.9991 3.20 0.9993 0.9993 0.9994 3.30 0.9995 0.9995 0.9995 3.40 0.9997 0.9997 0.9997 3.50 0.9998 0.9998 0.9998 3.60 0.9998 0.9998 0.9999 3.70 0.9999 0.9999 0.9999 3.80 0.9999 0.9999 0.9999 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS 3.90 1.0000 1.0000 1.0000 Suprayogi, 2006 4.00 1.0000 1.0000 1.0000

0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.9991 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.9992 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990 0.9993 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 25 1.0000 1.0000

0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

Contoh Perhitungan (1) Tinggi badan yang dinyatakan dengan variabel random X diketahui berdistribusi normal dengan rataan μ = 160 cm dan variansi σ2 = 16 cm2. Probabilitas bahwa tinggi badan antara 150 cm 165 − 160 ⎞ ⎛ 150 − 160 P (150 165 cm? < X < 165) = P ⎜
4

⎠

σX2 = 16

= P (Z < 1,25) − P (Z < 2,50 ) = 0,8944 − 0,0062 = 0,8862 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

150

μ X= 160

165

x 26

Contoh Perhitungan (2) Nilai ujian mahasiswa yang diasumsikan memiliki distribusi normal (rataan μ = 80, variansi σ2 = 100). Jika mahasiswa yang lulus diinginkan sebesar 99 persen, batas nilai kelulusan? Misal variabel random X Æ nilai ujian mahasiswa P ( X > x ) = 0,99 ⇒ P ( X < x ) = 0,01 x − 80 ⎞ ⎛ P⎜ Z < ⎟ = 0,01 10 ⎝ ⎠ x − 80 = −2,33 10 x = 80 − 10(2,33) x = 56,7

σX2 = 100

xα μX = 80


x

27

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

X − np Z= np(1 − p ) Z ∼ normal baku DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

28

Contoh Perhitungan Probabilitas bahwa seseorang pada suatu daerah terinfeksi virus demam berdarah adalah p = 0,4. Jika sebanyak n = 100 orang dipilih secara random dari daerah tersebut, probabilitas bahwa terdapat kurang dari 30 orang terinfeksi? Misal variabel random X Æ banyaknya orang yang terinfeksi

μ X = np = (100 )(0,4 ) = 40

σ X2 = np(1 − p ) = (100 )(0,4 )(1 − 0,4 ) = 24 30 − 40 ⎞ ⎛ P ( X < 30 ) = P ⎜ Z < ⎟ = P (Z < −2,14 ) = 0,0162 24 ⎠ ⎝ DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

29

Distribusi Lognormal X ∼ lognormal (μ, σ2) Fungsi distribusi probabilitas: ⎛ 1 ⎛ ln x − μ ⎞2 ⎞ ⎧ ⎜− ⎜ ⎟ ⎜ 2 ⎝ σ ⎟⎠ ⎟ 1 ⎠ ⎪ e⎝ ; x > 0 ⎪ xσ 2π f (x) = ⎨ ⎪ ⎪0, lainnya ⎩

Parameter:

μ bilangan ril; σ2 > 0 Rataan:

μ X = e μ +σ Variansi:

σ =e 2 X


2

2 μ +σ 2

2

(e

σ2

−1

)

30

Contoh Kurva Distribusi Lognormal 0.45 0.40 0.35

μ = 0,5;σ 2 = 1

f(x)

0.30 0.25 0.20 0.15

μ = 1;σ 2 = 1

0.10 0.05 0.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

x


31

Hubungan Distribusi Normal dengan Lognormal X ~ normal (μ, σ2)

X = ln Y

Y = eX

Y ~ lognormal (μ, σ2) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

32

Probabilitas dari Variabel Random Berdistribusi Lognormal X ∼ Lognormal(μ, σ2) P ( X < x ) = P (ln( X ) < ln(x )) ln(x ) − μ ⎞ ⎛ = P⎜ Z < ⎟ σ ⎝ ⎠


33

Contoh Soal Waktu perbaikan suatu mesin diketahui memiliki distribusi lognormal (μ = 2, σ2 = 1). Probabilitas bahwa waktu perbaikan mesin lebih dari 20 menit? Misal X Æ variabel random waktu perbaikan P ( X > 20 ) = P (ln( X ) > ln(20 )) ln(20 ) − μ ⎞ ⎛ = 1 − P ⎜ Z < ⎟ σ ⎝ ⎠ ln(20 ) − 2 ⎞ ⎛ = 1 − P ⎜ Z < ⎟ 1 ⎝ ⎠ = 1 − P (Z < 1,00 ) = 1 − 0,8413 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS = 0,1587 Suprayogi, 2006

34

Distribusi Gamma X ∼ gamma (α, β) Fungsi distribusi probabilitas: − ⎧ 1 α −1 x e β ; x > 0 ⎪ α ⎪ β Γ(α ) f (x ) = ⎨ ⎪0; x lainnya ⎪⎩

Parameter:

α, β > 0

x

Rataan:

μ X = αβ

Variansi:

σ X2 = αβ 2


35

Fungsi Gamma ∞

Γ(α ) = ∫ x α −1e − x dx 0

Γ(α ) = (α − 1)Γ(α − 1) Γ(n ) = (n − 1) !

⎛1⎞ Γ⎜ ⎟ = π ⎝2⎠ DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

36

Contoh Kurva Distribusi Gamma 1.80 1.60 1.40

f(x)

1.20

α = 0,5; β = 1

1.00 0.80 0.60

α = 1; β = 1 α = 2; β = 1

0.40 0.20 0.00 0.00

1.00

2.00

3.00

4.00

5.00

6.00

x


37

Distribusi Erlang X ∼ Erlang (n, β) Fungsi distribusi probabilitas: x − ⎧ 1 x n−1e β ; x > 0 ⎪ n ⎪ β (n − 1)! f (x ) = ⎨ ⎪0; x lainnya ⎪⎩

Parameter: n bulat > 0, β > 0 Rataan:

μ X = nβ

Variansi:

σ X2 = nβ 2 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

38

Contoh Kurva Distribusi Erlang 1.00 0.90 0.80

n = 1; β = 1

0.70

f(x)

0.60 0.50

n = 2; β = 1

0.40

n = 5; β = 1

0.30 0.20 0.10 0.00 0.00 ‐0.10

1.00

2.00

3.00

4.00

5.00

6.00

x


39

Hubungan Distribusi Erlang dan Gamma X ∼ gamma (α, β); α = n bulat

X ∼ Erlang (n, β)


40

Distribusi Eksponensial (1) X ∼ eksponensial (β) Parameter:


β > 0 β : rata‐rata

⎧ 1 − βx ⎪ e ; x > 0 ⎪β f (x ) = ⎨ ⎪0; x lainnya ⎪⎩ DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

Rataan:

μX = β

Variansi:

σ X2 = β 2

41

Distribusi Eksponensial (2) Fungsi distribusi probabilitas kumulatif: x − ⎧ ⎪1 − e β ; x > 0 F (x ) = ⎨ ⎪0; x lainnya ⎩


42

Contoh Kurva Distribusi Eksponensial 1.00 0.90 0.80

β = 1

0.70

f(x)

0.60 0.50 0.40 0.30 0.20

β = 4

0.10 0.00 0.00

1.00

β = 2

2.00

3.00

4.00


5.00

6.00

7.00

8.00

9.00

10.00

x

43

Probabilitas dari Variabel Random Berdistribusi Eksponensial X ∼ eksponensial (β)

β

x2

1

x1

β

P ( x1 < X < x 2 ) = ∫

x1

−

x

e β dx

x2


44

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?

1 − 6x P ( X > 8 ) = ∫ e dx 8 6 x 81 − 6 = 1 − ∫ e dx 06 = 0,2636 ∞


45

Hubungan Distribusi Erlang dan Eksponensial Xi ∼ eksponensial (β) Xi Æ saling independen n

Y = ∑ Xi i =1

Y ∼ Erlang (n, β) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

46

Hubungan Distribusi Gamma, Erlang dan Eksponensial X ∼ gamma (α, β)

α = 1 X ∼ eksponensial (β)

α = n bulat

n = 1 X ∼ Erlang (n, β) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

47

Hubungan Distribusi Eksponensial dan Poisson X ∼ Poisson (λt)

λ t

0

e − λt (λt ) = e − λt P(tidak ada kejadian sebelum t) = P(X = 0) = 0! P(tidak ada kejadian sebelum t) = P(kejadian pertama terjadi pada atau setelah saat t) 0

= e − λt t ∼ eksponensial(β = 1/λ); DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

e − λt = 1 − F (t ) F (t ) = 1 − e −λt dF (t ) = λ e − λt f (t ) = dt 1 1 f (t ) = e −t β ; β =

β

λ

48

Distribusi Khi‐kuadrat X ∼ khi‐kuadrat (v)

Parameter: v bilangan bulat > 0

Fungsi distribusi probabilitas: v x −1 − ⎧ 1 x 2 e 2 ; x > 0 ⎪ v ⎪ 2 2 Γ⎛⎜ v ⎞⎟ ⎪ ⎝2⎠ f (x ) = ⎨ ⎪0; x lainnya ⎪ ⎪ ⎩

v : derajat kebebasan (degree of freedom) Rataan:

μX = v

Variansi:

σ X2 = 2v


49

Contoh Kurva Distribusi Khi‐kuadrat 0.18 0.16 0.14

v = 5

0.12

v = 10

0.10 0.08

v = 15

0.06 0.04 0.02 0.00 0.00

5.00

10.00


15.00

20.00

25.00

30.00

50

Probabilitas dari Variabel Random Berdistribusi Khi‐kuadrat Simbol umum untuk variabel random khi‐kuadrat Æ Χ2 Χ2 ∼ khi‐kuadrat (v)

dengan fungsi distribusi probabilitas f(x)

P (Χ > χα ) = ∫ 2 f (x )dx = α 2

∞

2

χα

α

χ2α DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

51

Tabel nilai χ2 untuk derajat kebebasan v dan α

α

α

χ2α

Π

v

v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100

0.999 0.000 0.002 0.024 0.091 0.210 0.381 0.598 0.857 1.152 1.479 1.834 2.214 2.617 3.041 3.483 3.942 4.416 4.905 5.407 5.921 6.447 6.983 7.529 8.085 8.649 9.222 9.803 10.391 10.986 11.588 17.916 24.674 31.738 39.036 46.520 54.155 61.918

0.975 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 24.433 32.357 40.482 48.758 57.153 65.647 74.222

0.995 0.000 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.520 11.160 11.808 12.461 13.121 13.787 20.707 27.991 35.534 43.275 51.172 59.196 67.328

0.990 0.000 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.196 10.856 11.524 12.198 12.879 13.565 14.256 14.953 22.164 29.707 37.485 45.442 53.540 61.754 70.065

0.975 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 24.433 32.357 40.482 48.758 57.153 65.647 74.222

0.950 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.117 10.851 11.591 12.338 13.091 13.848 14.611 15.379 16.151 16.928 17.708 18.493 26.509 34.764 43.188 51.739 60.391 69.126 77.929


0.900 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 7.790 8.547 9.312 10.085 10.865 11.651 12.443 13.240 14.041 14.848 15.659 16.473 17.292 18.114 18.939 19.768 20.599 29.051 37.689 46.459 55.329 64.278 73.291 82.358

0.500 0.455 1.386 2.366 3.357 4.351 5.348 6.346 7.344 8.343 9.342 10.341 11.340 12.340 13.339 14.339 15.338 16.338 17.338 18.338 19.337 20.337 21.337 22.337 23.337 24.337 25.336 26.336 27.336 28.336 29.336 39.335 49.335 59.335 69.334 79.334 89.334 99.334

0.100 2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256 51.805 63.167 74.397 85.527 96.578 107.565 118.498

0.050 3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 55.758 67.505 79.082 90.531 101.879 113.145 124.342

0.025 5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 59.342 71.420 83.298 95.023 106.629 118.136 129.561

0.010 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 63.691 76.154 88.379 100.425 112.329 124.116 135.807

0.005 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169

0.025 5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 59.342 71.420 83.298 95.023 106.629 118.136 129.561

0.001 10.828 13.816 16.266 18.467 20.515 22.458 24.322 26.124 27.877 29.588 31.264 32.909 34.528 36.123 37.697 39.252 40.790 42.312 43.820 45.315 46.797 48.268 49.728 51.179 52.620 54.052 55.476 56.892 58.301 59.703 73.402 86.661 99.607 112.317 124.839 137.208 149.449

52

Contoh Perhitungan Variabel random Χ2 diketahui berdistribusi khi‐kuadrat dengan derajat kebebasan v = 10. Nilai χ2α agar probabilitas di sebelah kanan α = 0,05? α

α χ 2α

v

P (Χ 2 > χα2 ) = 0,05

χα2 = 18,307 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

53

Hubungan Distribusi Gamma dan Khi‐kuadrat X ∼ gamma (α, β); α = v/2; v bulat; β = 2

X ∼ khi‐kuadrat (v)


54

Hubungan Distribusi Normal Baku, Khi‐Kuadrat dan Gamma Xi ∼ normal baku

n

Y = ∑ X i2 i =1

Y ∼ khi‐kuadrat (v); v = n Y ∼ gamma (α, β); α = n/2; β = 2 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

55

Distribusi Weibull (1) X ∼ Weibull (α, β) Fungsi distribusi probabilitas: ⎧⎪αβ −α x α −1e −( x β ) ; x > 0 f (x ) = ⎨ ⎪⎩0; x lainnya α

Parameter:

α, β > 0 Rataan: μX =

β ⎛1⎞ Γ⎜ ⎟ α ⎝α ⎠

Variansi:

β 2 ⎛⎜ ⎛ 2 ⎞ 1 ⎛ ⎛ 1 ⎞ ⎞ 2Γ⎜ ⎟ − ⎜ Γ⎜ ⎟ ⎟ σ = α ⎜⎝ ⎝ α ⎠ α ⎝ ⎝ α ⎠ ⎠ 2 X


2

⎞ ⎟ ⎟ ⎠

56

Distribusi Weibull (2) Fungsi distribusi probabilitas kumulatif:

⎧⎪1 − e −( x β ) ; x > 0 F (x ) = ⎨ ⎪⎩0; x lainnya α


57

Contoh Kurva Distribusi Weibull 1.60 1.40 1.20

α = 0,5; β = 1

1.00 0.80 0.60

α = 1; β = 1

0.40 0.20 0.00 0.00

α = 2; β = 1 0.50

1.00

1.50


2.00

2.50

3.00

3.50

4.00

58

Probabilitas dari Variabel Random Berdistribusi Weibull X ∼ Weibull (α, β)

P (x1 < X < x2 ) = ∫ αβ x2

−α

x1

x

α −1 − ( x β )α

e

dx


59

Hubungan Distribusi Weibull dan Eksponensial

X ∼ Weibull (α, β)

α = 1 X ∼ eksponensial (β)


60

Distribusi Student t X ∼ student t (v)

Parameter: v > 0


1 Γ((v + 1) 2) ⎛ x2 ⎞ ⎜⎜ 1 + ⎟⎟ f (x ) = v ⎠ πv Γ(v 2) ⎝ ‐ ∞ < x < ∞

− (v + 1 ) 2

;

v : derajat kebebasan (degree of freedom) Rataan:

μX = 0 Variansi:

σ X2 =

v ;v > 2 v −2


61

Contoh Kurva Distribusi Student t 0.45 0.40

v = 100

0.35

v = 5

0.30

v = 2

f(x)

0.25 0.20 0.15 0.10 0.05 0.00 ‐5.00

‐4.00

‐3.00

‐2.00

‐1.00

0.00

1.00

2.00

3.00

4.00

5.00

x


62

Probabilitas dari Variabel Random Berdistribusi Student t Simbol umum untuk variabel random student‐t Æ T

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

P (T > tα ) = ∫ f (t )dt = α tα

α 0

Sifat simetris :

t1−α = −tα

tα


63

ٛ

v1

v

0.400 0.300 0.325 0.727 2 0.289 0.617 3 0.277 0.584 4 0.271 0.569 5 0.267 0.559 6 0.265 0.553 7 0.263 0.549 8 0.262 0.546 9 0.261 0.543 10 0.260 0.542 11 0.260 0.540 12 0.259 0.539 13 0.259 0.538 α 14 0.258 0.537 15 0.258 0.536 16 0.258 0.535 17 0.257 0.534 18 0.257 0.534 19 0.257 0.533 20 0.257 0.533 21 0.257 0.532 22 0.256 0.532 23 0.256 0.532 24 0.256 0.531 25 0.256 0.531 26 0.256 0.531 27 0.256 0.531 28 0.256 0.530 29 0.256 0.530 30 0.256 0.530 40 0.255 0.529 50 0.255 0.528 60 0.254 0.527 70 0.254 0.527 80 0.254 0.526 90 0.254 0.526 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS 100 0.254 0.526 Suprayogi, 2006 0.253 0.524 ∞

α

0

t

Tabel nilai t untuk derajat kebebasan v dan α

0.200 1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854 0.851 0.849 0.848 0.847 0.846 0.846 0.845 0.842

0.100 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.299 1.296 1.294 1.292 1.291 1.290 1.282

0.050 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.676 1.671 1.667 1.664 1.662 1.660 1.645

α 0.025 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.009 2.000 1.994 1.990 1.987 1.984 1.960

0.010 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.403 2.390 2.381 2.374 2.368 2.364 2.326

0.005 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.678 2.660 2.648 2.639 2.632 2.626 2.576

0.025 0.001 12.706 318.309 4.303 22.327 3.182 10.215 2.776 7.173 2.571 5.893 2.447 5.208 2.365 4.785 2.306 4.501 2.262 4.297 2.228 4.144 2.201 4.025 2.179 3.930 2.160 3.852 2.145 3.787 2.131 3.733 2.120 3.686 2.110 3.646 2.101 3.610 2.093 3.579 2.086 3.552 2.080 3.527 2.074 3.505 2.069 3.485 2.064 3.467 2.060 3.450 2.056 3.435 2.052 3.421 2.048 3.408 2.045 3.396 2.042 3.385 2.021 3.307 2.009 3.261 2.000 3.232 1.994 3.211 1.990 3.195 1.98764 3.183 1.984 3.174 1.960 3.090

Contoh Perhitungan (1) Variabel random T diketahui berdistribusi t dengan derajat kebebasan v = 10. Nilai tα agar v probabilitas di sebelah kanan α = 0,05?

α

0

tα

α

P (T < tα ) = 0,05 tα = 1,812 DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

65

Contoh Perhitungan (2) Variabel random T diketahui berdistribusi t dengan derajat kebebasan v = 10. Nilai tα agar v probabilitas di sebelah kiri 0,05?

α

0

tα

α

t1−0 ,05 = −t 0 ,05 P (T < t1−α ) = 0,05 t1−α = −1,812


66

Hubungan Distribusi Normal Baku, Khi‐kuadrat dan Student t Z ∼ normal baku Y ∼ khi‐kuadrat (v)

Z T= Y v T ∼ student t (v) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

67

Hubungan Distribusi Student t dan Normal Baku X ∼ student t (v); v → ∞

Z ∼ normal baku


68

Distribusi F Parameter:

X ∼ distribusi F (v1, v2)

v1, v2 > 0 v1, v2 : derajat kebebasan (degree of freedom)

Fungsi distribusi probabilitas: ⎧ Γ((v + v ) 2) ⎛ v ⎞v 2 x (v 2 )−1 1 2 1 ⎜⎜ ⎟⎟ ⎪ (v +v ) 2 ; x > 0 ⎪ Γ(v1 2)Γ(v2 2) ⎝ v2 ⎠ (1 + v1 x v2 ) 1

1

1

2

⎪ ⎪ f (x ) = ⎨ ⎪ ⎪ ⎪ x lainnya ⎪⎩

Rataan:

μX =

v2 ; v2 > 0 v2 − 2

Variansi: 2v22 (v1 + v2 − 2) σ = ; v2 > 4 v1 (v2 − 4 )(v2 − 2)2 2 X


69

Contoh Kurva Distribusi F 0.80

v1 = 10, v2 = 10

0.70 0.60

v1 = 10, v2 = 5

f(x)

0.50 0.40

v1 = 10, v2 = 2

0.30 0.20 0.10 0.00 0.00

0.50

1.00

1.50

2.00


2.50 x

3.00

3.50

4.00

4.50

5.00

70

Probabilitas dari Variabel Random Berdistribusi F Simbol umum untuk variabel random F Æ F

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

P (F > fα ) = ∫ f (x )dx = α fα

α fα

0


71

Tabel nilai f untuk derajat kebebasan v dan α α = 0,10

v2 v1

1 1

39.863

2 49.500

3 53.593

4 55.833

5 57.240

6 58.204

7 58.906

8 59.439

9 59.858

10

11

12

13

14

15

16

17

18

19

20

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


72


v2

v1

1

2

3

4

5

6

7

8

9

10

v2

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


73


vv1 1

v2 1

2

3

4

5

6

7

8

9

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


10

74

Contoh Perhitungan Variabel random F memiliki distribusi F (v1 = 10, v2 = 10) Nilai fα sehingga P(F > fα) = 0,05 ?

fα = 2,978

v1

Tabel nilai f untuk α = 0,05

v2


75

Hubungan Distribusi F dengan Khi‐kuadrat Χ12 ∼ khi‐kuadrat (v1) Χ22 ∼ khi‐kuadrat (v2)

independen

Χ 12 v1 F= 2 Χ 2 v2 F ∼ distribusi F (v1, v2) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

76

Distribusi Beta X ∼ beta (α1, α2) Fungsi distribusi probabilitas:

⎧ x α1 −1 (1 − x )α 2 −1 ; 0 ≤ x ≤ 1 ⎪ ⎪ Β(α 1 ,α 2 ) f (x ) = ⎨ ⎪0; x lainnya ⎪⎩

Parameter:

α1, α2 > 0 Rataan:

μX =

α1 α1 + α2

Variansi:

σ X2 =


α 1α 2 (α 1 + α 2 ) (α 1 + α 2 + 1)2 2

77

Fungsi Beta Β(α 1 ,α 2 ) = ∫ t α1 −1 (1 − t )α 2 −1 dt 1

0

Β(α 1 ,α 2 ) = Β(α 2 ,α 1 ) Β(α 1 ,α 2 ) =

Γ(α 1 )Γ(α 2 ) Γ(α 1 + α 2 )


78

Contoh Kurva Distribusi Beta (1) 3.00

α1 = 5; α2 = 1,5

α1 = 1,5; α2 = 5

2.50

f(x)

2.00

α1 = 3; α2 = 1,5

α1 = 1,5; α2 = 3

1.50 1.00 0.50 0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

x


79

Contoh Kurva Distribusi Beta (2) α1 = 10; α2 = 10

4.00 3.50 3.00

α1 = 5; α2 = 5

f(x)

2.50 2.00

α1 = 2; α2 = 2

1.50 1.00

α1 = 1; α2 = 1

0.50 0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

x DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

80

Contoh Kurva Distribusi Beta (3) 3.00

α1 = 2; α2 = 0,8

α1 = 0,8; α2 = 2

2.50

f(x)

2.00

α1 = 1; α2 = 2

1.50

α1 = 2; α2 = 1

1.00 0.50 0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

x


81

Contoh Kurva Distribusi Beta (4) 3.00 2.50

f(x)

2.00 1.50

α1 = 0,5; α2 = 0.5

1.00 0.50 0.00 0.00

α1 = 0,8; α2 = 0,2 0.10

0.20

0.30

α1 = 0,2; α2 = 0,8 0.40

0.50

0.60

0.70

0.80

0.90

1.00

x DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

82

Hubungan Gamma dan Beta X1 ∼ gamma (α1, β) X2 ∼ gamma (α2, β)

X=

X1 X1 + X2

X ∼ beta (α1, α2) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

83

Hubungan Distribusi Beta dan Seragam Kontinyu X ∼ beta (α1, α2);α1 = 1, α2 = 1

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


84

Hubungan Distribusi Beta dan Segitiga X ∼ segitiga kiri (left triangular) α1 = 1, α2 = 2

X ∼ beta (α1, α2) α1 = 2, α2 = 1

X ∼ segitiga kanan (right triangular) DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006

85

Distribusi Probabilitas Kontinyu Teoritis

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