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 ⎪⎩
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
6
Probabilitas dari Variabel Random Seragam Kontinyu P ( x1 < X < x 2 ) = ∫
x2
x1
1 dx b−a
f (x )
a
x1
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
11
Probabilitas dari Variabel Random Segitiga P ( x1 < X < x 2 ) = ∫
f(x)
x2
x1
a x1
x2 b
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
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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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
20
Distribusi Normal Baku X ∼ normal (μ, σ2)
Z=
X −μ
σ
Z ∼ normal baku (standard normal)
Rataan:
μZ = 0
Fungsi distribusi probabilitas:
1 f (z) = e 2π
⎛ 1 2⎞ ⎜− z ⎟ ⎝ 2 ⎠
Variansi:
σ Z2 = 1
; ‐∞ < z < ∞
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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 ⎝ ⎠
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
39
Hubungan Distribusi Erlang dan Gamma X ∼ gamma (α, β); α = n bulat
X ∼ Erlang (n, β)
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
40
Distribusi Eksponensial (1) X ∼ eksponensial (β) Parameter:
Fungsi distribusi probabilitas:
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 ⎩
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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 ∞
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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)
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
2
⎞ ⎟ ⎟ ⎠
56
Distribusi Weibull (2) Fungsi distribusi probabilitas kumulatif:
⎧⎪1 − e −( x β ) ; x > 0 F (x ) = ⎨ ⎪⎩0; x lainnya α
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
59
Hubungan Distribusi Weibull dan Eksponensial
X ∼ Weibull (α, β)
α = 1 X ∼ eksponensial (β)
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
60
Distribusi Student t X ∼ student t (v)
Parameter: v > 0
Fungsi distribusi probabilitas:
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
72
Tabel nilai f untuk derajat kebebasan v dan α α = 0,05
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
73
Tabel nilai f untuk derajat kebebasan v dan α α = 0,01
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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 =
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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 )
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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
DISTRIBUSI PROBABILITAS KONTINYU TEORITIS Suprayogi, 2006
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