Distribusi Probabilitas Kontinyu Teoritis Suprayogi Dist. Prob. Teoritis Kontinyu () 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 () Distribusi eksponensial (eponential distribution) Distribusi khi kuadrat (chisquare distribution) Distribusi Weibull (Weibull distribution) Distribusi student t (Student t distribution) Distribusi F (F distribution) Distribusi beta (beta distribution) 3 Distribusi Seragam Kontinyu () X seragam kontinyu (a, b) Fungsi distribusi probabilitas: f ( ) ; a < < b b a 0; lainnya Parameter: a, b bilangan riil (b > a) a : batas bawah b : batas atas Rataan: μ X Variansi: σ a + b ( b ) a X 4
Distribusi Seragam Kontinyu () Fungsi distribusi probabilitas kumulatif: F ( ) 0; < a a ; a < b a ; > b < b 5 Contoh Kurva Distribusi Seragam Kontinyu 0.5 0.0 0.5 a 0, b 0 f() 0.0 0.05 0.00 5.00 0.00 5.00 0.00 5.00 6
Probabilitas dari Variabel Random Seragam Kontinyu P ( < X < ) f ( ) d b a a b 7 Contoh Perhitungan Diameter komponen yang dinyatakan dengan variabel random X diketahui berdistribusi seragam kontinyu dengan batas bawah 0 mm dan batas atas 0 mm. Probabilitas bahwa diameter komponen kurang dari 5 mm? P ( X 5) 5 0 0 0 5 0 0 0,5 < d 8
Distribusi Segitiga X segitiga (a, b, c) Fungsi distribusi probabilitas: f ( ) ( a) ( c a)( b a) ( c ) ( c a)( c b) 0; lainnya ; a b ; b c Parameter: a, b, c bilangan ril (a < b < c) Rataan: a + b + c μ 3 Variansi: a + b + c ab ac bc σ 8 9 Distribusi Segitiga () Fungsi probabilitas kumulatif: F ( ) 0; < a ; > c ( a) ( c a)( b a) ( c ) ( c a)( c b) ; ; a < < b b < < c 0
Contoh Kurva Distribusi Segitiga 0.4 0. 0.0 a 5, b 0, c 0 f() 0.08 0.06 0.04 0.0 0.00 0.00 5.00 0.00 5.00 0.00 5.00 Probabilitas dari Variabel Random Segitiga f() P ( < X < ) ( a) ( c a)( b a) d a b c
Probabilitas dari Variabel Random Segitiga f() P ( < X < ) ( c ) ( c a)( c b) d a b c 3 Probabilitas dari Variabel Random Segitiga f() P ( < X < ) b ( a) ( c a)( b a) d + b ( c ) ( c a)( c b) d a b c 4
Contoh Perhitungan Misal variabel random X menyatakan waktu perjalanan yang diketahui berdistribusi segitiga dengan nilai optimistik a 5 menit, nilai pesimistik c 0 menit, dan most likely b 0 menit. Probabilitas bahwa waktu perjalanan antara 8 menit dan menit? P ( 8 < X < ) 0 8 0 ( 5) ( 0 ) d + ( 0 5)( 0 5) 0 ( 0 5)( 0 0) ( 5) ( 0 ) d + 8 75 0,33+ 0,400 0,4533 0 50 d d 5 8 0 5 5 Distribusi Segitiga Kanan dan Distribusi Segitiga Kiri X segitiga kiri (left triangular) X segitiga (a, b, c) b a f ( ) ( c ) ( c a) ; a < < 0; lainnya c b c X segitiga kanan (right triangular) f ( ) ( a) ( c a) ; a < < 0; lainnya c 6
Contoh Kurva Distribusi Segitiga Kiri 0.4 0. 0.0 a 5, b 5, c 0 0.08 f() 0.06 0.04 0.0 0.00 0.00 5.00 0.00 5.00 0.00 5.00 7 Contoh Kurva Distribusi Segitiga Kanan 0.4 0. 0.0 a 5, b 0, c 0 0.08 f() 0.06 0.04 0.0 0.00 0.00 5.00 0.00 5.00 0.00 5.00 8
Distribusi Normal X normal (μ, σ ) Parameter: Fungsi distribusi probabilitas: μ bilangan ril; σ > 0 f( ) e σ π μ σ ; < < μ :rataan σ : variansi Rataan: μ μ X Variansi: σ X σ 9 Contoh Kurva Distribusi Normal 0.45 0.40 0.35 0.30 μ 0, σ f() 0.5 0.0 0.5 0.0 μ 0, σ 4 0.05 0.00 0.00.00 4.00 6.00 8.00 0.00.00 4.00 6.00 8.00 0
Distribusi Normal Baku X normal (μ, σ ) Z X μ σ Z normal baku (standard normal) Fungsi distribusi probabilitas: f( z) e π z ; < z < Rataan: μ Z Variansi: σ Z 0 Kurva Distribusi Normal Baku 0.45 0.40 0.35 0.30 μ Z 0; σ Z f(z) 0.5 0.0 0.5 0.0 0.05 0.00 5.00 4.00 3.00.00.00 0.00.00.00 3.00 4.00 5.00 z
Probabilitas dari Variabel Random Berdistribusi Normal P ( < X < ) μ P < Z < σ μ σ P Z < μ σ P Z < μ σ f ( ) σ f () z μ z z 0 z 3 TabelDistribusiNormal Baku 0.00 0.0 0.0 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 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000-3.70 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000-3.60 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000-3.50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000-3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.000-3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003-3.0 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005-3.0 0.000 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007-3.00 0.003 0.003 0.003 0.00 0.00 0.00 0.00 0.00 0.000 0.000 -.90 0.009 0.008 0.008 0.007 0.006 0.006 0.005 0.005 0.004 0.004 -.80 0.006 0.005 0.004 0.003 0.003 0.00 0.00 0.00 0.000 0.009 -.70 0.0035 0.0034 0.0033 0.003 0.003 0.0030 0.009 0.008 0.007 0.006 -.60 0.0047 0.0045 0.0044 0.0043 0.004 0.0040 0.0039 0.0038 0.0037 0.0036 -.50 0.006 0.0060 0.0059 0.0057 0.0055 0.0054 0.005 0.005 0.0049 0.0048 -.40 0.008 0.0080 0.0078 0.0075 0.0073 0.007 0.0069 0.0068 0.0066 0.0064 -.30 0.007 0.004 0.00 0.0099 0.0096 0.0094 0.009 0.0089 0.0087 0.0084 -.0 0.039 0.036 0.03 0.09 0.05 0.0 0.09 0.06 0.03 0.00 -.0 0.079 0.074 0.070 0.066 0.06 0.058 0.054 0.050 0.046 0.043 -.00 0.08 0.0 0.07 0.0 0.007 0.00 0.097 0.09 0.088 0.083 -.90 0.087 0.08 0.074 0.068 0.06 0.056 0.050 0.044 0.039 0.033 -.80 0.0359 0.035 0.0344 0.0336 0.039 0.03 0.034 0.0307 0.030 0.094 -.70 0.0446 0.0436 0.047 0.048 0.0409 0.040 0.039 0.0384 0.0375 0.0367 -.60 0.0548 0.0537 0.056 0.056 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -.50 0.0668 0.0655 0.0643 0.0630 0.068 0.0606 0.0594 0.058 0.057 0.0559 -.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.07 0.0708 0.0694 0.068 -.30 0.0968 0.095 0.0934 0.098 0.090 0.0885 0.0869 0.0853 0.0838 0.083 -.0 0.5 0.3 0. 0.093 0.075 0.056 0.038 0.00 0.003 0.0985 -.0 0.357 0.335 0.34 0.9 0.7 0.5 0.30 0.0 0.90 0.70 -.00 0.587 0.56 0.539 0.55 0.49 0.469 0.446 0.43 0.40 0.379-0.90 0.84 0.84 0.788 0.76 0.736 0.7 0.685 0.660 0.635 0.6-0.80 0.9 0.090 0.06 0.033 0.005 0.977 0.949 0.9 0.894 0.867-0.70 0.40 0.389 0.358 0.37 0.96 0.66 0.36 0.06 0.77 0.48-0.60 0.743 0.709 0.676 0.643 0.6 0.578 0.546 0.54 0.483 0.45-0.50 0.3085 0.3050 0.305 0.98 0.946 0.9 0.877 0.843 0.80 0.776-0.40 0.3446 0.3409 0.337 0.3336 0.3300 0.364 0.38 0.39 0.356 0.3-0.30 0.38 0.3783 0.3745 0.3707 0.3669 0.363 0.3594 0.3557 0.350 0.3483-0.0 0.407 0.468 0.49 0.4090 0.405 0.403 0.3974 0.3936 0.3897 0.3859-0.0 0.460 0.456 0.45 0.4483 0.4443 0.4404 0.4364 0.435 0.486 0.447-0.00 0.5000 0.4960 0.490 0.4880 0.4840 0.480 0.476 0.47 0.468 0.464 4
TabelDistribusiNormal Baku 0.00 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.00 0.5000 0.5040 0.5080 0.50 0.560 0.599 0.539 0.579 0.539 0.5359 0.0 0.5398 0.5438 0.5478 0.557 0.5557 0.5596 0.5636 0.5675 0.574 0.5753 0.0 0.5793 0.583 0.587 0.590 0.5948 0.5987 0.606 0.6064 0.603 0.64 0.30 0.679 0.67 0.655 0.693 0.633 0.6368 0.6406 0.6443 0.6480 0.657 0.40 0.6554 0.659 0.668 0.6664 0.6700 0.6736 0.677 0.6808 0.6844 0.6879 0.50 0.695 0.6950 0.6985 0.709 0.7054 0.7088 0.73 0.757 0.790 0.74 0.60 0.757 0.79 0.734 0.7357 0.7389 0.74 0.7454 0.7486 0.757 0.7549 0.70 0.7580 0.76 0.764 0.7673 0.7704 0.7734 0.7764 0.7794 0.783 0.785 0.80 0.788 0.790 0.7939 0.7967 0.7995 0.803 0.805 0.8078 0.806 0.833 0.90 0.859 0.886 0.8 0.838 0.864 0.889 0.835 0.8340 0.8365 0.8389.00 0.843 0.8438 0.846 0.8485 0.8508 0.853 0.8554 0.8577 0.8599 0.86.0 0.8643 0.8665 0.8686 0.8708 0.879 0.8749 0.8770 0.8790 0.880 0.8830.0 0.8849 0.8869 0.8888 0.8907 0.895 0.8944 0.896 0.8980 0.8997 0.905.30 0.903 0.9049 0.9066 0.908 0.9099 0.95 0.93 0.947 0.96 0.977.40 0.99 0.907 0.9 0.936 0.95 0.965 0.979 0.99 0.9306 0.939.50 0.933 0.9345 0.9357 0.9370 0.938 0.9394 0.9406 0.948 0.949 0.944.60 0.945 0.9463 0.9474 0.9484 0.9495 0.9505 0.955 0.955 0.9535 0.9545.70 0.9554 0.9564 0.9573 0.958 0.959 0.9599 0.9608 0.966 0.965 0.9633.80 0.964 0.9649 0.9656 0.9664 0.967 0.9678 0.9686 0.9693 0.9699 0.9706.90 0.973 0.979 0.976 0.973 0.9738 0.9744 0.9750 0.9756 0.976 0.9767.00 0.977 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.98 0.987.0 0.98 0.986 0.9830 0.9834 0.9838 0.984 0.9846 0.9850 0.9854 0.9857.0 0.986 0.9864 0.9868 0.987 0.9875 0.9878 0.988 0.9884 0.9887 0.9890.30 0.9893 0.9896 0.9898 0.990 0.9904 0.9906 0.9909 0.99 0.993 0.996.40 0.998 0.990 0.99 0.995 0.997 0.999 0.993 0.993 0.9934 0.9936.50 0.9938 0.9940 0.994 0.9943 0.9945 0.9946 0.9948 0.9949 0.995 0.995.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.996 0.996 0.9963 0.9964.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.997 0.997 0.9973 0.9974.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.998.90 0.998 0.998 0.998 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.0 0.9990 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.9993 0.9993 3.0 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.50 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 3.60 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.70 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.80 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 3.90.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 4.00.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 5 Contoh Perhitungan () Tinggi badan yang dinyatakan dengan variabel random X diketahui berdistribusi normal dengan rataan μ 60 cm dan variansi σ 6 cm. Probabilitas bahwa tinggi badan antara 50 cm dan P( 5065 < X < 65 cm? ) 50 60 65 60 P < Z < 4 4 P P (,50 < Z <,5) ( Z <,5) P( Z <,50) 0,8944 0,006 0,886 σ X 6 50 65 μ X 60 6
Contoh Perhitungan () Nilai ujian mahasiswa yang diasumsikan memiliki distribusi normal (rataan μ 80, variansi σ 00). Jika mahasiswa yang lulus diinginkan sebesar 99 persen, batas nilai kelulusan? Misal variabel random X nilai ujian mahasiswa P ( X > ) 0,99 P( X < ) 80 P Z < 0,0 0 80,33 0 80 0 56,7 (,33) 0,0 μ X 80 σ X 00 7 Hampiran Distribusi Normal terhadap Distribusi Binomial X binomial (n, p); n, p 0,5 rataan μ np; variansi σ np( p) Z X np np ( p) Z normal baku 8
Contoh Perhitungan Probabilitas bahwa seseorang pada suatu daerah terinfeksi virus demam berdarah adalah p 0,4. Jika sebanyak n 00 orang dipilih secara random dari daerah tersebut, probabilitas bahwa terdapat kurang dari 30 orang terinfeksi? Misal variabel random X banyaknya orang yang terinfeksi μ σ X X np np ( 00)( 0,4) 40 ( p) ( 00)( 0,4)( 0,4) 4 P 30 40 4 ( X < 30) P Z < P( Z <,4) 0, 06 9 Distribusi Lognormal X lognormal (μ, σ ) Fungsi distribusi probabilitas: e f( ) σ π 0,lainnya ln μ σ ; > 0 Parameter: μ bilangan ril; σ > 0 Rataan: μ X e Variansi: σ X e μ + σ μ+ σ σ ( e ) 30
Contoh Kurva Distribusi Lognormal 0.45 0.40 0.35 f() 0.30 0.5 0.0 0.5 0.0 0.05 μ 0,5; σ μ ; σ 0.00 0.00.00 4.00 6.00 8.00 0.00.00 4.00 6.00 3 Hubungan Distribusi Normal dengan Lognormal X ~ normal (μ, σ ) X lny X Y e Y ~ lognormal (μ, σ ) 3
Probabilitas dari Variabel Random Berdistribusi Lognormal X Lognormal(μ, σ ) P ( X < ) P( ln( X) < ln( ) ) ln P Z < ( ) σ μ 33 Contoh Soal Waktu perbaikan suatu mesin diketahui memiliki distribusi lognormal (μ, σ ). Probabilitas bahwa waktu perbaikan mesin lebih dari 0 menit? Misal X variabel random waktu perbaikan P ( X > 0) P( ln( X) > ln( 0) ) ln( 0) μ P Z < σ ln( 0) P Z < P( Z <,00) 0,843 0,587 34
Distribusi Gamma X gamma (, β) Fungsi distribusi probabilitas: f ( ) β Γ( ) 0; lainnya e β ; > 0 Parameter:, β > 0 Rataan: μ β X Variansi: σ X β 35 Fungsi Gamma Γ Γ ( ) e 0 d ( ) ( ) Γ( ) Γ ( n) ( n )! Γ π 36
Contoh Kurva Distribusi Gamma.80.60 f().40.0.00 0.80 0.60 0,5; β ; β 0.40 0.0 ; β 0.00 0.00.00.00 3.00 4.00 5.00 6.00 37 Distribusi Erlang X Erlang (n, β) Fungsi distribusi probabilitas: f ( ) n β ( n )! 0; lainnya n β e ; > 0 Parameter: n bulat > 0, β > 0 Rataan: μx nβ Variansi: σ X nβ 38
Contoh Kurva Distribusi Erlang.00 0.90 0.80 0.70 n ; β 0.60 0.50 f() 0.40 0.30 0.0 n ; β n 5; β 0.0 0.00 0.00 0.0.00.00 3.00 4.00 5.00 6.00 39 Hubungan Distribusi Erlang dan Gamma X gamma (, β); n bulat X Erlang (n, β) 40
Distribusi Eksponensial () f X eksponensial (β) Fungsi distribusi probabilitas: ( ) β e ; > 0 β 0; lainnya Parameter: β > 0 Rataan: μ β X Variansi: β : rata rata σ X β 4 Distribusi Eksponensial () Fungsi distribusi probabilitas kumulatif: F ( ) β e ; > 0; lainnya 0 4
Contoh Kurva Distribusi Eksponensial.00 0.90 0.80 0.70 0.60 β f() 0.50 0.40 0.30 0.0 0.0 β 4 β 0.00 0.00.00.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 0.00 43 Probabilitas dari Variabel Random Berdistribusi Eksponensial β X eksponensial (β) P ( < X < ) e β β d 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? P ( X > 8) 8 6 e 6 8 e 0 6 0,636 d 6 d 45 Hubungan Distribusi Erlang dan Eksponensial X i eksponensial (β) X i saling independen Y n i X i Y Erlang (n, β) 46
Hubungan Distribusi Gamma, Erlang dan Eksponensial X gamma (, β) n bulat X eksponensial (β) n X Erlang (n, β) 47 Hubungan Distribusi Eksponensial dan Poisson X Poisson (λt) P(tidak ada kejadian sebelum t) P(X 0) 0 e 0 ( t) λt λ t λ e 0! P(tidak adakejadian sebelum t) P(kejadian pertama terjadi pada atau setelah saat t) λ t t eksponensial(β /λ); t e λ λt e F( t) λt F() t e df() t f () t dt f( t) e β t β λe λt ; β λ 48
Distribusi Khi kuadrat X khi kuadrat (v) Fungsi distribusi probabilitas: f ( ) v v v Γ 0; lainnya > 0 e ; Parameter: v bilangan bulat > 0 v : derajat kebebasan (degree of freedom) Rataan: μ X v Variansi: σ X v 49 Contoh Kurva Distribusi Khi kuadrat 0.8 0.6 0.4 0. 0.0 v 5 v 0 0.08 0.06 v 5 0.04 0.0 0.00 0.00 5.00 0.00 5.00 0.00 5.00 30.00 50
Probabilitas dari Variabel Random Berdistribusi Khi kuadrat Simbol umum untuk variabel random khi kuadrat Χ Χ khi kuadrat (v) dengan fungsi distribusi probabilitas f() P ( > χ ) f ( ) d Χ χ χ 5 Tabel nilai χ untuk derajat kebebasan v dan v v 0.999 0.975 0.995 0.990 0.975 0.950 0.900 0.500 0.00 0.050 0.05 0.00 0.005 0.05 0.00 0.000 0.00 0.000 0.000 0.00 0.004 0.06 0.455.706 3.84 5.04 6.635 7.879 5.04 0.88 0.00 0.05 0.00 0.00 0.05 0.03 0..386 4.605 5.99 7.378 9.0 0.597 7.378 3.86 3 0.04 0.6 0.07 0.5 0.6 0.35 0.584.366 6.5 7.85 9.348.345.838 9.348 6.66 4 0.09 0.484 0.07 0.97 0.484 0.7.064 3.357 7.779 9.488.43 3.77 4.860.43 8.467 5 0.0 0.83 0.4 0.554 0.83.45.60 4.35 9.36.070.833 5.086 6.750.833 0.55 6 0.38.37 0.676 0.87.37.635.04 5.348 0.645.59 4.449 6.8 8.548 4.449.458 7 0.598.690 0.989.39.690.67.833 6.346.07 4.067 6.03 8.475 0.78 6.03 4.3 8 0.857.80.344.646.80.733 3.490 7.344 3.36 5.507 7.535 0.090.955 7.535 6.4 9.5.700.735.088.700 3.35 4.68 8.343 4.684 6.99 9.03.666 3.589 9.03 7.877 0.479 3.47.56.558 3.47 3.940 4.865 9.34 5.987 8.307 0.483 3.09 5.88 0.483 9.588.834 3.86.603 3.053 3.86 4.575 5.578 0.34 7.75 9.675.90 4.75 6.757.90 3.64.4 4.404 3.074 3.57 4.404 5.6 6.304.340 8.549.06 3.337 6.7 8.300 3.337 3.909 3.67 5.009 3.565 4.07 5.009 5.89 7.04.340 9.8.36 4.736 7.688 9.89 4.736 34.58 4 3.04 5.69 4.075 4.660 5.69 6.57 7.790 3.339.064 3.685 6.9 9.4 3.39 6.9 36.3 5 3.483 6.6 4.60 5.9 6.6 7.6 8.547 4.339.307 4.996 7.488 30.578 3.80 7.488 37.697 6 3.94 6.908 5.4 5.8 6.908 7.96 9.3 5.338 3.54 6.96 8.845 3.000 34.67 8.845 39.5 7 4.46 7.564 5.697 6.408 7.564 8.67 0.085 6.338 4.769 7.587 30.9 33.409 35.78 30.9 40.790 8 4.905 8.3 6.65 7.05 8.3 9.390 0.865 7.338 5.989 8.869 3.56 34.805 37.56 3.56 4.3 9 5.407 8.907 6.844 7.633 8.907 0.7.65 8.338 7.04 30.44 3.85 36.9 38.58 3.85 43.80 0 5.9 9.59 7.434 8.60 9.59 0.85.443 9.337 8.4 3.40 34.70 37.566 39.997 34.70 45.35 6.447 0.83 8.034 8.897 0.83.59 3.40 0.337 9.65 3.67 35.479 38.93 4.40 35.479 46.797 6.983 0.98 8.643 9.54 0.98.338 4.04.337 30.83 33.94 36.78 40.89 4.796 36.78 48.68 3 7.59.689 9.60 0.96.689 3.09 4.848.337 3.007 35.7 38.076 4.638 44.8 38.076 49.78 4 8.085.40 9.886 0.856.40 3.848 5.659 3.337 33.96 36.45 39.364 4.980 45.559 39.364 5.79 5 8.649 3.0 0.50.54 3.0 4.6 6.473 4.337 34.38 37.65 40.646 44.34 46.98 40.646 5.60 6 9. 3.844.60.98 3.844 5.379 7.9 5.336 35.563 38.885 4.93 45.64 48.90 4.93 54.05 7 9.803 4.573.808.879 4.573 6.5 8.4 6.336 36.74 40.3 43.95 46.963 49.645 43.95 55.476 8 0.39 5.308.46 3.565 5.308 6.98 8.939 7.336 37.96 4.337 44.46 48.78 50.993 44.46 56.89 9 0.986 6.047 3. 4.56 6.047 7.708 9.768 8.336 39.087 4.557 45.7 49.588 5.336 45.7 58.30 30.588 6.79 3.787 4.953 6.79 8.493 0.599 9.336 40.56 43.773 46.979 50.89 53.67 46.979 59.703 40 7.96 4.433 0.707.64 4.433 6.509 9.05 39.335 5.805 55.758 59.34 63.69 66.766 59.34 73.40 50 4.674 3.357 7.99 9.707 3.357 34.764 37.689 49.335 63.67 67.505 7.40 76.54 79.490 7.40 86.66 60 3.738 40.48 35.534 37.485 40.48 43.88 46.459 59.335 74.397 79.08 83.98 88.379 9.95 83.98 99.607 70 39.036 48.758 43.75 45.44 48.758 5.739 55.39 69.334 85.57 90.53 95.03 00.45 04.5 95.03.37 80 46.50 57.53 5.7 53.540 57.53 60.39 64.78 79.334 96.578 0.879 06.69.39 6.3 06.6954.839 90 54.55 65.647 59.96 6.754 65.647 69.6 73.9 89.334 07.565 3.45 8.36 4.6 8.99 8.36 37.08 00 6.98 74. 67.38 70.065 74. 77.99 8.358 99.334 8.498 4.34 9.56 35.807 40.69 9.56 49.449 Π χ
Contoh Perhitungan Variabel random Χ diketahui berdistribusi khi kuadrat dengan derajat kebebasan v 0. Nilai χ agar probabilitas di sebelah kanan 0,05? v χ P ( Χ > χ ) χ 8,307 0,05 53 Hubungan Distribusi Gamma dan Khi kuadrat X gamma (, β); v/; v bulat; β X khi kuadrat (v) 54
Hubungan Distribusi Normal Baku, Khi Kuadrat dan Gamma X i normal baku Y n i X i Y khi kuadrat (v); v n Y gamma (, β); n/; β 55 Distribusi Weibull () X Weibull (, β) Fungsi distribusi probabilitas: f ( ) ( β ) > β e 0; lainnya ; 0 Parameter:, β > 0 Rataan: β μx Γ Variansi: β Γ Γ σ X 56
Distribusi Weibull () Fungsi distribusi probabilitas kumulatif: F ( ) ( β ) > e ; 0; lainnya 0 57 Contoh Kurva Distribusi Weibull.60.40.0 0,5; β.00 0.80 0.60 0.40 0.0 ; β ; β 0.00 0.00 0.50.00.50.00.50 3.00 3.50 4.00 58
Probabilitas dari Variabel Random Berdistribusi Weibull X Weibull (, β) P ( β ) ( < X < ) β e d 59 Hubungan Distribusi Weibull dan Eksponensial X Weibull (, β) X eksponensial (β) 60
Distribusi Student t X student t (v) Fungsi distribusi probabilitas: f ( ) πv < < Γ (( v + ) ) Γ( v ) + v ( v+ ) ; Parameter: v > 0 v : derajat kebebasan (degree of freedom) Rataan: μ X 0 Variansi: σ X v ; v v 6 > Contoh Kurva Distribusi Student t 0.45 f() 0.40 0.35 0.30 0.5 0.0 v 00 v 5 v 0.5 0.0 0.05 0.00 5.00 4.00 3.00.00.00 0.00.00.00 3.00 4.00 5.00 6
Probabilitas dari Variabel Random Berdistribusi Student t Simbol umum untuk variabel random student t T T student t (v) dengan fungsi distribusi probabilitas f(t) P ( T > t ) f () t dt t 0 t Sifat simetris : t t 63 0 t Tabel nilai t untuk derajat kebebasan v dan vv 0.400 0.300 0.00 0.00 0.050 0.05 0.00 0.005 0.05 0.00 0.35 0.77.376 3.078 6.34.706 3.8 63.657.706 38.309 0.89 0.67.06.886.90 4.303 6.965 9.95 4.303.37 3 0.77 0.584 0.978.638.353 3.8 4.54 5.84 3.8 0.5 4 0.7 0.569 0.94.533.3.776 3.747 4.604.776 7.73 5 0.67 0.559 0.90.476.05.57 3.365 4.03.57 5.893 6 0.65 0.553 0.906.440.943.447 3.43 3.707.447 5.08 7 0.63 0.549 0.896.45.895.365.998 3.499.365 4.785 8 0.6 0.546 0.889.397.860.306.896 3.355.306 4.50 9 0.6 0.543 0.883.383.833.6.8 3.50.6 4.97 0 0.60 0.54 0.879.37.8.8.764 3.69.8 4.44 0.60 0.540 0.876.363.796.0.78 3.06.0 4.05 0.59 0.539 0.873.356.78.79.68 3.055.79 3.930 3 0.59 0.538 0.870.350.77.60.650 3.0.60 3.85 4 0.58 0.537 0.868.345.76.45.64.977.45 3.787 5 0.58 0.536 0.866.34.753.3.60.947.3 3.733 6 0.58 0.535 0.865.337.746.0.583.9.0 3.686 7 0.57 0.534 0.863.333.740.0.567.898.0 3.646 8 0.57 0.534 0.86.330.734.0.55.878.0 3.60 9 0.57 0.533 0.86.38.79.093.539.86.093 3.579 0 0.57 0.533 0.860.35.75.086.58.845.086 3.55 0.57 0.53 0.859.33.7.080.58.83.080 3.57 0.56 0.53 0.858.3.77.074.508.89.074 3.505 3 0.56 0.53 0.858.39.74.069.500.807.069 3.485 4 0.56 0.53 0.857.38.7.064.49.797.064 3.467 5 0.56 0.53 0.856.36.708.060.485.787.060 3.450 6 0.56 0.53 0.856.35.706.056.479.779.056 3.435 7 0.56 0.53 0.855.34.703.05.473.77.05 3.4 8 0.56 0.530 0.855.33.70.048.467.763.048 3.408 9 0.56 0.530 0.854.3.699.045.46.756.045 3.396 30 0.56 0.530 0.854.30.697.04.457.750.04 3.385 40 0.55 0.59 0.85.303.684.0.43.704.0 3.307 50 0.55 0.58 0.849.99.676.009.403.678.009 3.6 60 0.54 0.57 0.848.96.67.000.390.660.000 3.3 70 0.54 0.57 0.847.94.667.994.38.648.994 3. 80 0.54 0.56 0.846.9.664.990.374.639.990 3.95 90 0.54 0.56 0.846.9.66.987.368.63.987 3.83 00 0.54 0.56 0.845.90.660.984.364.66.984 3.74 0.53 0.54 0.84.8.645.960.36.576.960 3.090 64
Contoh Perhitungan () Variabel random T diketahui berdistribusi t dengan derajat kebebasan v 0. Nilai t agar v probabilitas di sebelah kanan 0,05? 0 t P t ( T < t ),8 0,05 65 Contoh Perhitungan () Variabel random T diketahui berdistribusi t dengan derajat kebebasan v 0. Nilai t agar v probabilitas di sebelah kiri 0,05? 0 t t 0,05 t0,05 P t ( T < t ),8 0,05 66
Hubungan Distribusi Normal Baku, Khi kuadrat dan Student t Z normal baku Y khi kuadrat (v) T Z Y v T student t (v) 67 Hubungan Distribusi Student t dan Normal Baku X student t (v); v Z normal baku 68
Distribusi F f X distribusi F (v, v ) Fungsi distribusi probabilitas: ( ) (( v + v ) ) ( v ) Γ( v ) Γ Γ lainnya v v v ( v ) ( ) ( v + + ) v v v ; > 0 μ X σ X Parameter: v, v > 0 v, v : derajat kebebasan (degree of freedom) Rataan: v ; v > 0 v Variansi: v ( v + v ) ( v 4)( v ) ; v v > 4 69 Contoh Kurva Distribusi F 0.80 0.70 0.60 0.50 v 0, v 0 v 0, v 5 f() 0.40 0.30 v 0, v 0.0 0.0 0.00 0.00 0.50.00.50.00.50 3.00 3.50 4.00 4.50 5.00 70
Probabilitas dari Variabel Random Berdistribusi F Simbol umum untuk variabel random F F F distribusi F (v, v ) dengan fungsi distribusi probabilitas f() P ( F > f ) f ( ) d f 0 f 7 Tabel nilai f untuk derajat kebebasan v dan 0,0 v 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 39.863 49.500 53.593 55.833 57.40 58.04 58.906 59.439 59.858 60.95 60.473 60.705 60.903 6.073 6.0 6.350 6.464 6.566 6.658 6.740 8.56 9.000 9.6 9.43 9.93 9.36 9.349 9.367 9.38 9.39 9.40 9.408 9.45 9.40 9.45 9.49 9.433 9.436 9.439 9.44 3 5.538 5.46 5.39 5.343 5.309 5.85 5.66 5.5 5.40 5.30 5. 5.6 5.0 5.05 5.00 5.96 5.93 5.90 5.87 5.84 4 4.545 4.35 4.9 4.07 4.05 4.00 3.979 3.955 3.936 3.90 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 3.69 3.50 3.453 3.405 3.368 3.339 3.36 3.97 3.8 3.68 3.57 3.47 3.38 3.30 3.3 3.7 3. 3.07 6 3.776 3.463 3.89 3.8 3.08 3.055 3.04.983.958.937.90.905.89.88.87.863.855.848.84.836 7 3.589 3.57 3.074.96.883.87.785.75.75.703.684.668.654.643.63.63.65.607.60.595 8 3.458 3.3.94.806.76.668.64.589.56.538.59.50.488.475.464.455.446.438.43.45 9 3.360 3.006.83.693.6.55.505.469.440.46.396.379.364.35.340.39.30.3.305.98 0 3.85.94.78.605.5.46.44.377.347.33.30.84.69.55.44.33.4.5.08.0 3.5.860.660.536.45.389.34.304.74.48.7.09.93.79.67.56.47.38.30.3 3.77.807.606.480.394.33.83.45.4.88.66.47.3.7.05.094.084.075.067.060 3 3.36.763.560.434.347.83.34.95.64.38.6.097.080.066.053.04.03.03.04.007 4 3.0.76.5.395.307.43.93.54..095.073.054.037.0.00.998.988.978.970.96 5 3.073.695.490.36.73.08.58.9.086.059.037.07.000.985.97.96.950.94.93.94 6 3.048.668.46.333.44.78.8.088.055.08.005.985.968.953.940.98.97.908.899.89 7 3.06.645.437.308.8.5.0.06.08.00.978.958.940.95.9.900.889.879.870.86 8 3.007.64.46.86.96.30.079.038.005.977.954.933.96.900.887.875.864.854.845.837 9.990.606.397.66.76.09.058.07.984.956.93.9.894.878.865.85.84.83.8.84 0.975.589.380.49.58.09.040.999.965.937.93.89.875.859.845.833.8.8.80.794 v 7
Tabel nilai f untuk derajat kebebasan v dan 0,05 v 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 v v 6.448 99.500 5.707 4.583 30.6 33.986 36.768 38.883 40.543 4.88 4.983 43.906 44.690 45.364 45.950 46.464 46.98 47.33 47.686 48.03 8.53 9.000 9.64 9.47 9.96 9.330 9.353 9.37 9.385 9.396 9.405 9.43 9.49 9.44 9.49 9.433 9.437 9.440 9.443 9.446 3 0.8 9.55 9.77 9.7 9.03 8.94 8.887 8.845 8.8 8.786 8.763 8.745 8.79 8.75 8.703 8.69 8.683 8.675 8.667 8.660 4 7.709 6.944 6.59 6.388 6.56 6.63 6.094 6.04 5.999 5.964 5.936 5.9 5.89 5.873 5.858 5.844 5.83 5.8 5.8 5.803 5 6.608 5.786 5.409 5.9 5.050 4.950 4.876 4.88 4.77 4.735 4.704 4.678 4.655 4.636 4.69 4.604 4.590 4.579 4.568 4.558 6 5.987 5.43 4.757 4.534 4.387 4.84 4.07 4.47 4.099 4.060 4.07 4.000 3.976 3.956 3.938 3.9 3.908 3.896 3.884 3.874 7 5.59 4.737 4.347 4.0 3.97 3.866 3.787 3.76 3.677 3.637 3.603 3.575 3.550 3.59 3.5 3.494 3.480 3.467 3.455 3.445 8 5.38 4.459 4.066 3.838 3.687 3.58 3.500 3.438 3.388 3.347 3.33 3.84 3.59 3.37 3.8 3.0 3.87 3.73 3.6 3.50 9 5.7 4.56 3.863 3.633 3.48 3.374 3.93 3.30 3.79 3.37 3.0 3.073 3.048 3.05 3.006.989.974.960.948.936 0 4.965 4.03 3.708 3.478 3.36 3.7 3.35 3.07 3.00.978.943.93.887.865.845.88.8.798.785.774 4.844 3.98 3.587 3.357 3.04 3.095 3.0.948.896.854.88.788.76.739.79.70.685.67.658.646 4.747 3.885 3.490 3.59 3.06.996.93.849.796.753.77.687.660.637.67.599.583.568.555.544 3 4.667 3.806 3.4 3.79 3.05.95.83.767.74.67.635.604.577.554.533.55.499.484.47.459 4 4.600 3.739 3.344 3..958.848.764.699.646.60.565.534.507.484.463.445.48.43.400.388 5 4.543 3.68 3.87 3.056.90.790.707.64.588.544.507.475.448.44.403.385.368.353.340.38 6 4.494 3.634 3.39 3.007.85.74.657.59.538.494.456.45.397.373.35.333.37.30.88.76 7 4.45 3.59 3.97.965.80.699.64.548.494.450.43.38.353.39.308.89.7.57.43.30 8 4.44 3.555 3.60.98.773.66.577.50.456.4.374.34.34.90.69.50.33.7.03.9 9 4.38 3.5 3.7.895.740.68.544.477.43.378.340.308.80.56.34.5.98.8.68.55 0 4.35 3.493 3.098.866.7.599.54.447.393.348.30.78.50.5.03.84.67.5.37.4 73 Tabel nilai f untuk derajat kebebasan v dan 0,0 v v 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 405. 4999.5 5403.4 564.6 5763.6 5859.0 598.4 598. 60.5 6055.8 6083.3 606.3 65.9 64.7 657.3 670. 68.4 69.5 600.6 608.7 98.503 99.000 99.66 99.49 99.99 99.333 99.356 99.374 99.388 99.399 99.408 99.46 99.4 99.48 99.433 99.437 99.440 99.444 99.447 99.449 3 34.6 30.87 9.457 8.70 8.37 7.9 7.67 7.489 7.345 7.9 7.33 7.05 6.983 6.94 6.87 6.87 6.787 6.75 6.79 6.690 4.98 8.000 6.694 5.977 5.5 5.07 4.976 4.799 4.659 4.546 4.45 4.374 4.307 4.49 4.98 4.54 4.5 4.080 4.048 4.00 5 6.58 3.74.060.39 0.967 0.67 0.456 0.89 0.58 0.05 9.963 9.888 9.85 9.770 9.7 9.680 9.643 9.60 9.580 9.553 6 3.745 0.95 9.780 9.48 8.746 8.466 8.60 8.0 7.976 7.874 7.790 7.78 7.657 7.605 7.559 7.59 7.483 7.45 7.4 7.396 7.46 9.547 8.45 7.847 7.460 7.9 6.993 6.840 6.79 6.60 6.538 6.469 6.40 6.359 6.34 6.75 6.40 6.09 6.8 6.55 8.59 8.649 7.59 7.006 6.63 6.37 6.78 6.09 5.9 5.84 5.734 5.667 5.609 5.559 5.55 5.477 5.44 5.4 5.384 5.359 9 0.56 8.0 6.99 6.4 6.057 5.80 5.63 5.467 5.35 5.57 5.78 5. 5.055 5.005 4.96 4.94 4.890 4.860 4.833 4.808 0 0.044 7.559 6.55 5.994 5.636 5.386 5.00 5.057 4.94 4.849 4.77 4.706 4.650 4.60 4.558 4.50 4.487 4.457 4.430 4.405 9.646 7.06 6.7 5.668 5.36 5.069 4.886 4.744 4.63 4.539 4.46 4.397 4.34 4.93 4.5 4.3 4.80 4.50 4.3 4.099 9.330 6.97 5.953 5.4 5.064 4.8 4.640 4.499 4.388 4.96 4.0 4.55 4.00 4.05 4.00 3.97 3.939 3.909 3.883 3.858 3 9.074 6.70 5.739 5.05 4.86 4.60 4.44 4.30 4.9 4.00 4.05 3.960 3.905 3.857 3.85 3.778 3.745 3.76 3.689 3.665 4 8.86 6.55 5.564 5.035 4.695 4.456 4.78 4.40 4.030 3.939 3.864 3.800 3.745 3.698 3.656 3.69 3.586 3.556 3.59 3.505 5 8.683 6.359 5.47 4.893 4.556 4.38 4.4 4.004 3.895 3.805 3.730 3.666 3.6 3.564 3.5 3.485 3.45 3.43 3.396 3.37 6 8.53 6.6 5.9 4.773 4.437 4.0 4.06 3.890 3.780 3.69 3.66 3.553 3.498 3.45 3.409 3.37 3.339 3.30 3.83 3.59 7 8.400 6. 5.85 4.669 4.336 4.0 3.97 3.79 3.68 3.593 3.59 3.455 3.40 3.353 3.3 3.75 3.4 3. 3.86 3.6 8 8.85 6.03 5.09 4.579 4.48 4.05 3.84 3.705 3.597 3.508 3.434 3.37 3.36 3.69 3.7 3.90 3.58 3.8 3.0 3.077 9 8.85 5.96 5.00 4.500 4.7 3.939 3.765 3.63 3.53 3.434 3.360 3.97 3.4 3.95 3.53 3.6 3.084 3.054 3.07 3.003 0 8.096 5.849 4.938 4.43 4.03 3.87 3.699 3.564 3.457 3.368 3.94 3.3 3.77 3.30 3.088 3.05 3.08.989.96.938 74
Contoh Perhitungan Variabel random F memiliki distribusi F (v 0, v 0) Nilai f sehingga P(F > f ) 0,05? f,978 v Tabel nilai f untuk 0,05 v 75 Hubungan Distribusi F dengan Khi kuadrat Χ khi kuadrat (v ) Χ khi kuadrat (v ) independen F Χ Χ v v F distribusi F (v, v ) 76
Distribusi Beta X beta (, ) Fungsi distribusi probabilitas: Parameter:, > 0 f ( ) ( ) (, ) Β 0; lainnya ; 0 Rataan: μx + Variansi: σ X ( + ) ( + + ) 77 Fungsi Beta Β Β ( ) t ( t), dt 0 (, ) Β( ), Β ( ), Γ Γ ( ) Γ( ) ( + ) 78
Contoh Kurva Distribusi Beta () 3.00,5; 5 5;,5.50.00,5; 3 3;,5 f().50.00 0.50 0.00 0.00 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90.00 79 Contoh Kurva Distribusi Beta () f() 4.00 3.50 3.00.50.00.50.00 0.50 0; 0 5; 5 ; ; 0.00 0.00 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90.00 80
Contoh Kurva Distribusi Beta (3) 3.00 0,8; ; 0,8.50 f().00.50 ; ;.00 0.50 0.00 0.00 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90.00 8 Contoh Kurva Distribusi Beta (4) 3.00.50.00 f().50.00 0,5; 0.5 0.50 0.00 0,8; 0, 0,; 0,8 0.00 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90.00 8
Hubungan Gamma dan Beta X gamma (, β) X gamma (, β) X X X + X X beta (, ) 83 Hubungan Distribusi Beta dan Seragam Kontinyu X beta (, );, X seragam kontinyu (a, b); a 0, b 84
Hubungan Distribusi Beta dan Segitiga X segitiga kiri (left triangular), X beta (, ), X segitiga kanan (right triangular) 85