Pertemuan-3
Farid
2025-04-11
Pendahuluan
Variabel Random adalah variabel yang nilainya ditentukan oleh hasil dari suatu percobaan acak. Variabel random dapat dibagi menjadi dua jenis: - Variabel Random Diskrit: Variabel yang hanya dapat mengambil nilai-nilai tertentu (biasanya bilangan bulat). Contoh: jumlah pelanggan yang datang ke restoran dalam sehari. - Variabel Random Kontinu: Variabel yang dapat mengambil nilai apa pun dalam suatu interval. Contoh: tinggi badan seseorang, pendapatan bulanan.
Dalam pemodelan statistika dan simulasi, variabel random digunakan untuk memodelkan fenomena acak. Simulasi variabel random membantu kita memahami distribusi data, menghitung probabilitas, dan membuat prediksi berdasarkan model yang telah dibuat.
Simulasi Sederhana: Variabel Random Uniform
Distribusi uniform adalah distribusi di mana semua nilai dalam interval tertentu memiliki probabilitas yang sama.
# Simulasi 1000 variabel random dari distribusi uniform
set.seed(123) # Set seed untuk reproducibility
n <- 1000
uniform_data <- runif(n, min = 0, max = 1)uniform_data## [1] 0.2875775201 0.7883051354 0.4089769218 0.8830174040 0.9404672843
## [6] 0.0455564994 0.5281054880 0.8924190444 0.5514350145 0.4566147353
## [11] 0.9568333453 0.4533341562 0.6775706355 0.5726334020 0.1029246827
## [16] 0.8998249704 0.2460877344 0.0420595335 0.3279207193 0.9545036491
## [21] 0.8895393161 0.6928034062 0.6405068138 0.9942697766 0.6557057991
## [26] 0.7085304682 0.5440660247 0.5941420204 0.2891597373 0.1471136473
## [31] 0.9630242325 0.9022990451 0.6907052784 0.7954674177 0.0246136845
## [36] 0.4777959711 0.7584595375 0.2164079358 0.3181810076 0.2316257854
## [41] 0.1428000224 0.4145463358 0.4137243263 0.3688454509 0.1524447477
## [46] 0.1388060634 0.2330340995 0.4659624503 0.2659726404 0.8578277153
## [51] 0.0458311667 0.4422000742 0.7989248456 0.1218992600 0.5609479838
## [56] 0.2065313896 0.1275316502 0.7533078643 0.8950453592 0.3744627759
## [61] 0.6651151946 0.0948406609 0.3839696378 0.2743836446 0.8146400389
## [66] 0.4485163414 0.8100643530 0.8123895095 0.7943423211 0.4398316876
## [71] 0.7544751586 0.6292211316 0.7101824014 0.0006247733 0.4753165741
## [76] 0.2201188852 0.3798165377 0.6127710033 0.3517979092 0.1111354243
## [81] 0.2436194727 0.6680555874 0.4176467797 0.7881958340 0.1028646443
## [86] 0.4348927415 0.9849569800 0.8930511144 0.8864690608 0.1750526503
## [91] 0.1306956916 0.6531019250 0.3435164723 0.6567581280 0.3203732425
## [96] 0.1876911193 0.7822943013 0.0935949867 0.4667790416 0.5115054599
## [101] 0.5999889593 0.3328235403 0.4886130337 0.9544738275 0.4829023972
## [106] 0.8903502221 0.9144381869 0.6087349823 0.4106897765 0.1470946909
## [111] 0.9352998033 0.3012289000 0.0607205715 0.9477269400 0.7205962734
## [116] 0.1422942956 0.5492846561 0.9540912386 0.5854833531 0.4045102817
## [121] 0.6478934793 0.3198206171 0.3077200109 0.2197676313 0.3694888658
## [126] 0.9842192035 0.1542023008 0.0910439999 0.1419069078 0.6900071015
## [131] 0.6192564834 0.8913941171 0.6729990926 0.7370777379 0.5211357258
## [136] 0.6598384497 0.8218054601 0.7862815517 0.9798219174 0.4394315362
## [141] 0.3117022021 0.4094749526 0.0104671118 0.1838495240 0.8427293189
## [146] 0.2311617821 0.2390999557 0.0766911653 0.2457236780 0.7321352055
## [151] 0.8474531651 0.4975272671 0.3879090298 0.2464489941 0.1110964613
## [156] 0.3899944352 0.5719353140 0.2168927628 0.4447680020 0.2179906687
## [161] 0.5022995633 0.3539045718 0.6499851588 0.3747139566 0.3554453808
## [166] 0.5336879455 0.7403343604 0.2211029378 0.4127461186 0.2656866868
## [171] 0.6299730535 0.1838284908 0.8636441114 0.7465680041 0.6682846497
## [176] 0.6180178733 0.3722380602 0.5298356859 0.8746823429 0.5817500998
## [181] 0.8397677648 0.3124481649 0.7082903222 0.2650178061 0.5943431940
## [186] 0.4812898005 0.2650327315 0.5645904348 0.9131882230 0.9018743895
## [191] 0.2741666215 0.3214827564 0.9856408844 0.6199933102 0.9373140892
## [196] 0.4665327023 0.4068325933 0.6592303242 0.1523466168 0.5728670582
## [201] 0.2387260268 0.9623589364 0.6013657260 0.5150297272 0.4025733422
## [206] 0.8802465412 0.3640918648 0.2882392807 0.1706452351 0.1721717464
## [211] 0.4820426055 0.2529649285 0.2162547896 0.6743763881 0.0476636274
## [216] 0.7008530875 0.3518886385 0.4089439979 0.8209513240 0.9188573482
## [221] 0.2825283301 0.9611047937 0.7283944283 0.6863750820 0.0528439428
## [226] 0.3952201346 0.4778453798 0.5602532637 0.6982615949 0.9156835384
## [231] 0.6183512274 0.4284215088 0.5420803672 0.0584784886 0.2608568571
## [236] 0.3971519533 0.1977447367 0.8319275628 0.1528872228 0.8034185420
## [241] 0.5468261566 0.6623176420 0.1716984939 0.6330553598 0.3118697468
## [246] 0.7245543464 0.3989398247 0.9693564111 0.9673983706 0.7267025390
## [251] 0.2572167465 0.2217879347 0.5930456517 0.2675214321 0.5310703989
## [256] 0.7852916713 0.1680608105 0.4043991810 0.4715762776 0.8681068069
## [261] 0.9257079558 0.8819775593 0.6741868425 0.9501669793 0.5164448943
## [266] 0.5765190213 0.3363312059 0.3473246314 0.0200243006 0.5028130456
## [271] 0.8710434136 0.0063007839 0.0720571240 0.1642112250 0.7703340743
## [276] 0.7351843058 0.9718756357 0.4664723768 0.0743845133 0.6488181243
## [281] 0.7585931695 0.1371060810 0.3965845946 0.2249853292 0.0579585605
## [286] 0.3958926883 0.0649283000 0.2258864329 0.0546291091 0.6702820396
## [291] 0.2977417826 0.1007215823 0.0719040972 0.8804405688 0.7542474021
## [296] 0.8166058876 0.9821403737 0.1035996450 0.0990418294 0.7988316112
## [301] 0.7845752665 0.0094299051 0.7790658828 0.7293906519 0.6301318528
## [306] 0.4809108300 0.1566368514 0.0082155198 0.4524583942 0.4922933287
## [311] 0.3895871115 0.4646659419 0.7132790007 0.0553019263 0.3547830975
## [316] 0.8028122729 0.8357088370 0.2377494050 0.3539861026 0.8568854202
## [321] 0.8537633738 0.2958954554 0.1470483246 0.7039920613 0.1038066882
## [326] 0.0337277730 0.9994045279 0.0348748041 0.3383912838 0.9150637616
## [331] 0.6172352696 0.2862853487 0.7377974030 0.8340543092 0.3142707804
## [336] 0.4925665478 0.6973737662 0.6414623540 0.6439229150 0.9778534048
## [341] 0.4147353345 0.1194047800 0.5260296601 0.2250733508 0.4864117641
## [346] 0.3702147985 0.9833501808 0.3883191152 0.2292448403 0.6232975463
## [351] 0.1365401971 0.9674694943 0.5150718079 0.1630703292 0.6219022952
## [356] 0.9859541652 0.6687715175 0.4189158969 0.3233449932 0.8352553200
## [361] 0.1438170436 0.1928159469 0.8967386826 0.3081195543 0.3633005435
## [366] 0.7839464787 0.1933786799 0.0177658119 0.4066078663 0.4831676693
## [371] 0.4218449502 0.3428088017 0.8664833147 0.4551080510 0.5337648736
## [376] 0.9638433319 0.7745915421 0.2088763486 0.3087868327 0.9713424502
## [381] 0.5849000935 0.7608236254 0.3727093944 0.7691939112 0.5376771830
## [386] 0.9139954497 0.1852964419 0.2822184174 0.0949624132 0.2104870791
## [391] 0.9770989963 0.2963021754 0.7259830269 0.7856878343 0.1054177461
## [396] 0.2395946290 0.2705448724 0.1010584941 0.1179138413 0.9912365559
## [401] 0.9860542973 0.1370674714 0.9053095817 0.5763018376 0.3954488591
## [406] 0.4498024841 0.7065019011 0.0825027458 0.3393125802 0.6807875512
## [411] 0.3169492481 0.8315685980 0.2151720827 0.4979489362 0.2760496726
## [416] 0.1920233187 0.9506212641 0.3217255378 0.4784563838 0.0279925719
## [421] 0.5474594680 0.6442402205 0.5962635446 0.3219373757 0.8911143125
## [426] 0.6262569474 0.3029049153 0.3882046638 0.1604750925 0.8625518975
## [431] 0.9531012138 0.5636446835 0.3295474134 0.9966172187 0.2348196758
## [436] 0.6126719653 0.1081785294 0.4870325699 0.0994458233 0.1611657627
## [441] 0.2829928708 0.5838723360 0.7317076589 0.1655209111 0.8664677632
## [446] 0.7085741372 0.7603995362 0.1470841086 0.3580569634 0.6733324814
## [451] 0.5238226017 0.3498017928 0.2405307016 0.0581917963 0.2366197421
## [456] 0.8900779132 0.8118274161 0.7475163222 0.1549117258 0.1247420886
## [461] 0.9747258085 0.4361299961 0.4640166268 0.1652980812 0.5849365573
## [466] 0.2707780194 0.2300969204 0.6912078315 0.2828523996 0.8103980662
## [471] 0.0939166071 0.8220300712 0.4274282807 0.7558872597 0.6623855066
## [476] 0.4445273969 0.6271461844 0.0004653491 0.2172435140 0.7048722457
## [481] 0.2151664882 0.8139337213 0.3077638743 0.6877426857 0.9326810790
## [486] 0.1157796648 0.1277056879 0.6782238574 0.4289485959 0.8344009779
## [491] 0.9714380668 0.0704889777 0.4597851932 0.7015851184 0.0869401679
## [496] 0.9929440161 0.2530989940 0.0495384356 0.6863249468 0.7869273527
## [501] 0.3536060760 0.3664414450 0.2871001307 0.0799729123 0.3654542696
## [506] 0.1780138146 0.5360537206 0.5039487120 0.9450351035 0.3413212840
## [511] 0.4647137742 0.0825311816 0.8601068461 0.3956606400 0.7358993504
## [516] 0.1717434060 0.4547616313 0.7702047594 0.0626499983 0.8150814951
## [521] 0.3011425245 0.3646710280 0.3121127919 0.0373699649 0.5188049234
## [526] 0.6790134157 0.9032335603 0.0255266984 0.9890782707 0.3028876246
## [531] 0.9391384006 0.6876075817 0.4470369949 0.8164781388 0.0394991979
## [536] 0.7390316855 0.3487194937 0.8292508235 0.5355413917 0.2745453636
## [541] 0.8009482119 0.0916446948 0.8321077742 0.2768550725 0.7531100996
## [546] 0.9641525701 0.0814665600 0.8543647465 0.8022382171 0.3851736046
## [551] 0.3275973990 0.2049387030 0.5693826627 0.8880551904 0.5297140914
## [556] 0.5869586568 0.6657351425 0.5298924586 0.5098428994 0.0161604809
## [561] 0.0477099849 0.9293506825 0.7692855387 0.2010806161 0.6502615004
## [566] 0.6537666824 0.3952537614 0.8123048285 0.5470234177 0.8851326515
## [571] 0.5533139876 0.9060481116 0.5874613670 0.4234636510 0.9495853230
## [576] 0.7090379035 0.4133054081 0.0183640837 0.5667340828 0.4900634531
## [581] 0.8786740212 0.8128526879 0.8540999775 0.3678959478 0.8739499128
## [586] 0.1513381714 0.2818116697 0.6667051578 0.9773835768 0.5827397367
## [591] 0.5265900795 0.0607822249 0.9690388637 0.1202371286 0.0883633010
## [596] 0.8807641149 0.5083706297 0.3374949973 0.8943346255 0.0319716281
## [601] 0.2372296974 0.6864903511 0.2258184233 0.3184945881 0.1739838168
## [606] 0.8014295837 0.1462820580 0.8227173917 0.3309978286 0.3741693879
## [611] 0.6297454238 0.0966337430 0.0219937153 0.9930447803 0.5839388457
## [616] 0.7818230651 0.8918961226 0.7548654536 0.9792037301 0.0441470679
## [621] 0.9034008847 0.8655008140 0.7754075914 0.3768164169 0.0421080487
## [626] 0.3644110817 0.2737512698 0.8504674807 0.3624017125 0.3044777906
## [631] 0.7591353394 0.8448346243 0.4579215611 0.7296316645 0.1040786360
## [636] 0.2199831777 0.9539512477 0.7520375412 0.8189553833 0.4177771017
## [641] 0.5938384931 0.7988235578 0.8884086504 0.3851915933 0.0903220843
## [646] 0.6257436371 0.7456362173 0.0853784573 0.3005176112 0.6145621573
## [651] 0.7529320579 0.9170571330 0.4757505057 0.5671194661 0.7366203461
## [656] 0.8574474244 0.9091465045 0.0563822254 0.5029083041 0.3505447730
## [661] 0.8455561229 0.8064351638 0.1173312273 0.7126865550 0.2352688580
## [666] 0.0749567512 0.9356461472 0.1571678251 0.6470573500 0.1735177462
## [671] 0.0200740092 0.5213138463 0.0862778933 0.2830023461 0.4204349010
## [676] 0.5872781242 0.8066944296 0.2019952089 0.4594246717 0.4481431139
## [681] 0.7337477582 0.7147548106 0.8312219789 0.8865661209 0.9526445740
## [686] 0.5506167219 0.5223357661 0.1706982553 0.4792715600 0.2538207965
## [691] 0.0397806370 0.6348005098 0.5395609140 0.1401035499 0.2837117768
## [696] 0.5830305952 0.1650258922 0.0963011107 0.4286130914 0.3557615657
## [701] 0.8449335399 0.2601324737 0.0231444922 0.8623995397 0.3345879612
## [706] 0.6317888717 0.5464262587 0.3764444932 0.1858730151 0.4289405912
## [711] 0.6307735986 0.5208423513 0.6596213372 0.7293609313 0.4868228734
## [716] 0.3844566157 0.0068335088 0.0036842276 0.9949364401 0.1078868802
## [721] 0.4186691584 0.7178845466 0.7425391322 0.8719988237 0.6078679611
## [726] 0.7562033513 0.8472412981 0.6127796490 0.7932241678 0.0228546835
## [731] 0.4169145497 0.8748823875 0.6473470156 0.9244789868 0.1722250960
## [736] 0.3158127891 0.8042562793 0.9889634112 0.3223855519 0.0011916284
## [741] 0.9921706193 0.1483523918 0.0493642972 0.5973534826 0.3213284435
## [746] 0.5279290003 0.7954220204 0.0689325968 0.6950206973 0.9486315073
## [751] 0.2777693174 0.2494065852 0.1594354801 0.0716300560 0.5574375831
## [756] 0.4898370516 0.4942923619 0.8883715852 0.0367043598 0.2037438136
## [761] 0.5137811392 0.2363398890 0.5754356699 0.4822579115 0.5693526887
## [766] 0.1441523151 0.1457087626 0.4122831130 0.6830102825 0.6534278204
## [771] 0.9159863330 0.8131131325 0.6758900012 0.8074991093 0.1280644196
## [776] 0.2507785340 0.3315469916 0.4072692899 0.6353675770 0.8086120449
## [781] 0.2588012014 0.8194671399 0.0154776690 0.6544007666 0.8117195843
## [786] 0.4613689268 0.2034171016 0.0187682172 0.2831326972 0.9152726256
## [791] 0.9336965205 0.5425316137 0.2194225460 0.4894278040 0.8011487362
## [796] 0.4076306578 0.1037115669 0.2805651748 0.3614662206 0.2592448411
## [801] 0.4706818336 0.3658454733 0.1212720543 0.0469936805 0.2627963042
## [806] 0.9686411680 0.4884954824 0.4778220297 0.7487928811 0.6676402313
## [811] 0.0494155968 0.6951052444 0.3632615667 0.8841336074 0.7752972296
## [816] 0.1392036413 0.2950092712 0.1260827854 0.5899016238 0.5616756089
## [821] 0.6887211201 0.3112708090 0.6055868415 0.9910343189 0.7432049152
## [826] 0.0758571303 0.4511689064 0.0535369345 0.3395555140 0.7339521488
## [831] 0.0041069405 0.7719094823 0.4629752112 0.7208403500 0.6665057172
## [836] 0.5720737206 0.7038129501 0.6572210605 0.2893521450 0.0972394557
## [841] 0.9624213211 0.7363340291 0.6127234686 0.1199288808 0.5502590465
## [846] 0.2627562778 0.8983608312 0.0091799458 0.2362349359 0.1300445953
## [851] 0.3262331467 0.7263988883 0.9917452354 0.7151335278 0.5044398110
## [856] 0.4360476718 0.9488252506 0.1201814876 0.0751451766 0.8890214928
## [861] 0.4244520951 0.0423606788 0.6474421436 0.4686191624 0.6179261233
## [866] 0.2708154325 0.1572952867 0.1142542702 0.5076828632 0.5480322875
## [871] 0.1406461895 0.1697690501 0.7619853232 0.5273949406 0.8609893576
## [876] 0.6735549879 0.0130410297 0.6931989014 0.8917137322 0.6318501753
## [881] 0.1072946549 0.9210652038 0.6753623812 0.1485937126 0.7453418423
## [886] 0.9425988619 0.4207833756 0.2977230782 0.2594266701 0.2228813169
## [891] 0.5656543435 0.7566501328 0.6696035459 0.5465227943 0.8114636578
## [896] 0.7591668020 0.0200684662 0.3809037001 0.0508801136 0.7979029373
## [901] 0.9236992132 0.5425983705 0.8523646002 0.5835628633 0.6683236435
## [906] 0.5113145965 0.7627505893 0.9033622879 0.8204745122 0.0714318496
## [911] 0.0038963431 0.0521986159 0.8665601797 0.5762451689 0.3138425720
## [916] 0.9594657838 0.5911937570 0.5314093358 0.3839366715 0.3195532295
## [921] 0.8083862553 0.0419194985 0.3637426947 0.8565969670 0.6979465785
## [926] 0.6844864730 0.3480150527 0.5546818294 0.1372436176 0.7849315563
## [931] 0.8868625716 0.2040958833 0.7706229615 0.5963629608 0.9576697424
## [936] 0.1586883976 0.5259742741 0.8731513675 0.8697060703 0.0236886432
## [941] 0.9758896935 0.4902241982 0.3891703230 0.4175549599 0.0929258205
## [946] 0.1618092102 0.4054164952 0.3418144442 0.4152574532 0.3040524691
## [951] 0.5602805375 0.1558741506 0.9565797641 0.0439666254 0.3721577427
## [956] 0.9626153414 0.6454275157 0.0612626143 0.4099459285 0.4259051341
## [961] 0.5081580847 0.4495999108 0.6232613835 0.1399779310 0.9079464008
## [966] 0.5694432748 0.5482805739 0.1168276225 0.7620283323 0.4783694467
## [971] 0.7819689147 0.0460265486 0.8198446678 0.2694084474 0.2828504145
## [976] 0.6432215595 0.9481188378 0.0069984545 0.3516176720 0.4190457249
## [981] 0.4578767971 0.7116925793 0.9198480577 0.6271107006 0.9021817783
## [986] 0.7573291592 0.1378583657 0.1532548780 0.1913121869 0.4331851406
## [991] 0.0872193018 0.2237800010 0.5721486795 0.4001691784 0.5654653807
## [996] 0.8296238787 0.6421138195 0.3914987517 0.7095798547 0.1088240731# Plot histogram
hist(uniform_data, breaks = 30, main = "Histogram Distribusi Uniform", xlab = "Nilai", col = "lightyellow")
runif(n, min, max) digunakan untuk menghasilkan n variabel random dari distribusi uniform dengan rentang min hingga max. Histogram menunjukkan bahwa nilai-nilai tersebar merata antara 0 dan 1, sesuai dengan sifat distribusi uniform.
Simulasi Distribusi Diskrit: Distribusi Binomial
Distribusi binomial menggambarkan jumlah sukses dalam n percobaan independen dengan probabilitas sukses p.
# Simulasi 1000 variabel random dari distribusi binomial
n_trials <- 10 # Jumlah percobaan
p_success <- 0.5 # Probabilitas sukses
binomial_data <- rbinom(n, size = n_trials, prob = p_success)binomial_data## [1] 4 5 3 7 7 5 6 4 3 5 5 4 4 5 2 6 5 6 5 4 5 6 5 5
## [25] 6 4 5 5 4 5 4 5 5 3 5 5 3 5 5 5 7 6 5 5 4 3 5 2
## [49] 7 9 6 6 7 5 5 4 6 4 4 6 6 4 4 4 3 6 5 3 8 5 6 7
## [73] 7 6 6 6 4 2 5 4 5 4 4 5 6 4 3 2 6 5 6 3 3 3 6 6
## [97] 9 6 4 4 6 6 4 6 7 5 5 7 8 6 3 6 5 5 4 4 5 7 4 5
## [121] 4 4 6 4 3 3 7 5 3 5 5 6 6 5 6 6 5 4 5 4 7 5 3 8
## [145] 4 5 6 6 4 3 7 5 6 7 5 4 3 7 5 4 4 4 5 4 3 3 6 4
## [169] 3 8 5 5 6 7 4 5 6 5 3 6 1 4 6 7 6 3 5 6 6 5 3 5
## [193] 5 6 2 4 7 3 8 5 7 7 5 6 5 8 3 3 6 5 5 9 4 5 3 6
## [217] 5 6 4 8 4 5 3 7 8 2 5 4 7 5 1 4 4 5 4 2 3 6 7 4
## [241] 3 4 6 5 6 4 5 2 4 6 6 5 5 1 3 7 3 4 8 5 4 5 2 3
## [265] 6 3 5 7 3 8 2 6 5 5 7 5 6 6 4 5 4 7 5 6 5 4 6 5
## [289] 4 5 6 5 4 7 4 4 6 6 7 5 2 5 5 7 6 5 3 3 6 3 4 4
## [313] 7 4 6 3 6 4 5 2 6 4 7 4 8 5 5 5 2 6 5 4 5 5 5 3
## [337] 6 6 7 5 4 5 5 4 6 8 4 6 5 5 2 1 3 2 3 5 4 6 4 5
## [361] 6 3 7 5 4 4 3 5 3 4 4 4 5 5 6 6 4 8 2 5 5 4 7 3
## [385] 7 4 4 7 3 4 9 6 5 5 5 6 6 5 2 4 4 4 3 5 4 6 5 8
## [409] 3 8 6 7 5 3 4 6 5 4 6 4 7 5 5 7 5 3 5 8 7 5 7 4
## [433] 6 6 5 5 6 7 6 5 3 5 2 5 6 6 3 5 5 4 5 6 6 3 5 6
## [457] 4 6 7 3 8 6 6 4 2 7 7 3 5 5 6 3 6 5 5 6 5 10 7 7
## [481] 6 3 7 6 9 2 6 5 5 7 2 2 9 6 4 6 9 4 6 5 7 8 5 4
## [505] 6 5 5 5 5 5 5 8 7 4 5 5 2 5 5 4 6 2 6 5 6 2 2 5
## [529] 6 4 7 6 7 5 5 4 4 4 7 5 2 6 4 5 6 4 2 5 5 8 8 3
## [553] 5 5 5 4 6 4 5 9 6 4 6 4 5 3 3 7 3 7 4 2 6 4 2 6
## [577] 2 5 4 5 6 3 4 3 6 6 6 5 5 4 6 7 6 5 4 3 5 4 6 4
## [601] 6 3 4 6 6 2 7 3 5 5 5 6 5 8 8 5 5 4 5 4 1 2 3 6
## [625] 5 7 5 4 5 7 6 5 5 6 7 7 4 4 3 6 3 7 6 7 5 7 6 7
## [649] 5 3 4 6 4 5 2 5 3 7 3 6 7 2 4 6 6 5 7 5 5 4 5 5
## [673] 2 6 4 6 5 6 5 4 4 5 9 4 4 2 5 5 6 8 5 3 4 7 7 7
## [697] 8 6 5 4 6 3 5 7 7 6 3 5 9 5 4 3 3 7 4 8 5 5 8 3
## [721] 6 5 5 4 3 6 4 6 5 4 9 5 2 7 7 5 4 6 8 7 3 8 5 1
## [745] 3 1 6 6 7 4 8 6 6 5 8 5 7 5 5 7 6 2 3 5 5 9 7 4
## [769] 6 7 6 6 3 5 4 5 4 4 6 5 3 4 6 5 3 3 8 4 6 7 4 2
## [793] 5 7 4 4 6 8 5 5 3 7 4 6 5 2 8 5 7 5 4 5 6 6 5 5
## [817] 5 5 5 1 9 5 3 7 5 7 2 7 6 6 7 5 7 4 2 4 6 4 7 6
## [841] 4 5 5 6 5 2 6 4 6 4 7 7 5 3 5 5 9 4 6 7 5 6 3 7
## [865] 5 4 5 7 5 6 4 6 4 4 4 7 5 3 7 4 7 3 4 4 2 6 4 5
## [889] 4 5 5 9 4 6 8 5 5 6 3 5 5 4 3 7 5 5 5 6 6 5 2 5
## [913] 5 6 5 3 3 6 7 3 6 5 9 5 5 5 5 3 5 8 5 5 2 2 2 3
## [937] 5 6 4 6 5 4 2 6 3 7 3 6 7 4 3 6 4 4 5 5 4 6 5 3
## [961] 9 4 3 4 5 4 4 2 7 5 4 7 5 5 6 6 3 4 8 4 6 5 3 4
## [985] 6 3 5 5 5 4 5 3 7 3 3 7 4 6 5 5summary(binomial_data)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 4.000 5.000 5.001 6.000 10.000# Plot histogram
hist(binomial_data, breaks = 30, main = "Histogram Distribusi Binomial", xlab = "Jumlah Sukses", col = "lightgreen")
rbinom(n, size, prob) digunakan untuk menghasilkan n variabel random dari distribusi binomial dengan size percobaan dan probabilitas sukses prob. Histogram menunjukkan distribusi jumlah sukses, yang berbentuk simetris karena p = 0.5.
Simulasi Distribusi Kontinu: Distribusi Normal
Distribusi normal adalah distribusi kontinu yang berbentuk lonceng, dengan mean mu dan standar deviasi sigma.
# Simulasi 1000 variabel random dari distribusi normal
mu <- 0 # Mean
sigma <- 1 # Standar deviasi
normal_data <- rnorm(n, mean = mu, sd = sigma)normal_data## [1] -0.9957987249 -1.0399550438 -0.0179802406 -0.1321751329 -2.5493427748
## [6] 1.0405734557 0.2497257360 2.4162073731 0.6851982381 -0.4469593090
## [11] 2.7973911470 2.8322260239 -1.2187118163 0.4690319560 -0.2112469186
## [16] 0.1870511466 0.2275427286 -1.2619004629 0.2855895802 1.7492473631
## [21] -0.1640900041 -0.1629267107 1.3985720063 0.8983962407 -1.6484948184
## [26] 0.2285569733 1.6535472347 1.4152763499 0.4199516038 0.7212208071
## [31] -1.1969352122 0.3001315669 -0.9544489352 -0.4580180681 0.9356036838
## [36] -1.1368931117 0.2669182509 0.4283320395 0.0549119698 1.8221888218
## [41] -1.0223473286 0.6061302614 -0.0889305675 -0.2608322438 0.4640912339
## [46] -1.0204005907 -1.3134509165 -0.4944808755 1.7517571498 0.0557647746
## [51] 0.3314343982 -0.1898466377 0.4704927320 -0.9516795430 1.1579104666
## [56] 0.5847052602 -0.8064528227 0.0545532458 0.7163316195 0.5577309821
## [61] 1.4819340159 -0.6129877549 1.1161366155 1.0365480099 -0.1624831319
## [66] -0.9759266928 -1.0891451909 0.4577869579 -0.0711267337 1.7791026669
## [71] 0.5351379605 -0.3719448752 -1.0255422485 -0.5824016746 0.3428883929
## [76] -0.4509346471 0.5142301202 -0.3343380522 -0.1055599088 -0.7305096728
## [81] 1.9050435849 0.3326217315 0.2306336405 -1.6918624149 0.6597918995
## [86] -1.0236235889 -0.8915215744 0.9183411710 -0.4527006465 -1.7483722800
## [91] 1.7699041099 -2.3774069254 0.5728115299 1.0172492491 -0.6309678667
## [96] 0.4442870514 0.4391303884 1.0406231529 0.4840993880 -0.2448837791
## [101] 0.9159920579 0.8006223565 -0.9365690341 -1.4007874340 0.1602775400
## [106] -0.2739623748 -0.9855391126 0.0839306795 -1.3199965265 0.1612263513
## [111] -0.6249283875 0.9571642742 2.4244891412 -0.9159792437 1.0576641709
## [116] 0.8251497278 -0.0701942243 -0.4536463741 1.5753077068 -2.0054578182
## [121] -0.6431947916 -1.4368434437 1.3953134389 -0.1907034326 -0.5246711995
## [126] 3.1840444741 -0.0500372679 -0.4437493119 0.2998652501 -1.5684246208
## [131] 0.4903026427 -0.0961632011 0.4685251225 -0.9823706359 -1.0229838421
## [136] -0.6934146633 -0.7679895731 1.2990499667 1.5791455618 -0.1568919530
## [141] -0.3589365606 -0.3290388304 0.0692364779 0.0969042337 0.2900343878
## [146] -0.7466789410 -0.8468963888 1.1970776637 -0.5486273611 0.3030456952
## [151] -0.0569705338 -0.9578493922 0.5910619094 0.1731048735 1.3997833562
## [156] 0.1174595846 -0.3315457582 0.2782949133 -1.1855916490 -0.8358940534
## [161] 0.5102732511 -0.3331209012 -0.0659609464 -0.1152217094 -0.6505126188
## [166] -2.0186886591 0.3488349702 0.7616395076 -1.2887162354 1.4824027185
## [171] 0.3851548237 1.3416402950 -0.9571704712 0.1667812884 -0.1000139589
## [176] 0.7685074319 -0.5758595679 -0.0100976720 -1.7786591487 -0.7776214429
## [181] 0.1250338837 -0.7063214924 -0.0435694858 -0.4679259739 0.6069301362
## [186] 1.1684883087 -0.8225014128 -0.3070365586 1.4397612584 -2.1989232542
## [191] -0.3198377917 2.0647042841 2.1935900691 0.1565953234 -0.8636089506
## [196] 0.1654574195 -0.6527743959 1.4528172823 -0.8064826557 0.3729115952
## [201] 0.6198500746 -0.7575101607 0.8515246815 -0.7479299709 0.6302398297
## [206] 1.0966616331 -0.9884429224 1.1079950410 -0.4895328712 0.2943533884
## [211] 0.2018374723 -0.4271963863 0.2681028700 -1.2304309254 -0.1361368722
## [216] 0.8257908317 -2.1741246492 -1.4879261874 -1.1619375576 -1.5890896941
## [221] 0.4195830393 -0.9929283498 -2.1645470853 -0.6375687651 -0.3906352515
## [226] 0.8567854738 -1.1037521360 1.1612892557 0.3983627156 0.3623521576
## [231] -0.8525256703 1.9536678799 -0.1642708321 -1.8248975842 -0.2038564714
## [236] -1.9344440707 -0.3105101158 -0.4222270032 0.6818296904 1.0094961857
## [241] -0.7261049649 0.8061088684 1.4243231097 -0.7841440048 -0.6524043665
## [246] 0.6507783632 0.1830479699 0.5487749604 1.4046842945 0.3870831196
## [251] 1.0517012702 0.6229054623 0.4336203898 0.3860844382 1.2913233045
## [256] -1.0022598737 -1.1051827292 0.5919460034 -0.1196896641 0.0740052117
## [261] 0.7412773772 0.7532950478 -0.2626705034 -0.3125438690 0.0735986090
## [266] 1.0630177893 0.4260204867 1.4330075108 -0.0076368704 1.1256676052
## [271] 0.8830023110 0.6120834609 0.4147007068 -0.2798823971 -0.1090375092
## [276] 0.2293954957 0.0488888907 0.9432244684 -0.1093171232 -0.0703769216
## [281] -0.4843190876 -0.1383363285 -0.0687656423 -2.3137357739 -1.3648316972
## [286] -0.0724869100 -0.2652837749 -1.2008693282 -1.9915381760 -0.3543692199
## [291] 0.6534957657 1.7732386319 -0.0384567937 1.4931848406 0.0830221576
## [296] 0.1155321043 0.3248253133 -0.8705772453 -0.0517182129 0.9084476988
## [301] -0.7497257869 -0.3216060699 -1.1477707505 0.3543521964 0.4247997824
## [306] 0.6483473512 -1.2198100315 0.1072350348 -0.9440576916 -0.0003846487
## [311] 1.3426239200 -0.5035252869 0.7166833209 -0.7496685841 -0.4785282105
## [316] 0.4387217506 -0.6791122705 -1.7029648351 1.2651684352 0.3603572379
## [321] -0.5836394406 -1.9940787873 1.9022097714 3.3903708213 0.2074804074
## [326] 0.8498066475 1.2245603121 -0.7018044335 -0.3511962296 -1.7271210366
## [331] -0.7365782323 0.6224097829 -0.2907159892 -0.2142115342 -0.1125595515
## [336] -1.8636669825 0.8376299342 -1.4434928889 -0.2085701624 -0.4385634621
## [341] -0.2185938169 1.4599659447 -0.5820599179 -0.7830975957 -1.5196539949
## [346] -0.8056980816 -1.1661847074 0.4079461962 -0.8630042460 0.3040420350
## [351] -0.1464274878 -1.4335621799 -0.7906077857 0.8851124551 0.9030760860
## [356] 2.0055732743 -0.0035803084 -1.4958268140 -0.7684170270 0.4084885048
## [361] 1.9001363349 0.1100091234 1.1403868251 0.7680813047 -1.1680916221
## [366] -0.1711126523 1.3052615363 0.8760961096 0.4637961416 0.4771142454
## [371] -0.4914053002 -1.3193853133 1.2954257908 -1.4202194917 -0.9388959197
## [376] 0.6289649925 -1.2621945494 -0.5518704133 -1.1827995068 0.6206635577
## [381] 0.4463130166 0.4218846933 0.4424647721 0.5572457464 0.6393564920
## [386] -1.9686615567 -0.1488163614 0.1124638126 0.7246762026 -1.1874860760
## [391] -0.4996001898 -1.0736429908 1.0572402127 1.2790725832 0.7876767254
## [396] -1.2224033826 0.4519521167 1.1504491864 0.1679409807 -0.5661093329
## [401] -1.0861182406 -0.6653027956 0.7148483559 -0.4316611004 0.2276149399
## [406] 1.2949457957 0.5783349405 1.3646727815 -1.7015798027 -0.2806762797
## [411] 0.0650680195 0.5785892916 -1.1692066215 0.8061848554 0.3073900762
## [416] 0.2638060136 0.5084847916 -0.1163584399 0.9255460985 0.6482297737
## [421] -0.1502093742 1.0403770193 0.2925586849 0.6687513994 -0.5941776416
## [426] 1.5804318370 -0.0039889443 0.8478427689 -0.1001165259 -0.2796299070
## [431] 0.7844382453 -1.5846166446 0.4783661478 0.3935663730 -2.6953293691
## [436] 0.3683773285 -2.1684177473 0.6598043769 -0.4539137334 -0.6949368252
## [441] -0.0068463032 1.3730520450 -0.6353230772 0.5581032939 0.3411578684
## [446] -1.1795186291 -1.7410220173 -1.9925857712 0.5512742115 -0.0347420615
## [451] 1.8505717036 0.5736751083 0.8496958911 1.3343835853 -0.5007190980
## [456] 0.5100979282 0.8687932702 1.3693516880 0.7626511463 0.4211471730
## [461] -0.8682240473 0.7295603610 0.5002658724 0.6342502537 0.4236450456
## [466] -0.2018380447 -0.0768658984 0.6873641133 0.1716315069 -0.8301085743
## [471] -0.2901591198 -1.3191257242 -0.9670319027 -0.1446110701 -1.7981325564
## [476] -1.6885424746 1.1025651994 -0.5766189242 -1.8516917296 -0.1128632394
## [481] 1.3210692672 0.6622542969 0.4413831984 1.1837459123 -0.7715014411
## [486] 0.7296891914 -0.5870856158 0.0007641864 2.2144653193 0.9694343957
## [491] 0.7680077137 -1.1083279118 -0.7862359200 2.2841164803 -1.0933007640
## [496] 0.2144793753 0.8925710596 1.0187579723 1.0891120109 -0.1631289899
## [501] -0.8209866971 -0.3072572330 -0.9020980085 0.6270687431 1.1203550284
## [506] 2.1272135525 0.3661143829 -0.8747813772 1.0244748630 0.9047588937
## [511] -0.2382486959 -1.5578549039 0.7613098954 1.1291443956 -0.2951078307
## [516] 0.5362428184 -0.2758904748 0.6823152451 -0.1172907147 -0.3446758639
## [521] 0.1116204982 -0.2834053150 -0.5910171645 -0.3159369312 -0.0081521523
## [526] 0.2074951407 1.5324236216 -1.3579978310 -0.1996190506 0.6315231278
## [531] 1.7620209034 0.4260143626 -0.0137534165 -0.3075569096 0.4143081641
## [536] 0.9890579202 -0.1838583105 0.1637614068 0.2169363441 0.7292776342
## [541] 1.1113804070 0.2791608169 -0.0761706715 1.3946631322 0.1645341176
## [546] 1.5778519788 -0.0619226576 0.6139229642 -1.5460885942 -0.1123919613
## [551] -0.0217945400 -0.7583454174 -1.0358928844 0.9481593026 0.9141587339
## [556] -1.2987319948 0.4243787949 -1.1125453200 -1.0510732262 0.5254124483
## [561] -0.6860240001 0.9934799821 0.0385235991 0.5361489758 -0.5236266979
## [566] -1.1512213349 0.9147522410 0.2380714916 -0.2390677595 0.0692353265
## [571] 1.3259083426 -0.6981666346 -0.7494084445 -0.6196150527 -1.5849912682
## [576] 0.8196281375 0.1923696467 0.2071719742 -0.0433473543 -0.5101604414
## [581] -0.8234186138 0.8518564033 -1.4261846729 0.4402989420 -0.7926116508
## [586] 0.2823102145 -0.7406905222 -0.5233416827 1.7693659170 0.6682826189
## [591] -2.1448970243 0.1264124161 -0.4518129363 -1.1366261878 0.2097858904
## [596] 0.1299655160 -0.3285065726 1.9727035673 -2.2486900668 0.8382193873
## [601] -0.2890232705 0.6565134106 -0.4539977006 -0.5938645624 -1.7103796664
## [606] -0.2094484283 2.4787458014 0.9897022081 1.6755721558 0.9149653183
## [611] 1.1442627080 0.9028764145 0.4753924318 -0.5825287740 -0.5329347373
## [616] -1.6008399964 -0.0058177139 0.8993556759 1.0319225565 0.0951327045
## [621] -0.5476276172 3.2905174433 0.7366855313 1.4205753049 -0.3376806407
## [626] -0.0379576271 0.4486070977 1.6765223119 -0.3114745455 0.8536156668
## [631] -2.0948146343 -0.5072544341 -1.2920090768 1.1133627174 -0.1644530883
## [636] -0.3903740820 1.3690998462 1.1162728582 -0.8980212035 0.4278664876
## [641] -1.2284445685 -0.4756150242 1.6165776371 1.4501279513 1.1090187555
## [646] -0.5709038863 -1.8814314703 -1.1756981840 0.9525565246 -0.2905678862
## [651] -2.1626081456 -0.1801874877 1.4102392211 0.6434686405 -0.8212585435
## [656] -1.5459166521 -0.8265472256 0.0345276713 0.8880737009 -1.9399401548
## [661] 1.0232017553 0.0054577266 0.5697789696 -1.6532555632 -0.6666543801
## [666] -0.4482341893 1.0438913483 1.0281740474 0.4350904592 1.6042121816
## [671] -0.5154111997 1.0125371938 -0.0359400298 -0.6673420962 0.9233800383
## [676] 1.3811003308 0.8782504162 -0.5094034554 -0.4697876342 1.3776758468
## [681] 0.3528264064 0.8295739793 -0.3387019837 1.2610349365 -0.8087551450
## [686] 0.6253515209 -0.8171749656 -2.4625750166 -1.3429575107 0.1362951992
## [691] 0.8829227496 -1.7513020831 -1.2514244689 1.7645459966 -0.4338993498
## [696] 0.5057001319 -0.5269353214 -0.2985828852 0.0872442066 0.0109618428
## [701] -0.1925602110 -0.4697964851 -3.0478608898 1.8686555003 1.7904242082
## [706] -1.1010817417 -0.1681075221 1.3752753047 0.9982900231 1.2766016187
## [711] -1.0717469182 2.5772680962 -1.1334599598 0.7539163419 0.1412759804
## [716] -0.4037103180 -0.3794157982 -0.9913968125 1.6226597958 0.0895132278
## [721] 0.2592179492 0.2096328331 -0.3751707531 -1.1340212448 0.2537263097
## [726] -2.0936394500 -1.4185669428 -1.0763966949 -1.0786788627 0.1071888166
## [731] 1.5984875483 -1.5153241441 0.4336760206 0.8995447533 -0.9895322023
## [736] -0.0527994015 0.8236109026 -0.2555091025 -0.2206843473 0.3077267912
## [741] -0.0600132533 -0.5556528899 -0.1386150195 1.8828397924 0.8736686796
## [746] -0.9145970728 -1.2449176223 -0.3599822407 1.3287747009 0.2926791184
## [751] -0.7015052368 0.8822345678 -0.1333703887 -1.1206784993 0.4611924542
## [756] 1.5241428102 0.4344682981 0.1920003711 -0.6562431284 0.5683985309
## [761] -1.0705705349 -1.6531490245 -0.0433527685 -0.0345935058 2.3650555320
## [766] -1.2163473079 0.1709063233 0.8050530936 1.0505928441 -0.0107244847
## [771] -0.7432561414 -0.0657840519 1.9397559919 0.4827390082 -2.0444770726
## [776] 1.4234591287 0.5405026609 -0.0335717718 -0.0178636211 -0.1497897198
## [781] 0.2565594810 -0.5038669330 0.2770112517 -0.9313560248 0.2001468753
## [786] 1.1068374207 0.5092061137 1.0337496765 -1.0908687618 0.0547927845
## [791] 0.6172503027 -1.0680048677 1.5658143371 -1.0348080110 0.1645187085
## [796] 0.1518323304 0.1216703020 -0.2104245836 0.4499367871 -1.0311644919
## [801] -1.2893641882 -0.6545686381 -0.0573241038 1.2567478204 1.5874541399
## [806] 0.3194814625 0.3815916227 -0.2436448836 0.0480530841 -1.4045458611
## [811] 0.2899337289 -0.5355535823 0.3346787731 -0.3459813392 -0.6616157348
## [816] -0.2191113768 -0.3669049106 1.0945782078 0.2092080824 0.4324914262
## [821] -1.2408535858 1.4968217096 0.1593704412 -0.8562814034 0.3090466454
## [826] 0.8704340298 -1.3836771383 1.6901069700 -0.1580307052 1.1211707806
## [831] 0.0722613191 -0.3324228446 -1.8349200466 -1.1001722186 -0.0413402999
## [836] 0.8278525451 -1.8816786542 1.3754411123 1.3989904643 -1.1433162559
## [841] 0.4723005622 -1.0336392134 -0.1251999787 0.9286627388 0.8683396477
## [846] -0.8491746036 -0.3866364545 -0.9761635710 0.3395436599 -1.5590751644
## [851] -2.6293254416 1.4698122819 2.2734729130 -0.4550335402 0.7611024871
## [856] -0.0075027837 1.4743137997 0.5541439326 0.2036639651 -1.7991364518
## [861] 1.0829556808 -0.3508536150 -1.4034900847 -0.2017966649 -0.1267781598
## [866] 1.0592068727 -1.1673960323 -0.5576436273 1.4881199284 1.3586657688
## [871] 1.1632145439 1.6615239453 0.2040309798 -0.5818836869 0.5552040619
## [876] 1.0587231258 2.4136332715 -1.9649823327 0.2732357031 0.6547945829
## [881] -0.0545986554 -1.5578222485 0.7415008921 -0.7790857413 0.5058614988
## [886] 0.9075517060 1.2839570105 -1.5578637974 1.0817418482 -0.7569813569
## [891] -1.2890194738 1.3143206659 1.1462599733 -0.2425832677 0.7595407062
## [896] -0.8603257414 -0.1510315791 -0.0937232338 -0.2807400545 0.7340987365
## [901] 1.5373275355 -0.4557710628 -0.0326584524 1.6367573492 -0.3290419705
## [906] -2.6040381706 0.5139837864 -0.8864680111 -0.9985384060 1.4208168136
## [911] 2.4479980070 -1.0397825414 1.0310251817 -0.0941478384 0.1418074592
## [916] 1.2222367015 0.2136745184 -0.8513653539 -0.4704088747 0.6861352553
## [921] -2.3359473319 1.0952443818 -1.5671500988 0.0219310628 -0.1903589776
## [926] 1.2930694860 0.1888493151 0.1019391349 0.6981358089 -0.8270145562
## [931] -0.1958988561 1.1775844138 0.6834736210 -1.2754967068 0.6379563674
## [936] -1.3775896246 -0.5983108037 1.2109203839 -2.2510451754 -1.7790141909
## [941] 1.3013726728 -0.8147927841 1.2437070186 -0.1682502008 0.4277756784
## [946] 0.8132788896 -0.6512118694 -0.3045909231 -0.4150971696 2.8160842760
## [951] 0.1261470691 0.4728004231 -0.3407535409 -0.2417906427 1.3787546689
## [956] -0.3388836695 0.0201363028 0.3769621641 -0.4317237450 1.9590641562
## [961] -1.4284596122 2.0112929798 -0.3515918889 1.3571196468 -1.9991774110
## [966] 0.9560806214 0.8764312650 -1.2712169716 -0.7683238785 0.1935248455
## [971] 1.1438354288 -0.7659993040 -0.2241259991 1.5713469350 -1.1273472409
## [976] 0.9477939807 0.4487681892 -1.1058145286 -0.6678678422 0.7832775103
## [981] 0.2489594327 1.4250982775 -0.6017839621 -1.7144876964 1.0478269265
## [986] -0.6086216184 0.1203405268 1.7190418139 -0.2504140523 1.5495553274
## [991] -1.0971396499 0.9255112361 0.2467992119 -0.7367715394 -1.2800089389
## [996] 0.0766436636 0.2551647625 0.2774468170 0.5368560222 -0.4604855660# Plot histogram
hist(normal_data, breaks = 30, main = "Histogram Distribusi Normal", xlab = "Nilai", col = "lightpink")
rnorm(n, mean, sd) digunakan untuk menghasilkan n variabel random dari distribusi normal dengan mean mu dan standar deviasi sigma. Histogram menunjukkan distribusi berbentuk lonceng, yang khas untuk distribusi normal.
Distribusi Poisson (Diskrit)
lambda <- 3 # Parameter lambda
poisson_data <- rpois(n, lambda)poisson_data## [1] 2 5 3 4 2 1 5 0 3 7 2 7 0 2 2 6 7 4 2 2 2 4 4 3
## [25] 4 4 4 3 2 4 5 3 1 3 3 6 2 4 4 4 3 3 5 3 7 1 3 1
## [49] 2 5 2 3 2 2 3 4 7 1 1 3 1 2 2 2 4 4 4 2 1 3 3 3
## [73] 3 1 5 8 5 5 2 3 3 0 4 5 0 5 2 4 4 4 2 2 3 0 3 3
## [97] 2 6 1 3 2 1 2 9 4 2 6 4 3 4 4 2 3 3 4 2 3 3 3 5
## [121] 5 5 1 2 6 6 3 4 3 0 2 3 3 6 3 0 0 2 2 5 4 4 2 1
## [145] 2 4 2 2 3 1 1 0 4 4 0 3 2 3 2 6 1 1 1 2 3 2 3 3
## [169] 2 7 0 5 1 1 3 4 1 1 2 1 5 4 3 5 1 2 2 0 2 4 2 4
## [193] 1 4 3 1 4 5 3 3 2 0 4 3 4 3 3 5 4 1 1 5 2 3 2 3
## [217] 3 2 3 2 3 2 2 2 1 3 3 2 5 2 2 3 2 6 3 3 2 0 3 2
## [241] 3 4 2 1 4 1 4 0 5 5 5 2 2 6 2 6 3 3 4 2 2 3 2 4
## [265] 2 5 6 3 0 6 3 5 3 3 1 1 4 2 0 1 5 4 0 4 0 3 2 3
## [289] 3 3 2 2 2 1 3 0 1 3 3 3 4 12 6 6 4 4 6 2 4 1 3 4
## [313] 0 1 1 4 2 3 4 2 2 2 4 3 1 2 2 3 3 2 2 6 3 3 2 2
## [337] 3 2 3 1 3 6 0 2 2 3 1 4 1 2 2 3 3 8 3 2 5 2 4 5
## [361] 5 1 2 3 2 5 5 4 3 4 0 2 1 5 5 5 2 0 3 4 3 2 2 2
## [385] 2 3 4 2 3 4 5 6 2 3 1 1 3 5 4 3 8 8 2 1 2 3 4 4
## [409] 5 4 2 2 4 2 2 5 1 1 4 3 3 4 4 4 2 7 6 0 4 2 2 0
## [433] 4 1 1 4 3 0 6 4 3 1 5 4 2 1 0 2 1 3 3 2 6 4 2 4
## [457] 3 2 1 0 1 3 2 0 5 5 5 3 4 4 5 4 2 3 4 2 3 2 1 2
## [481] 2 2 3 6 3 2 2 3 1 4 0 0 4 2 2 1 4 3 1 2 1 2 3 3
## [505] 1 2 0 2 4 4 5 1 1 4 2 1 1 1 3 3 3 5 1 4 2 1 1 5
## [529] 4 3 4 4 3 5 2 3 0 1 4 4 3 2 3 0 5 4 1 3 5 2 2 4
## [553] 2 1 2 2 4 3 2 1 1 3 2 1 0 5 2 6 4 5 3 6 4 2 4 6
## [577] 4 4 4 1 0 2 3 6 6 3 2 3 4 5 3 1 4 5 2 1 2 0 3 3
## [601] 3 5 3 2 3 1 4 4 2 1 5 2 7 4 6 4 2 2 0 5 3 2 4 4
## [625] 4 3 1 5 1 5 3 3 4 2 4 5 4 2 1 4 1 1 0 4 4 1 4 4
## [649] 3 6 1 1 6 5 2 2 6 1 4 3 5 4 3 1 3 3 1 5 2 3 3 3
## [673] 3 1 3 1 3 4 3 2 2 0 2 4 1 7 4 5 3 2 3 1 2 2 2 4
## [697] 1 3 1 3 0 4 5 2 3 3 1 4 5 6 4 0 4 2 2 0 2 2 3 5
## [721] 4 2 4 2 6 4 2 2 1 5 3 5 9 3 2 4 5 7 2 4 2 3 2 4
## [745] 2 4 6 1 3 4 4 5 4 5 7 3 4 3 1 3 1 3 1 4 0 1 5 7
## [769] 2 3 4 3 5 4 2 1 3 3 3 4 3 3 2 3 4 2 4 4 2 2 1 3
## [793] 4 3 6 0 6 3 4 3 2 2 2 1 2 2 2 0 4 3 3 1 0 1 0 6
## [817] 3 4 3 1 0 3 2 4 3 3 6 1 0 3 1 2 1 1 3 1 3 4 2 4
## [841] 1 3 2 1 2 3 2 3 6 3 2 4 3 1 2 1 1 4 4 3 1 5 9 0
## [865] 3 4 3 2 3 2 7 4 4 0 2 3 1 2 1 2 2 3 5 3 3 2 2 1
## [889] 4 6 5 2 2 2 2 3 3 1 3 0 4 6 6 5 3 2 2 6 4 2 3 4
## [913] 4 3 2 2 4 4 1 3 3 1 2 2 1 5 2 1 3 4 3 3 4 5 5 3
## [937] 4 1 2 2 1 0 2 1 5 3 3 3 1 4 4 1 3 4 2 4 3 2 5 0
## [961] 5 5 2 2 3 3 1 2 2 1 7 2 6 4 2 4 2 4 1 2 7 0 3 4
## [985] 2 3 5 3 1 2 4 0 6 4 2 1 6 2 4 0hist(poisson_data, breaks = 30, main = "Histogram Distribusi Poisson", xlab = "Jumlah Kejadian", col = "lightyellow")
Distribusi Poisson digunakan untuk memodelkan jumlah kejadian langka dalam interval waktu atau ruang. Histogram menunjukkan distribusi yang miring ke kanan, yang khas untuk distribusi Poisson dengan lambda kecil.
Distribusi Eksponensial (Kontinu)
rate <- 1 # Parameter rate
exp_data <- rexp(n, rate)exp_data## [1] 1.553035218 2.468125617 0.879852302 0.084137604 0.561296626 0.780513311
## [7] 0.961835421 0.339819463 0.691800873 0.079011615 0.283089773 0.271604763
## [13] 0.326924743 0.722076448 0.932341057 0.680912765 0.376546263 0.549864237
## [19] 0.924507310 0.283125876 0.226783255 1.556918928 0.569539452 4.108050128
## [25] 0.510666214 0.082786562 0.179522779 0.666050538 0.746347203 0.987793947
## [31] 1.960074432 1.447855834 0.127498755 1.115389571 0.648991768 0.853896972
## [37] 4.119256144 1.316849202 0.192069228 0.057479305 0.365141817 0.015625815
## [43] 0.108743591 0.682480210 0.943187822 2.922928572 0.631988121 0.557153579
## [49] 0.006103936 0.183361751 0.026045009 1.334309514 1.154012025 0.785805540
## [55] 0.750397948 1.040412077 3.250382641 3.672457984 2.950655021 0.155644464
## [61] 0.580071719 0.470522067 0.767298057 0.610732849 0.807665626 0.651520712
## [67] 0.561901855 2.383071931 2.211575435 0.062586170 1.232326610 0.249447808
## [73] 1.245310821 1.536333168 0.234716057 0.026625927 1.099581148 0.646960788
## [79] 0.614877820 0.819439118 0.017297199 0.874064942 0.251162786 0.164607826
## [85] 0.601252237 0.993258641 0.762290514 2.195939517 0.141034741 4.919765757
## [91] 0.816753786 0.813495975 0.636458673 2.261160176 0.220161007 1.438530814
## [97] 0.943847345 0.998185847 3.410538092 0.402635587 0.326952604 0.327732776
## [103] 0.725255940 0.838699308 0.505262477 1.465604454 1.464342667 0.020835642
## [109] 0.153846169 0.775044423 0.214510050 1.053941530 0.513938583 2.276840174
## [115] 0.534065478 0.123837480 1.585094819 2.406106820 0.044846019 0.215650008
## [121] 3.154420249 1.135842834 0.083768488 0.004703472 0.886198577 0.181100604
## [127] 1.885970451 0.614563497 0.188674822 0.782507674 0.605169706 0.209813308
## [133] 0.002937459 1.180191358 0.542888348 0.146255728 1.749970158 2.509768443
## [139] 0.100920508 1.117283793 2.184544215 0.033616582 0.379139593 1.025251864
## [145] 0.763781032 1.327514948 0.599233748 0.227472894 0.156578282 1.028648106
## [151] 1.654259451 0.020247590 2.939726710 1.526118584 1.761713255 0.989105131
## [157] 1.107438386 0.247064909 0.913307090 1.035017111 1.677516116 0.249879052
## [163] 1.853567611 0.078456511 1.955661504 2.743604811 1.014204727 0.203136326
## [169] 0.380817804 2.025386030 1.353249093 1.541737251 2.046755042 0.531858000
## [175] 0.859151616 0.807585009 0.362035599 0.576160377 0.384715960 0.097823602
## [181] 0.245312035 1.001646349 0.290784892 0.042685724 0.125133081 0.875570170
## [187] 0.293397989 0.332111243 0.108829783 0.925993629 0.486774253 0.255440474
## [193] 0.283391591 0.007423815 3.038839482 1.274283722 1.079180986 0.364558517
## [199] 0.363213272 0.937259512 4.141617169 1.426626669 0.003855170 0.021847712
## [205] 2.193468445 0.810732450 1.606564000 0.747209631 0.140596213 0.959362685
## [211] 1.014379286 0.538459054 0.146122624 1.683238983 0.626737911 0.175006001
## [217] 2.021002788 0.658486649 0.901204665 0.533217198 2.648669486 0.010089667
## [223] 0.888573334 0.999909196 2.275802547 0.416164389 4.077664440 0.648120930
## [229] 0.022862800 0.533415330 0.375658741 0.960925848 0.840321732 0.168376023
## [235] 0.393571205 0.522364222 0.788068366 5.185306476 1.381546931 0.195633837
## [241] 0.930577932 1.401835915 0.056733040 1.367000187 0.034357921 5.056831049
## [247] 1.869652938 0.268501814 0.036944580 0.356116834 0.922037104 1.282704461
## [253] 1.612663753 1.212382972 0.774563807 0.174853586 2.162361571 1.176383155
## [259] 1.679718014 0.977289409 1.343591632 1.637462253 0.258253740 0.538793211
## [265] 0.422923934 0.324128359 0.179189930 0.856519977 0.258756852 0.679944483
## [271] 2.534299222 4.346810031 2.375531258 1.034216384 0.871891405 0.281884649
## [277] 1.479360468 0.291099848 2.354625473 0.500459602 0.757429209 0.058687221
## [283] 3.214877605 0.538472425 0.067520921 0.123746263 0.342536365 1.366433711
## [289] 0.507015876 1.352848762 0.188063089 0.972612564 0.292962443 1.125253649
## [295] 0.281730362 0.006894395 0.311117757 1.696230866 0.337262157 2.674654038
## [301] 2.040097430 0.282466181 0.159591021 1.442411700 0.822280494 0.095961135
## [307] 0.085233874 0.811972408 0.201213297 0.006877973 0.730700186 0.975567835
## [313] 1.457464071 0.271949677 1.414410498 0.354886161 0.262861232 0.548496144
## [319] 0.322923740 1.157157970 1.091936295 1.079990232 3.467170929 0.734894925
## [325] 0.484210134 1.611359620 1.069640529 1.215961341 0.441634141 1.344632569
## [331] 0.409073096 1.964519331 0.055627059 0.201459948 1.164853759 0.860892313
## [337] 1.227432236 2.241103763 0.363792635 1.851117676 0.363620679 0.488157181
## [343] 0.591643473 0.002853942 0.041309516 0.409424766 0.896997631 0.178491879
## [349] 0.320232774 0.183505994 0.717615007 0.964424027 0.136013242 1.185258089
## [355] 1.195063130 0.919474385 0.248744692 1.703423193 1.468601281 2.391771296
## [361] 2.298978468 0.380074862 0.453761603 1.075840611 0.288109577 1.252166872
## [367] 2.368281964 0.298001587 0.152347838 0.274216248 0.925339042 2.633530935
## [373] 0.452952198 0.091204378 0.090255007 0.300461666 1.362060921 0.949781351
## [379] 1.516039835 0.580938138 0.681652136 1.930054488 0.750637468 0.538109604
## [385] 0.800660030 0.010156279 1.462354690 0.702898940 0.173457311 2.907185867
## [391] 0.220319203 1.294211796 0.006882564 2.158826217 1.258398121 0.112678877
## [397] 1.419165148 1.664425498 0.202871650 0.409948065 0.589842578 2.483378203
## [403] 2.725517174 1.429466955 1.618641338 1.957371045 0.819811898 2.373530315
## [409] 0.276878203 0.418878757 0.326334196 0.493825065 0.836786376 0.716113006
## [415] 0.266817289 0.075419495 0.643646488 0.323015501 1.229962350 1.222492641
## [421] 0.148070381 0.934480870 0.386060306 1.749779581 0.828426102 0.024239325
## [427] 0.052682838 0.227232729 1.060826354 0.092426731 0.706268740 0.152262782
## [433] 0.138640138 3.479452094 3.919554969 0.535693087 0.561879803 0.386604854
## [439] 3.407015212 0.318613754 0.525341908 0.974482785 1.309073509 2.816516355
## [445] 0.791348741 0.301493165 0.597657797 1.158352327 1.217097520 0.347095296
## [451] 0.994886397 0.330743778 0.046128329 0.650082695 1.676638995 1.662970237
## [457] 0.062749664 0.062663895 1.814435055 1.455022140 0.240549569 0.137826571
## [463] 1.331981364 0.030138656 0.440126808 2.790490329 0.018426302 0.209063318
## [469] 0.130471929 0.959680759 1.073109240 0.064121366 0.886604845 1.648310723
## [475] 2.136355953 0.286415619 0.211189597 0.308228367 0.911031467 3.843314519
## [481] 1.221726364 0.705067208 0.065387626 0.336442803 0.861075770 0.762421029
## [487] 1.811901370 0.690021020 1.276629670 1.775082851 0.449190161 1.098781449
## [493] 0.558767061 0.887108277 6.913914663 1.904897675 0.486730716 0.269962313
## [499] 0.011273422 3.508060850 0.030643609 0.773884283 2.360069701 0.264295645
## [505] 3.043354258 1.174715791 1.614447486 0.105010427 0.985711546 0.863946968
## [511] 0.851114983 0.406221442 0.290028877 0.508447916 0.365140777 0.726781664
## [517] 0.432430475 1.023938577 1.106219130 0.206587442 0.188466378 1.046167112
## [523] 0.703653164 1.334487346 2.048021210 0.007184121 0.015312859 0.714203096
## [529] 0.418853519 0.326705429 0.215291700 0.234452931 0.191111194 0.353744257
## [535] 0.052676036 1.682166783 0.762755696 0.663324865 0.255527909 0.993801328
## [541] 0.482666608 0.818289843 0.748150095 3.343874690 0.253396397 1.484284404
## [547] 1.229218208 0.653512456 0.032320001 0.851422464 0.207722165 0.868853079
## [553] 0.090178926 0.540614723 1.522788349 0.082324972 0.315002738 0.152263275
## [559] 0.009982317 0.009542417 0.805018115 0.939373421 0.093547865 1.444919412
## [565] 2.999925190 1.028731432 0.611898188 0.595884708 0.275451250 1.179337796
## [571] 0.057617147 2.260008132 0.433699537 1.459082149 1.167049509 1.198459028
## [577] 1.042410887 1.326478491 1.599270420 3.171625539 0.467682219 1.317321964
## [583] 0.757728708 0.254089445 0.042094110 0.347776283 1.638277685 0.531915124
## [589] 0.590886130 0.985273119 2.559615051 0.311930387 1.753950281 3.821610292
## [595] 1.606994293 3.414207242 1.396476554 0.154711188 2.550641527 0.473291897
## [601] 0.787454541 0.798988737 2.930657173 0.434372534 2.011298498 2.957167705
## [607] 1.014232732 0.003870609 2.428646477 0.323497290 0.350230479 0.343391426
## [613] 0.013530413 2.426137486 0.095411629 1.127213225 0.076443654 0.237575476
## [619] 1.396455688 2.621201856 1.486429686 1.354714650 0.125580367 0.171411897
## [625] 2.206398269 0.541045983 0.170954720 0.544982498 1.130965112 0.389560234
## [631] 0.044896741 0.920175614 1.001472034 3.510942673 1.724236222 0.799283248
## [637] 0.892405530 0.001221019 0.327666050 0.663648949 0.537716309 0.207044991
## [643] 0.116719578 0.050875649 1.813457947 0.528726649 1.889309706 1.182615791
## [649] 0.512795420 2.483031807 0.312786317 0.431184944 0.145439061 0.670712261
## [655] 0.815644748 0.173024023 0.240155634 3.386491159 0.372606962 0.386485986
## [661] 0.600558908 0.753555671 0.866569940 1.887957845 1.612392856 0.385010628
## [667] 3.178200319 0.811174001 0.194665515 0.040510894 1.327023338 0.018574630
## [673] 3.034031913 0.181056901 1.878758516 1.290626207 0.814116009 0.406901503
## [679] 1.589823347 1.091722032 2.166624246 0.035843973 0.021328700 0.262985813
## [685] 1.225945321 0.548129113 1.354417407 0.305142350 0.331245507 0.660171941
## [691] 0.319184593 5.341488110 3.121107772 0.298250941 0.225764026 0.146661989
## [697] 1.195087667 0.326627303 1.445711330 0.161544734 0.875200223 2.007736832
## [703] 0.447957782 1.383930847 0.286754823 0.258327143 3.104308677 1.003975922
## [709] 0.411978014 0.272255884 0.729888000 1.231600305 0.742992936 0.329314662
## [715] 1.367988617 2.271748012 2.562035082 1.829922594 0.048325880 1.921490693
## [721] 0.030779289 1.118065087 0.652738731 0.948141651 0.042789645 1.611757416
## [727] 3.149804297 1.049853368 0.472512591 0.759269592 2.675101524 1.186928761
## [733] 2.340450669 1.468344992 0.189287710 0.384564055 0.042421814 1.295897773
## [739] 1.378443279 0.058368030 0.377738497 0.202789381 1.665848713 1.023938063
## [745] 0.866356032 1.154110781 0.952806686 1.424881778 0.122474311 0.074749563
## [751] 0.233864762 1.746447408 3.907212696 1.147675984 0.217035970 1.845235484
## [757] 0.485298559 1.377922595 2.189554816 2.336025007 0.818343296 1.239021504
## [763] 0.716431124 3.896715691 1.003276746 1.062389669 0.086491317 2.853304320
## [769] 0.186334381 0.297361657 0.079686103 0.551928530 0.123513044 1.355256060
## [775] 1.384431900 0.199940535 1.340315145 2.324097903 0.310747195 2.835645050
## [781] 1.621206777 0.195916037 5.928787931 0.987223253 0.111157073 0.114556853
## [787] 0.977050248 1.595412660 0.658595042 0.751691665 0.760163669 0.831639825
## [793] 0.731685732 0.081154484 0.450770405 1.863194868 1.869467357 0.004470726
## [799] 0.783400507 0.311796418 0.015866147 0.944208318 1.924683802 0.072693936
## [805] 0.370078505 0.205107463 0.301165293 0.507970278 0.195612118 2.512251625
## [811] 0.395625348 3.767331039 0.172815875 0.155721801 1.067908792 0.499149081
## [817] 0.582046979 0.484523243 0.994926162 0.038526449 0.966945003 0.531848948
## [823] 0.245262534 1.260578373 0.381004519 0.031621955 0.298605137 0.737839528
## [829] 0.848268746 0.199706658 1.137158709 0.340871417 1.249553143 0.188402232
## [835] 0.343022641 1.296873004 2.323477173 1.716248352 2.977953836 1.216451450
## [841] 0.426876098 0.717738716 1.314057820 0.608802274 0.589857700 1.235274937
## [847] 0.492920472 0.834465496 3.454822101 0.170387822 0.001101778 0.484944848
## [853] 1.162882688 4.779267180 0.072724886 0.501675976 3.767823731 0.396806959
## [859] 0.180251282 0.590650414 1.129031577 1.053600450 0.367040028 0.677024949
## [865] 1.272302362 0.017703613 0.368600645 0.816604764 0.361036696 0.316263078
## [871] 1.377409234 0.208228895 1.295767948 0.072732708 0.171924513 1.326270586
## [877] 0.080543731 0.122273415 1.062499761 0.692721805 2.363344701 2.079689881
## [883] 2.347790736 0.241071353 0.767746503 0.845300911 1.381386184 0.478546026
## [889] 0.249703533 0.148412415 2.674806869 1.464116676 1.526479682 1.853327747
## [895] 3.906786581 0.752318604 3.167315803 2.890103620 1.701728875 0.169602477
## [901] 1.388475226 0.412998126 0.483784550 0.165322918 0.374335250 1.573604645
## [907] 3.883621758 1.470287960 0.384173875 0.931703064 1.088071425 0.330835144
## [913] 0.272081090 0.626265683 0.064873331 0.734829997 0.221942827 0.255880286
## [919] 1.637197228 1.353286900 0.823171206 1.215466491 0.573536313 0.352374416
## [925] 1.331808099 0.019748565 1.241920888 0.824305838 0.678015933 0.151650460
## [931] 0.290718329 1.142886595 0.214460525 0.590568435 0.945969671 2.153900487
## [937] 1.292190619 0.853468463 0.552925405 0.056283902 2.150040493 0.381703402
## [943] 0.552974408 0.284516611 1.060204199 0.551070163 0.820456910 1.387860937
## [949] 0.291834390 0.624202040 1.852705782 0.454960676 0.196523476 0.933050219
## [955] 0.557982556 0.514025330 0.304553792 0.287577556 1.218413685 2.120534640
## [961] 0.131755749 2.559634948 2.392109845 1.206953078 0.916019946 0.407663456
## [967] 0.398831311 0.480872198 0.835490766 0.822110308 0.504823956 0.109540232
## [973] 0.544454308 0.024155050 0.219814293 0.321366773 0.357387638 1.386412343
## [979] 0.294911402 1.758060484 0.640128634 1.220713076 0.572765465 0.160177202
## [985] 0.417226025 0.604713706 0.788782945 0.040393844 0.944200873 0.102527181
## [991] 1.923594519 0.207728734 0.051125568 1.381293993 0.431669028 1.609824568
## [997] 1.173211854 0.760633629 0.626049954 0.277850234hist(exp_data, breaks = 30, main = "Histogram Distribusi Eksponensial", xlab = "Nilai", col = "lightcoral")
Distribusi eksponensial digunakan untuk memodelkan waktu antara kejadian dalam proses Poisson. Histogram menunjukkan distribusi yang miring ke kanan, dengan nilai-nilai yang semakin kecil semakin sering muncul.
Latihan Studi Kasus
Studi Kasus 1: Simulasi Pendapatan Bulanan
# Simulasi pendapatan bulanan
set.seed(123)
n_employees <- 500
mean_income <- 10000000
sd_income <- 2000000income_data <- rnorm(n_employees, mean = mean_income, sd = sd_income)
income_data## [1] 8879049 9539645 13117417 10141017 10258575 13430130 10921832 7469878
## [9] 8626294 9108676 12448164 10719628 10801543 10221365 8888318 13573826
## [17] 10995701 6066766 11402712 9054417 7864353 9564050 7947991 8542218
## [25] 8749921 6626613 11675574 10306746 7723726 12507630 10852928 9409857
## [33] 11790251 11756267 11643162 11377281 11107835 9876177 9388075 9239058
## [41] 8610586 9584165 7469207 14337912 12415924 7753783 9194230 9066689
## [49] 11559930 9833262 10506637 9942906 9914259 12737205 9548458 13032941
## [57] 6902494 11169227 10247708 10431883 10759279 8995353 9333585 7962849
## [65] 7856418 10607057 10896420 10106008 11844535 14100169 9017938 5381662
## [73] 12011477 8581598 8623983 12051143 9430454 7558565 10362607 9722217
## [81] 10011528 10770561 9258680 11288753 9559027 10663564 12193678 10870363
## [89] 9348137 12297615 11987008 11096794 10477463 8744188 12721305 8799481
## [97] 14374666 13065221 9528599 7947158 8579187 10513767 9506616 9304915
## [105] 8096763 9909945 8430191 6664116 9239547 11837993 8849306 11215929
## [113] 6764235 9888876 11038814 10602307 10211352 8718588 8300591 7951742
## [121] 10235293 8105051 9018885 9487816 13687724 8696100 10470773 10155922
## [129] 8076287 9857384 12889102 10903008 10082466 9155006 5893506 12262674
## [137] 7078720 11479895 13818207 7112214 11403569 9475605 6855712 6970665
## [145] 6796928 8938187 7076489 11375834 14200218 7425939 11575478 11538084
## [153] 10664405 7983247 9761095 9439209 11125979 9255122 11953947 9250838
## [161] 12105423 7901646 7479690 16482080 9166285 10596455 11273139 9032439
## [169] 11033724 10737929 9569239 10130586 9931865 14256904 8517328 7808007
## [177] 10075577 10620961 10873047 9083269 7873348 12526370 9300699 8268974
## [185] 9527441 9605648 12219841 10169475 11508108 9001416 10428891 9350628
## [193] 10189167 8209273 7378397 13994427 11201418 7497457 8777668 7629040
## [201] 14397621 12624826 9469710 11086388 9171320 9047506 8422794 8810765
## [209] 13301815 9891944 10238490 10487375 12464952 8967872 8014986 13351394
## [217] 9117674 8553868 7527454 7430569 8852053 11235972 12219696 11415177
## [225] 9272685 10119500 8590807 8565564 11769301 7968815 13910588 9819361
## [233] 10429078 8522945 8851223 7365968 9634149 10837965 10648609 8436927
## [241] 8422756 8995603 12992121 7725393 9641897 13804724 9798050 7280319
## [249] 8670461 10970920 9248794 8876247 9312166 10180993 13197018 9822870
## [257] 12161599 11261508 9772720 6934196 8957765 9020259 10094309 12600397
## [265] 14586158 13095162 9733698 6486945 9222440 10178414 11690026 11925056
## [273] 11368619 7209451 11699286 9106886 10349605 10149102 10856334 10049350
## [281] 6665050 11472992 10772053 9468697 10236289 10268077 10442039 13281692
## [289] 9561899 10336131 12336768 12108362 12290526 8845064 14004965 10133402
## [297] 13733704 7298195 10041967 12499829 8569516 8494622 8122923 7894973
## [305] 9125681 10662358 5971579 10423961 12473350 14075148 12602352 11513550
## [313] 6546539 8796987 9295907 11407048 9788657 7482703 13368871 11822783
## [321] 10474861 12436217 7322451 11321641 8954175 11367491 9878356 11265921
## [329] 12671035 10014580 12035117 7623132 8556791 13038435 10754776 5895554
## [337] 7271925 9598438 11731559 9796233 11248375 11918011 13342110 10112033
## [345] 9896036 6493525 10198655 8856300 8051981 9640188 12029886 6014503
## [353] 9145441 10233275 8213585 10667806 10822860 9933928 5068204 15142916
## [361] 9589401 11302387 10547533 12049346 11635319 9580414 10756336 8109182
## [369] 11713846 9077923 14833547 6697902 9072026 11650760 11020265 8821038
## [377] 8006439 10288951 9971385 6419438 10069102 10380461 10349453 7889966
## [385] 10952267 12757140 10912473 7728823 9128709 10692207 8705909 5684707
## [393] 11768502 8341045 8852879 13007801 8451710 11691463 7478634 9290915
## [401] 9852888 7662697 8730503 9942317 11341392 6698907 9300492 11512813
## [409] 8922382 10454584 10984457 10535670 11306515 9754583 9172647 4713702
## [417] 9814118 10860569 11070798 8889443 13559006 10572849 10252632 12544534
## [425] 8563068 9099323 14794905 10022258 13267137 7122987 9618966 10756848
## [433] 10600077 7988727 10038519 7845159 11425407 12169550 5550025 12471387
## [441] 7517911 10909539 11319805 9600220 8709772 10330642 10877637 11766606
## [449] 5895326 6727241 12860805 12093258 10870578 11430357 11834350 4678154
## [457] 12220554 9030025 10461234 9409684 11743930 9303055 11037008 9218630
## [465] 7814426 12420021 11481800 13448524 10130308 12250005 13950838 9437036
## [473] 7354098 9521297 9571918 10303361 13424610 9347712 10746009 9544632
## [481] 10040901 10628115 12656429 10242637 11425685 11557720 11829547 8851211
## [489] 13253762 9238087 9788432 12808101 12588168 7820016 8253858 7283842
## [497] 10363694 10329682 10728229 11104315# 1. Rata-rata pendapatan simulasi
mean_simulated <- mean(income_data)
cat("Rata-rata pendapatan simulasi:", mean_simulated, "\n")## Rata-rata pendapatan simulasi: 10069181## Rata-rata pendapatan simulasi: 10069181
# 2. Probabilitas pendapatan di atas Rp 12.000.000
prob_above_12m <- sum(income_data > 12000000) / n_employees
cat("Probabilitas pendapatan di atas Rp 12.000.000:", prob_above_12m, "\n")## Probabilitas pendapatan di atas Rp 12.000.000: 0.164Probabilitas pendapatan di atas Rp 12.000.000: 0.164
- Simulasi ini memodelkan pendapatan bulanan karyawan dengan distribusi normal.
- Rata-rata pendapatan simulasi mendekati mean yang diberikan (Rp 10.000.000).
- Probabilitas pendapatan di atas Rp 12.000.000 dihitung dengan menghitung proporsi data yang melebihi nilai tersebut.
Studi Kasus 2: Simulasi Jumlah Pelanggan
# Simulasi jumlah pelanggan
set.seed(123)
n_days <- 30
lambda_customers <- 50customers_data <- rpois(n_days, lambda_customers)
customers_data## [1] 46 58 38 50 62 53 41 37 58 52 52 50 46 59 55 49 44 42 47 44 57 48 41 41 47
## [26] 47 49 56 48 52# 1. Rata-rata jumlah pelanggan simulasi
mean_customers <- mean(customers_data)
cat("Rata-rata jumlah pelanggan simulasi:", mean_customers, "\n")## Rata-rata jumlah pelanggan simulasi: 48.96667## Rata-rata jumlah pelanggan simulasi: 48.96667
# 2. Probabilitas jumlah pelanggan lebih dari 60
prob_above_60 <- sum(customers_data > 60) / n_days
cat("Probabilitas jumlah pelanggan lebih dari 60:", prob_above_60, "\n")## Probabilitas jumlah pelanggan lebih dari 60: 0.03333333Probabilitas jumlah pelanggan lebih dari 60: 0.03333333
- Simulasi ini memodelkan jumlah pelanggan yang datang ke restoran setiap hari dengan distribusi Poisson.
- Rata-rata jumlah pelanggan simulasi mendekati lambda (50).
- Probabilitas jumlah pelanggan lebih dari 60 dihitung dengan menghitung proporsi data yang melebihi nilai tersebut.
Tugas Tambahan
1.Buat simulasi untuk distribusi diskrit dan distribusi kontinu. 2.Buat studi kasus sendiri yang melibatkan simulasi variabel random dari distribusi yang telah dipelajari.
1. Simulasi Distribusi Diskrit dan Kontinu
# A. Simulasi Distribusi Binomial
set.seed(123)
n <- 1000 # jumlah simulasi
size <- 10 # jumlah percobaan
prob <- 0.3 # peluang suksesbinom_data <- rbinom(n, size, prob)
binom_data## [1] 2 4 3 5 5 1 3 5 3 3 6 3 4 3 1 5 2 1 2 6 5 4 3 7 4 4 3 3 2 1 6 5 4 4 0 3 4
## [38] 2 2 2 1 3 3 2 2 1 2 3 2 5 1 3 4 1 3 2 1 4 5 2 4 1 3 2 4 3 4 4 4 3 4 3 4 0
## [75] 3 2 2 3 2 1 2 4 3 4 1 3 6 5 5 2 1 4 2 4 2 2 4 1 3 3 3 2 3 6 3 5 5 3 3 1 5
## [112] 2 1 5 4 1 3 6 3 3 3 2 2 2 2 6 2 1 1 4 3 5 4 4 3 4 4 4 6 3 2 3 0 2 4 2 2 1
## [149] 2 4 4 3 3 2 1 3 3 2 3 2 3 2 4 2 2 3 4 2 3 2 3 2 5 4 4 3 2 3 5 3 4 2 4 2 3
## [186] 3 2 3 5 5 2 2 6 3 5 3 3 4 2 3 2 6 3 3 3 5 2 2 2 2 3 2 2 4 1 4 2 3 4 5 2 6
## [223] 4 4 1 3 3 3 4 5 3 3 3 1 2 3 2 4 2 4 3 4 2 3 2 4 3 6 6 4 2 2 3 2 3 4 2 3 3
## [260] 5 5 5 4 5 3 3 2 2 0 3 5 0 1 2 4 4 6 3 1 3 4 1 3 2 1 3 1 2 1 4 2 1 1 5 4 4
## [297] 6 1 1 4 4 0 4 4 3 3 2 0 3 3 3 3 4 1 2 4 4 2 2 5 5 2 1 4 1 1 8 1 2 5 3 2 4
## [334] 4 2 3 4 3 3 6 3 1 3 2 3 2 6 3 2 3 1 6 3 2 3 6 4 3 2 4 1 2 5 2 2 4 2 0 3 3
## [371] 3 2 5 3 3 6 4 2 2 6 3 4 2 4 3 5 2 2 1 2 6 2 4 4 1 2 2 1 1 7 6 1 5 3 3 3 4
## [408] 1 2 4 2 4 2 3 2 2 5 2 3 0 3 3 3 2 5 3 2 3 2 5 6 3 2 7 2 3 1 3 1 2 2 3 4 2
## [445] 5 4 4 1 2 4 3 2 2 1 2 5 4 4 2 1 6 3 3 2 3 2 2 4 2 4 1 4 3 4 4 3 3 0 2 4 2
## [482] 4 2 4 5 1 1 4 3 4 6 1 3 4 1 7 2 1 4 4 2 2 2 1 2 2 3 3 5 2 3 1 5 3 4 2 3 4
## [519] 1 4 2 2 2 1 3 4 5 0 6 2 5 4 3 4 1 4 2 4 3 2 4 1 4 2 4 6 1 5 4 3 2 2 3 5 3
## [556] 3 4 3 3 0 1 5 4 2 4 4 3 4 3 5 3 5 3 3 5 4 3 0 3 3 5 4 5 2 5 2 2 4 6 3 3 1
## [593] 6 1 1 5 3 2 5 1 2 4 2 2 2 4 1 4 2 2 3 1 0 7 3 4 5 4 6 1 5 5 4 2 1 2 2 5 2
## [630] 2 4 4 3 4 1 2 6 4 4 3 3 4 5 3 1 3 4 1 2 3 4 5 3 3 4 5 5 1 3 2 4 4 1 4 2 1
## [667] 5 2 3 2 0 3 1 2 3 3 4 2 3 3 4 4 4 5 5 3 3 2 3 2 1 3 3 1 2 3 2 1 3 2 4 2 0
## [704] 5 2 3 3 2 2 3 3 3 4 4 3 3 0 0 7 1 3 4 4 5 3 4 4 3 4 0 3 5 3 5 2 2 4 6 2 0
## [741] 7 1 1 3 2 3 4 1 4 5 2 2 2 1 3 3 3 5 1 2 3 2 3 3 3 1 1 3 4 4 5 4 4 4 1 2 2
## [778] 3 3 4 2 4 0 4 4 3 2 0 2 5 5 3 2 3 4 3 1 2 2 2 3 2 1 1 2 6 3 3 4 4 1 4 2 5
## [815] 4 1 2 1 3 3 4 2 3 7 4 1 3 1 2 4 0 4 3 4 4 3 4 4 2 1 6 4 3 1 3 2 5 0 2 1 2
## [852] 4 7 4 3 3 5 1 1 5 3 1 3 3 3 2 2 1 3 3 1 2 4 3 5 4 0 4 5 3 1 5 4 1 4 5 3 2
## [889] 2 2 3 4 4 3 4 4 0 2 1 4 5 3 5 3 4 3 4 5 4 1 0 1 5 3 2 6 3 3 3 2 4 1 2 5 4
## [926] 4 2 3 1 4 5 2 4 3 6 2 3 5 5 0 6 3 3 3 1 2 3 2 3 2 3 2 6 1 2 6 3 1 3 3 3 3
## [963] 3 1 5 3 3 1 4 3 4 1 4 2 2 3 5 0 2 3 3 4 5 3 5 4 1 2 2 3 1 2 3 3 3 4 3 3 4
## [1000] 1# Visualisasi
hist(binom_data, breaks=10, col="skyblue", main="Simulasi Distribusi Binomial", xlab="Jumlah Sukses")
# B. Simulasi Distribusi Normal
set.seed(123)
n <- 1000
mean <- 50
sd <- 10normal_data <- rnorm(n, mean, sd)
normal_data## [1] 44.39524 47.69823 65.58708 50.70508 51.29288 67.15065 54.60916 37.34939
## [9] 43.13147 45.54338 62.24082 53.59814 54.00771 51.10683 44.44159 67.86913
## [17] 54.97850 30.33383 57.01356 45.27209 39.32176 47.82025 39.73996 42.71109
## [25] 43.74961 33.13307 58.37787 51.53373 38.61863 62.53815 54.26464 47.04929
## [33] 58.95126 58.78133 58.21581 56.88640 55.53918 49.38088 46.94037 46.19529
## [41] 43.05293 47.92083 37.34604 71.68956 62.07962 38.76891 45.97115 45.33345
## [49] 57.79965 49.16631 52.53319 49.71453 49.57130 63.68602 47.74229 65.16471
## [57] 34.51247 55.84614 51.23854 52.15942 53.79639 44.97677 46.66793 39.81425
## [65] 39.28209 53.03529 54.48210 50.53004 59.22267 70.50085 45.08969 26.90831
## [73] 60.05739 42.90799 43.11991 60.25571 47.15227 37.79282 51.81303 48.61109
## [81] 50.05764 53.85280 46.29340 56.44377 47.79513 53.31782 60.96839 54.35181
## [89] 46.74068 61.48808 59.93504 55.48397 52.38732 43.72094 63.60652 43.99740
## [97] 71.87333 65.32611 47.64300 39.73579 42.89593 52.56884 47.53308 46.52457
## [105] 40.48381 49.54972 42.15096 33.32058 46.19773 59.18997 44.24653 56.07964
## [113] 33.82117 49.44438 55.19407 53.01153 51.05676 43.59294 41.50296 39.75871
## [121] 51.17647 40.52525 45.09443 47.43908 68.43862 43.48050 52.35387 50.77961
## [129] 40.38143 49.28692 64.44551 54.51504 50.41233 45.77503 29.46753 61.31337
## [137] 35.39360 57.39948 69.09104 35.56107 57.01784 47.37803 34.27856 34.85332
## [145] 33.98464 44.69093 35.38244 56.87917 71.00109 37.12970 57.87739 57.69042
## [153] 53.32203 39.91623 48.80547 47.19605 55.62990 46.27561 59.76973 46.25419
## [161] 60.52711 39.50823 37.39845 82.41040 45.83142 52.98228 56.36570 45.16219
## [169] 55.16862 53.68965 47.84619 50.65293 49.65933 71.28452 42.58664 39.04004
## [177] 50.37788 53.10481 54.36523 45.41635 39.36674 62.63185 46.50350 41.34487
## [185] 47.63720 48.02824 61.09920 50.84737 57.54054 45.00708 52.14445 46.75314
## [193] 50.94584 41.04637 36.89198 69.97213 56.00709 37.48729 43.88834 38.14520
## [201] 71.98810 63.12413 47.34855 55.43194 45.85660 45.23753 42.11397 44.05383
## [209] 66.50907 49.45972 51.19245 52.43687 62.32476 44.83936 40.07493 66.75697
## [217] 45.58837 42.76934 37.63727 37.15284 44.26027 56.17986 61.09848 57.07588
## [225] 46.36343 50.59750 42.95404 42.82782 58.84650 39.84407 69.55294 49.09680
## [233] 52.14539 42.61472 44.25611 36.82984 48.17075 54.18982 53.24304 42.18464
## [241] 42.11378 44.97801 64.96061 38.62696 48.20948 69.02362 48.99025 36.40159
## [249] 43.35231 54.85460 46.24397 44.38124 46.56083 50.90497 65.98509 49.11435
## [257] 60.80799 56.30754 48.86360 34.67098 44.78883 45.10130 50.47154 63.00199
## [265] 72.93079 65.47581 48.66849 32.43473 46.11220 50.89207 58.45013 59.62528
## [273] 56.84309 36.04726 58.49643 45.53443 51.74803 50.74551 54.28167 50.24675
## [281] 33.32525 57.36496 53.86027 47.34348 51.18145 51.34039 52.21019 66.40846
## [289] 47.80950 51.68065 61.68384 60.54181 61.45263 44.22532 70.02483 50.66701
## [297] 68.66852 36.49097 50.20984 62.49915 42.84758 42.47311 40.61461 39.47487
## [305] 45.62840 53.31179 29.85790 52.11980 62.36675 70.37574 63.01176 57.56775
## [313] 32.73270 43.98493 46.47954 57.03524 48.94329 37.41351 66.84436 59.11391
## [321] 52.37430 62.18109 36.61226 56.60820 44.77088 56.83746 49.39178 56.32961
## [329] 63.35518 50.07290 60.17559 38.11566 42.78396 65.19218 53.77388 29.47777
## [337] 36.35963 47.99219 58.65779 48.98117 56.24187 59.59005 66.71055 50.56017
## [345] 49.48018 32.46763 50.99328 44.28150 40.25990 48.20094 60.14943 30.07252
## [353] 45.72721 51.16637 41.06792 53.33903 54.11430 49.66964 25.34102 75.71458
## [361] 47.94701 56.51193 52.73766 60.24673 58.17659 47.90207 53.78168 40.54591
## [369] 58.56923 45.38962 74.16773 33.48951 45.36013 58.25380 55.10133 44.10519
## [377] 40.03219 51.44476 49.85693 32.09719 50.34551 51.90230 51.74726 39.44983
## [385] 54.76133 63.78570 54.56236 38.64412 45.64355 53.46104 43.52954 28.42354
## [393] 58.84251 41.70522 44.26440 65.03901 42.25855 58.45732 37.39317 46.45458
## [401] 49.26444 38.31349 43.65252 49.71158 56.70696 33.49453 46.50246 57.56406
## [409] 44.61191 52.27292 54.92229 52.67835 56.53258 48.77291 45.86323 23.56851
## [417] 49.07059 54.30285 55.35399 44.44722 67.79503 52.86424 51.26316 62.72267
## [425] 42.81534 45.49661 73.97452 50.11129 66.33568 35.61493 48.09483 53.78424
## [433] 53.00039 39.94364 50.19259 39.22579 57.12703 60.84775 27.75012 62.35693
## [441] 37.58956 54.54769 56.59903 48.00110 43.54886 51.65321 54.38819 58.83303
## [449] 29.47663 33.63621 64.30402 60.46629 54.35289 57.15178 59.17175 23.39077
## [457] 61.10277 45.15012 52.30617 47.04842 58.71965 46.51528 55.18504 46.09315
## [465] 39.07213 62.10011 57.40900 67.24262 50.65154 61.25003 69.75419 47.18518
## [473] 36.77049 47.60648 47.85959 51.51681 67.12305 46.73856 53.73005 47.72316
## [481] 50.20451 53.14058 63.28215 51.21318 57.12842 57.78860 59.14773 44.25605
## [489] 66.26881 46.19043 48.94216 64.04050 62.94084 39.10008 41.26929 36.41921
## [497] 51.81847 51.64841 53.64115 55.52158 43.98107 40.06301 60.26785 57.51061
## [505] 34.90833 49.04853 41.04052 29.29249 51.50120 49.20788 49.02631 52.16153
## [513] 58.82465 52.05598 43.83564 42.65201 48.68197 53.10017 39.60320 48.15691
## [521] 59.67267 48.91720 43.01579 47.24055 61.14649 55.50044 62.36676 51.39098
## [529] 54.10275 44.41543 56.05371 44.93666 35.79434 51.27993 69.45851 58.00914
## [537] 61.65253 53.58856 43.91443 47.97759 47.26752 45.31300 57.04167 38.02636
## [545] 58.66366 58.64152 38.01378 56.39492 74.30227 44.42785 58.44904 42.17798
## [553] 61.10711 52.49825 66.51915 35.41029 49.48702 44.73075 48.02735 43.70421
## [561] 41.66156 55.78722 39.12419 64.84031 38.13793 51.01079 55.32989 55.86735
## [569] 46.98253 50.79502 59.61264 35.43534 42.18260 53.20402 45.55218 63.70004
## [577] 56.73254 50.72167 34.92243 50.26100 46.83584 48.97653 38.18441 54.98658
## [585] 39.61044 47.73778 53.81426 42.16484 55.82991 36.83490 21.90225 54.64968
## [593] 58.40540 47.14155 55.04126 38.44083 48.72851 30.58482 61.81181 68.59911
## [601] 60.74012 49.72653 49.66670 34.83932 57.90385 47.89266 43.43257 35.87974
## [609] 47.00238 41.50939 46.02969 37.82400 66.87589 49.83997 60.74945 23.98300
## [617] 45.46802 43.24518 37.77074 65.46609 35.84718 53.18390 58.46436 51.78190
## [625] 41.24745 59.41166 51.70588 39.36502 36.11951 70.86717 43.21497 31.44428
## [633] 55.33259 53.10230 36.46166 30.57044 48.83697 61.39396 56.36124 45.07063
## [641] 41.65812 52.71067 51.57353 56.29712 46.04202 58.99354 41.69188 46.69455
## [649] 57.40815 59.89972 30.61495 51.07190 56.08779 35.49176 54.80626 41.71826
## [657] 60.20253 55.38482 57.69052 51.20719 58.63648 63.80515 69.66248 49.71605
## [665] 27.50949 50.31526 52.05561 48.44655 55.68289 60.10678 44.82018 47.05905
## [673] 53.97842 44.49776 50.91267 30.38292 38.80100 36.72245 41.46376 43.06695
## [681] 53.82305 59.82113 42.72616 40.03161 39.58311 45.85411 47.60971 54.83618
## [689] 46.78675 29.21511 49.08566 61.87187 61.91601 42.11037 34.52223 74.58060
## [697] 48.37578 49.02549 54.20574 33.85961 42.71781 34.59558 43.06905 51.18849
## [705] 36.35291 55.89983 52.89344 40.95785 52.26325 57.48081 60.61095 47.87152
## [713] 49.06363 49.13286 64.41462 61.25072 58.34402 47.12659 53.73241 54.03290
## [721] 39.58327 32.71695 56.41830 34.70689 50.01684 52.50248 55.63867 51.89426
## [729] 42.67146 59.86366 67.38634 58.81179 30.56349 63.99576 49.43944 55.24914
## [737] 56.22033 49.03314 49.24737 60.19157 57.11602 59.90262 73.82927 56.64416
## [745] 52.07381 27.89367 76.91714 45.17323 73.74735 53.74644 65.38430 48.90290
## [753] 55.11471 52.13958 48.13879 48.79606 60.12834 47.98542 29.62318 48.04111
## [761] 55.39791 56.16456 56.16568 33.07898 53.68742 59.67859 62.76579 47.75039
## [769] 46.78107 64.87838 33.32072 45.63170 54.57462 33.82226 52.79628 68.77864
## [777] 49.95939 47.21546 54.74912 47.20928 58.13400 59.04435 50.02692 38.23308
## [785] 36.81779 44.07003 57.97381 30.41795 31.13675 43.46220 53.94395 40.86434
## [793] 58.86749 53.33370 48.29360 58.18828 53.88365 45.54065 52.31115 56.47513
## [801] 53.56283 43.41990 58.55202 61.52936 52.76275 51.44105 49.24375 71.61416
## [809] 52.76316 48.41706 24.92082 34.34718 49.22327 52.06294 52.76872 58.21507
## [817] 48.05848 62.14589 40.78484 37.91557 37.71014 57.42297 49.17080 57.89818
## [825] 47.32294 44.08108 46.31647 31.47383 38.30385 35.57965 60.54322 44.02670
## [833] 57.89460 65.16491 48.08225 52.83879 32.48932 41.81330 50.56215 52.99087
## [841] 42.40602 76.84859 45.41610 50.64244 56.49792 49.73981 43.56433 60.45306
## [849] 66.15545 49.70306 55.62267 49.02588 60.16455 38.43833 73.20860 43.96469
## [857] 35.41151 46.49082 51.46708 66.23621 59.11210 51.42458 36.10516 41.33962
## [865] 48.36715 75.53026 31.39772 61.31055 44.72766 66.65991 38.60799 51.43623
## [873] 39.00449 59.03516 64.83779 69.50721 57.97601 68.43266 62.46424 48.68125
## [881] 54.77037 40.28006 48.14798 62.20964 55.41284 54.57357 39.61869 43.95487
## [889] 42.35394 53.95296 40.09492 55.62041 38.83584 68.28530 54.60591 42.98996
## [897] 52.41046 46.47547 53.71148 52.43533 39.85886 42.08686 52.99594 66.39052
## [905] 60.84617 43.75433 58.25923 49.51432 53.01314 52.60361 75.75450 38.14711
## [913] 51.00920 32.20023 55.89836 60.96608 64.45662 30.74855 54.12769 65.93370
## [921] 45.85984 47.87849 49.63463 53.65019 56.65160 63.17821 49.04512 51.96278
## [929] 74.87998 54.31099 51.88753 36.57757 50.02856 47.78674 49.88954 44.24582
## [937] 43.13184 42.79226 47.85495 63.68133 60.49087 46.40025 33.14084 41.55417
## [945] 45.42239 51.03638 43.37393 70.06681 47.27732 37.86056 48.58738 39.94622
## [953] 51.56156 52.33634 53.55588 33.78142 52.20711 53.10450 35.78892 59.55366
## [961] 57.84171 72.99619 51.56703 50.46734 50.96586 50.69766 31.51527 33.28873
## [969] 49.22461 44.18933 50.54737 28.88792 35.01302 38.98517 59.86058 39.01510
## [977] 42.00486 50.79874 46.77254 51.46417 73.05062 38.75396 46.94530 44.83241
## [985] 65.12395 42.30515 49.17913 57.87134 39.41409 66.55176 56.75762 39.25793
## [993] 54.54578 47.86693 53.13229 49.10025 60.70516 36.48900 44.77383 47.50809# Visualisasi
hist(normal_data, breaks=30, col="lightgreen", main="Simulasi Distribusi Normal", xlab="Nilai")
2. Studi Kasus Simulasi Variabel Random
Studi Kasus: Simulasi Nilai Mahasiswa dalam Ujian dan Penilaian Akhir
- Ujian terdiri dari 10 soal pilihan ganda (Benar/Salah), peluang menjawab benar: 0.7 (Distribusi Binomial)
- Nilai ujian akhir terdiri dari gabungan tugas proyek (Distribusi Normal, rata-rata 75, SD 8)
set.seed(456)
# Simulasi jawaban soal pilihan ganda
ujian_pg <- rbinom(1000, size=10, prob=0.7) * 10 # dikalikan 10 untuk skor per soal
ujian_pg## [1] 90 80 60 50 60 80 90 80 80 70 80 80 60 60 70 60 60 70
## [19] 60 80 80 60 50 70 60 60 50 70 90 80 40 80 40 70 60 50
## [37] 40 50 50 70 80 70 90 90 90 90 70 70 70 50 50 60 80 70
## [55] 80 50 50 70 90 80 80 70 80 70 100 80 70 70 70 70 40 60
## [73] 80 90 70 100 80 80 70 70 70 70 60 50 70 60 70 70 40 60
## [91] 70 60 70 30 60 60 90 70 60 60 80 90 70 80 60 60 60 60
## [109] 60 80 60 80 80 60 50 50 60 60 60 60 100 90 80 60 70 60
## [127] 60 70 70 90 50 80 50 60 90 60 80 50 80 40 70 80 70 70
## [145] 90 70 50 30 70 50 70 50 70 90 80 40 80 60 70 70 90 60
## [163] 80 70 40 90 70 70 60 40 60 70 90 80 60 70 70 60 70 50
## [181] 80 50 80 90 50 60 50 60 80 80 70 80 60 70 60 90 50 50
## [199] 70 70 70 60 60 90 70 80 70 70 90 80 50 80 90 60 80 80
## [217] 80 90 70 90 80 60 70 80 90 80 70 90 60 80 60 80 60 40
## [235] 50 50 90 50 90 70 70 80 80 70 60 40 70 60 50 80 90 70
## [253] 70 70 90 70 70 90 70 80 70 60 50 100 80 50 80 40 90 100
## [271] 80 70 100 100 80 90 80 80 70 60 80 70 70 70 50 80 50 80
## [289] 90 70 60 70 50 70 90 60 50 80 80 60 50 90 70 60 80 70
## [307] 90 90 80 50 80 50 50 60 70 80 90 70 60 50 70 90 90 60
## [325] 80 90 70 70 60 90 90 90 40 40 70 80 100 60 70 80 80 80
## [343] 60 60 50 80 70 70 90 70 80 50 90 60 80 50 100 60 60 80
## [361] 90 80 80 70 70 60 70 90 50 70 70 80 50 70 50 60 80 70
## [379] 60 50 80 50 60 70 50 60 60 70 80 100 50 100 70 60 90 70
## [397] 70 90 80 80 70 80 50 70 70 80 80 50 70 90 90 60 80 70
## [415] 70 60 70 80 60 70 50 80 60 50 60 90 80 90 70 60 50 80
## [433] 60 80 60 60 70 80 60 100 90 60 50 80 80 80 80 60 80 50
## [451] 60 70 60 70 90 40 50 50 60 60 50 60 80 100 60 80 70 70
## [469] 70 100 60 80 80 70 60 40 40 60 90 60 40 60 80 60 90 70
## [487] 60 60 80 100 90 70 60 70 60 70 80 90 60 90 70 80 70 60
## [505] 70 50 90 80 70 90 80 80 90 70 70 80 80 80 70 90 50 80
## [523] 80 60 60 40 40 100 60 60 70 70 80 80 40 90 70 80 60 60
## [541] 100 50 70 70 90 50 50 60 60 30 70 50 60 80 70 60 90 40
## [559] 70 70 60 60 70 70 40 80 70 80 30 80 80 70 80 90 70 60
## [577] 80 40 80 60 80 60 60 60 60 60 70 50 80 70 70 60 70 90
## [595] 80 90 80 40 50 80 90 50 60 40 80 60 40 70 70 60 70 40
## [613] 90 50 90 80 50 60 70 70 70 70 60 40 70 50 60 60 60 50
## [631] 70 60 60 80 70 60 60 80 40 70 70 60 80 80 60 50 70 80
## [649] 50 70 70 70 80 50 70 50 60 90 70 80 50 60 70 70 50 80
## [667] 80 50 40 60 80 70 40 90 70 90 90 30 80 70 70 80 80 70
## [685] 80 30 50 60 90 70 70 90 70 90 80 70 70 70 70 60 50 100
## [703] 70 60 80 60 50 60 50 90 60 70 60 50 60 80 60 80 70 90
## [721] 60 70 60 70 80 50 60 90 60 50 80 90 70 50 80 80 80 40
## [739] 90 70 70 90 90 80 60 70 40 70 70 70 70 90 50 80 80 50
## [757] 70 50 60 90 90 50 50 70 80 90 90 60 70 70 70 60 60 80
## [775] 60 70 70 70 80 60 80 70 80 70 60 80 90 100 50 50 70 80
## [793] 80 60 60 70 70 90 90 60 60 50 80 70 70 90 80 80 70 60
## [811] 50 80 60 90 50 70 50 70 60 90 90 90 40 80 40 70 70 70
## [829] 30 70 70 90 60 40 100 60 80 80 80 90 80 60 80 60 70 80
## [847] 70 70 60 70 70 60 80 70 70 70 60 90 80 70 60 70 50 90
## [865] 60 80 50 90 60 90 60 80 70 90 70 80 70 60 70 70 40 70
## [883] 60 80 60 70 80 40 70 80 50 60 50 50 70 50 80 80 60 60
## [901] 80 80 50 70 70 60 80 100 70 70 60 70 60 60 70 90 60 90
## [919] 70 70 60 70 20 80 50 80 80 60 80 80 70 60 60 90 30 40
## [937] 30 60 90 80 90 60 70 60 90 60 70 80 60 70 50 70 60 80
## [955] 80 60 90 80 40 60 50 80 60 80 100 70 70 80 80 40 70 60
## [973] 60 70 70 90 70 70 80 80 80 70 90 70 80 70 60 60 70 60
## [991] 70 60 40 60 70 70 80 70 80 90# Simulasi nilai proyek
nilai_proyek <- rnorm(1000, mean=75, sd=8)
nilai_proyek## [1] 76.16232 71.94545 70.50199 85.15494 65.25189 65.49057 68.68122 76.74384
## [9] 60.55126 80.57942 74.79275 66.86542 70.61714 71.06187 89.24123 84.35760
## [17] 66.87028 51.98971 77.77452 64.04685 84.45840 83.99006 68.44209 78.28144
## [25] 81.36996 82.82443 76.65003 87.12676 79.67575 60.08842 79.54618 79.50030
## [33] 74.57648 75.67073 70.89939 73.18293 83.80935 65.64434 79.04952 62.94274
## [41] 76.21470 83.45681 74.23177 68.32027 73.22114 79.96558 73.88155 74.63253
## [49] 64.67535 62.16727 88.14897 77.21687 84.27571 76.59127 82.98986 74.42443
## [57] 56.66084 72.84011 84.32926 82.99775 76.02863 69.64855 80.93815 86.19388
## [65] 79.57646 75.41907 71.76435 71.89535 67.14401 59.37610 66.53026 71.45399
## [73] 86.07624 78.59786 78.89573 67.77406 70.63880 77.57281 79.45620 71.38931
## [81] 81.52427 76.75467 74.22943 69.42811 67.66970 73.46843 64.71478 80.61597
## [89] 84.89145 82.89759 72.81664 75.98647 80.33507 82.86475 82.50328 89.93665
## [97] 91.13873 78.99814 76.61452 77.77841 73.89927 78.11050 67.57866 69.94151
## [105] 83.91605 72.80795 70.16138 99.45646 78.00765 63.90969 83.53953 63.23432
## [113] 75.27949 74.07102 81.31839 81.16802 68.02806 88.98179 71.35865 72.67575
## [121] 51.28592 70.81011 66.05939 85.83347 75.43451 73.39755 70.12643 74.71902
## [129] 81.68636 71.06792 59.53934 65.39983 75.51700 68.76841 84.53933 80.59541
## [137] 57.12509 62.18593 73.65279 71.98297 84.12650 67.91277 72.12854 72.49469
## [145] 90.02031 62.88765 77.55767 71.98883 79.24061 74.82373 83.44088 83.51240
## [153] 80.28818 87.33345 77.83869 84.81240 74.66227 77.94691 77.24954 82.17584
## [161] 68.31970 82.02776 72.47525 61.62776 63.93550 80.71598 78.62572 69.51047
## [169] 77.60747 74.27499 69.68546 75.08110 62.73874 73.37315 66.08970 82.54073
## [177] 93.68047 83.82348 81.99483 78.60759 75.40280 68.57448 72.31988 80.23842
## [185] 65.21580 65.93910 74.27210 69.76571 77.30858 64.90549 72.88124 74.45003
## [193] 59.30163 68.52971 70.27568 74.00108 77.73750 76.92203 71.55734 81.53636
## [201] 81.85047 68.48314 67.84294 95.15101 76.35461 67.31303 63.06362 68.68535
## [209] 73.39936 87.24903 82.64603 82.96545 66.75242 73.62279 81.07354 78.93933
## [217] 70.61350 83.34684 86.98706 64.90135 69.74908 77.94684 80.04497 75.38968
## [225] 82.23676 70.96733 66.53384 76.86598 75.28399 77.34028 62.37614 83.57593
## [233] 74.05881 67.54736 73.42055 64.66891 75.88325 78.91441 82.61305 74.31453
## [241] 81.10736 73.28222 84.29489 66.40595 69.84122 88.75940 63.32852 78.02764
## [249] 68.64918 71.60267 74.35717 76.71766 61.60174 83.64704 82.47746 59.70551
## [257] 83.22440 70.05385 74.86502 64.41214 78.33994 84.62798 77.84081 89.57118
## [265] 78.11511 81.96429 92.10935 61.78521 72.29539 74.56438 92.50888 68.72028
## [273] 63.33278 72.96864 85.67769 76.25390 89.03598 75.33648 70.22431 82.26168
## [281] 71.01922 93.11518 75.40686 75.60558 78.92942 75.87242 76.87480 75.04046
## [289] 65.43819 71.35218 55.87289 79.67487 71.60878 65.80864 85.85455 73.85732
## [297] 70.00744 72.62226 74.53670 67.02722 78.41599 86.84094 87.62827 76.77046
## [305] 73.68764 73.98802 77.22269 64.16868 92.83262 78.84243 84.01130 74.14779
## [313] 73.85366 61.20392 83.44782 72.68785 72.62042 68.80748 86.63616 76.84893
## [321] 76.66801 77.61389 70.05452 85.47035 69.08202 88.83824 83.40480 57.14013
## [329] 74.60589 74.10520 68.40685 69.79202 56.49254 67.39824 78.80440 95.36396
## [337] 81.80249 75.99148 80.89025 70.38213 62.99617 75.18233 72.54339 83.01384
## [345] 66.81152 74.79779 78.18916 72.03249 80.92291 84.66500 94.82654 87.69968
## [353] 61.12471 79.93158 76.41948 67.19276 64.01643 73.08650 66.68005 64.60953
## [361] 81.25491 63.77922 61.13411 56.56197 72.29649 85.63799 77.94244 85.88677
## [369] 81.50931 72.69994 76.25205 79.29404 65.53979 63.94851 64.27670 76.72412
## [377] 72.77981 80.34819 73.11126 61.79754 78.24947 63.59833 69.46281 65.46503
## [385] 63.89651 75.66705 70.35250 72.43663 76.01955 81.74771 89.11846 73.20560
## [393] 68.74589 83.30701 90.59209 69.33126 67.71517 79.56740 81.59464 80.39592
## [401] 68.60020 60.74717 76.33740 71.48565 84.32738 68.17182 81.69056 82.16548
## [409] 72.90879 91.01835 64.10422 74.99287 77.59326 75.56978 71.78533 81.87614
## [417] 76.43127 70.90165 66.02080 75.10966 73.16014 66.84029 82.28292 67.97003
## [425] 75.88036 84.12487 66.13372 79.66941 63.24466 73.83711 80.87585 74.86143
## [433] 77.66009 72.40727 78.72214 72.05474 78.39650 72.02141 77.84031 75.86801
## [441] 78.28454 81.29454 83.27209 71.80994 62.02211 78.18382 85.83286 85.21050
## [449] 73.11216 69.42832 80.89753 84.01172 77.88177 91.19574 74.01951 74.58582
## [457] 79.30977 64.22145 64.94788 76.46329 67.49350 80.28249 73.94765 80.97910
## [465] 82.05013 81.15425 69.18373 72.41701 67.24540 80.47611 77.93979 86.92817
## [473] 79.23993 61.52544 63.43664 77.14496 77.14072 82.90123 77.57669 67.91447
## [481] 91.34170 85.15615 74.36239 63.33024 76.11678 73.13593 78.56927 71.89024
## [489] 74.72544 73.33184 78.57615 75.49260 87.84378 81.42166 67.19002 72.41668
## [497] 88.72934 70.49452 77.15010 85.36792 73.73348 75.15051 78.98597 61.59790
## [505] 99.02540 70.12881 65.43790 78.93397 81.19640 79.39989 69.73988 71.36426
## [513] 80.69188 60.64836 88.84698 84.83069 62.09799 77.24211 67.55196 68.43954
## [521] 61.86016 65.27766 81.45553 70.00686 68.91212 65.84511 80.60071 80.84105
## [529] 70.77201 71.58514 78.91001 67.19424 83.62257 75.73741 71.01927 81.88587
## [537] 74.60090 78.30479 92.84309 78.57477 77.91250 73.99289 82.16733 78.84490
## [545] 65.45151 69.50493 76.26570 74.91840 79.40417 70.64703 84.97548 75.85622
## [553] 73.07169 86.45656 86.16048 84.06659 70.58149 86.01631 71.05714 81.18455
## [561] 74.75026 79.66028 68.57404 71.78935 86.23074 71.91316 74.60172 64.82111
## [569] 91.42103 70.60473 76.29359 71.86620 82.61388 88.34188 87.92363 87.40097
## [577] 74.89757 74.73468 66.73925 74.30276 89.80917 82.04155 88.68464 77.72477
## [585] 81.67577 73.09527 73.07080 86.94892 72.70023 81.05322 79.41742 92.09848
## [593] 75.85272 77.15393 67.14787 83.22649 67.30329 67.60387 82.23799 72.92038
## [601] 88.27421 79.91868 78.27756 79.49355 70.55013 78.87544 74.91733 64.40299
## [609] 81.87380 81.07141 70.76368 72.30865 68.76291 79.56743 68.37762 86.72823
## [617] 75.37650 64.00080 69.14723 68.71466 78.63232 73.73362 66.31519 80.56595
## [625] 73.79457 77.02168 74.10442 75.18469 88.88535 87.23929 83.99418 83.71047
## [633] 88.43622 81.43254 70.75157 74.00413 69.17375 73.70514 69.44178 81.48497
## [641] 80.68555 73.12083 86.72783 74.83581 73.28700 82.72897 62.85010 85.92691
## [649] 75.79431 66.37172 77.55198 77.28629 66.28118 69.13145 74.49785 77.42604
## [657] 68.70881 69.54086 89.57638 70.69967 75.86074 71.28181 67.19123 78.20000
## [665] 74.10738 62.54554 70.76255 67.55669 67.92317 81.89929 88.85454 71.89880
## [673] 72.35904 76.16848 84.28309 65.70442 74.81886 76.47561 66.86615 74.72137
## [681] 72.19877 75.87472 76.47918 67.50638 85.23580 65.44130 67.25081 80.19494
## [689] 79.95547 76.28651 59.89702 84.85220 86.77151 70.28948 66.24172 77.66289
## [697] 82.12977 63.33401 60.33889 70.82757 72.20880 83.54624 74.95365 79.48329
## [705] 95.26654 78.49390 79.67805 68.74678 78.18806 73.00754 85.03043 89.55573
## [713] 69.69867 73.19372 75.31254 75.32966 69.57524 76.82420 70.22248 71.02663
## [721] 68.90964 63.60530 75.20498 76.94298 72.18903 76.15215 76.00924 78.71473
## [729] 74.77587 73.26644 79.05656 81.34169 85.64116 70.54994 83.05492 61.48216
## [737] 73.35930 71.59091 72.82366 72.02468 77.55449 80.93139 66.41718 72.41998
## [745] 64.87145 70.25467 73.48602 70.74109 85.12842 66.26827 85.02504 77.37691
## [753] 78.63798 83.85242 83.64560 83.87223 82.17418 65.85255 76.77559 78.23083
## [761] 78.34879 78.40283 71.69184 80.74370 60.14059 70.07932 75.79161 68.04073
## [769] 84.26021 95.62267 61.19004 80.93758 88.32583 59.91363 63.58135 67.80546
## [777] 77.01525 78.27137 68.23442 81.89581 65.62325 82.01691 83.09606 66.94383
## [785] 66.26812 79.25835 75.23368 72.21389 83.35335 69.55216 78.55406 70.75599
## [793] 75.41607 62.57603 84.34966 77.86228 58.63460 68.21836 82.22035 85.34727
## [801] 80.21976 74.30501 82.38207 79.65117 63.54552 74.59392 67.18655 79.80655
## [809] 65.19486 87.71815 73.73498 65.54924 81.38002 77.41194 63.49813 77.26216
## [817] 83.93467 80.50896 82.71191 73.01755 76.94171 80.10818 65.38227 68.96186
## [825] 82.16530 72.09622 80.82664 97.51991 71.50867 76.23851 76.42037 71.22985
## [833] 74.62198 73.25830 87.87878 80.98227 71.79697 90.05194 86.17297 87.26999
## [841] 82.79514 68.21983 78.31170 85.14714 72.03605 63.78038 71.78972 64.92310
## [849] 84.51037 73.10549 74.20369 66.68905 75.46721 76.96956 77.86569 72.66528
## [857] 62.67280 70.05910 79.84583 78.02609 81.06065 80.10414 71.29009 77.59211
## [865] 74.02915 75.27162 65.52622 87.23316 71.98622 76.24513 85.49957 98.58066
## [873] 70.78637 78.44990 82.20229 93.11170 59.62791 90.42710 76.72582 73.11131
## [881] 71.70295 70.87084 77.81617 71.48254 88.59748 74.12830 70.78612 65.88542
## [889] 96.66525 62.33212 83.47599 84.06665 72.31410 72.43459 77.70367 78.14585
## [897] 57.04405 68.83540 58.29929 78.08343 89.63965 72.91180 73.88616 70.26521
## [905] 62.73790 63.80293 67.20418 67.11530 75.78795 78.74223 78.05570 76.61062
## [913] 82.62379 95.57474 70.44930 77.81021 89.79775 63.75646 82.33602 62.69798
## [921] 66.76457 61.86690 71.67351 80.23796 77.40571 69.59053 78.61823 59.34332
## [929] 72.28809 79.48165 71.62029 76.90393 79.14296 85.65143 73.65169 83.24963
## [937] 85.81773 72.80032 62.79536 53.45183 77.40173 59.40282 82.81529 80.43859
## [945] 74.13128 68.98913 79.16717 64.80967 81.02588 73.01714 78.02240 78.16275
## [953] 71.77345 69.79481 81.15319 83.45289 85.65528 79.10719 73.89616 72.90892
## [961] 72.27369 73.44022 64.28792 86.80788 78.76696 81.05231 72.84404 71.37776
## [969] 74.70347 94.59180 83.71230 71.67279 68.98970 82.56290 64.89262 81.15411
## [977] 81.94847 69.88225 73.51864 81.28431 78.99182 85.36216 72.78627 60.59731
## [985] 81.65858 64.81876 87.66025 86.56590 61.08641 65.87990 72.85006 64.63822
## [993] 56.81288 77.52843 75.24938 96.89834 57.15814 72.64382 73.37104 62.77275# Gabungan nilai akhir (40% PG, 60% Proyek)
nilai_akhir <- 0.4 * ujian_pg + 0.6 * nilai_proyek# Visualisasi
hist(nilai_akhir, breaks=30, col="orange", main="Distribusi Nilai Akhir Mahasiswa", xlab="Nilai Akhir")
# Ringkasan statistik
summary(nilai_akhir)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 49.90 67.61 72.96 72.85 77.70 93.74