#install_github("rubenfcasal/simres")
library(simres)
#install.packages("devtools")
library(devtools)
library(remotes)
library(simres)
library(ggplot2)
#install.packages("snpar")
#library(snpar)
#install.packages("randtoolbox")

Problema 1

Los números pseudoaleatorios son secuencias de números que se comportan como una secuencia proveniente de una \(U(0,1)\) y verifican las varias propiedades, entre las que se encuentran:

I.Tienen un adistribucion uniforme II.Son estadisticamente independiente III.Son reproducibles Iv.Tienen un ciclo no repetitivo tan largo como se quiera V.Ocupan poca memopria en el ordenador

Problema 2

Generar una secuencia de números aleatorios a partir de

\[ x_{n+1}=\left(a^kx_n+\frac{a^k-1}{a-1}c\right)\mod m \] Probar:

set.seed(10)

DatR=runif(100,0,1)#u(0,1) se genera 100 datos
re <-acf(DatR,type="correlation");re

  
  Autocorrelations of series 'DatR', by lag
  
       0      1      2      3      4      5      6      7      8      9     10 
   1.000  0.029  0.086 -0.071 -0.093 -0.149  0.003 -0.091  0.026 -0.096 -0.046 
      11     12     13     14     15     16     17     18     19     20 
   0.040 -0.022  0.008  0.167  0.224  0.036 -0.041  0.062 -0.158 -0.114

No hay correlación en los datos de la muestra generada de una uniforme Como en el gráfico se ve que las barras salen del rango, Por lo tanto se concluye , que la muesta es independiente.

Se rechaza \(H_{0}\) si \(p-valor <\alpha\)

shapiro.test(DatR)
  
    Shapiro-Wilk normality test
  
  data:  DatR
  W = 0.96717, p-value = 0.01349

De acuerdo a los resultados, p-valor = 0.01, Note que \(p-valor > \alpha\), llegamos a conclusion ,que se rechaza \(H0\) Por lo tanto La muestra generada no es aleatoria

Problema 3

El Generador RANDU de IBM, está dado por: \(c=0\), \(a=2^{16}+3\) y \(m=2^{31}\)

########datos
seed<- 526
   n<- 1000# numeros aleatorios a generar (100)
   a<- 2^16+3#valor de operacion
   m<- 2^31#valor del modulo
   c<- 0

   u <- numeric(n)
   for(i in 1:n) 
     {
       seed <- (a*seed +c)%%m
       u[i] <- seed/m 
   }
#las funciones de ggplot unicamente trabaja con data frame,primero se descaga
## guardar en los resultados de muestr/m como dataframe 
## en datos  
u
     [1] 0.0160529809 0.0963156810 0.4334172579 0.7336624181 0.5012191879
     [6] 0.4043533644 0.9151474955 0.8517046934 0.8739007013 0.5780619672
    [11] 0.6032654913 0.4170352435 0.0728220390 0.6836150428 0.4462919058
    [16] 0.5252160495 0.1346691446 0.0810704222 0.2744002314 0.9167675888
    [21] 0.0310034500 0.9351124009 0.3316433551 0.5738485223 0.4583009379
    [26] 0.5851689270 0.3863051208 0.0513103819 0.8311162042 0.5249037882
    [31] 0.6693768920 0.2921272581 0.7283715205 0.7410837999 0.8911591144
    [36] 0.6772004878 0.0427708970 0.1618209919 0.5859878780 0.0595383411
    [41] 0.0833391445 0.9641897967 0.0350864800 0.5328107094 0.8810859369
    [46] 0.4912192365 0.0175419869 0.6842787927 0.9477948742 0.5282601109
    [51] 0.6394067975 0.0820997870 0.7379375445 0.6887271842 0.4909252049
    [56] 0.7470065719 0.0637125866 0.6592163732 0.3818849595 0.3583623981
    [61] 0.7132097529 0.0539969346 0.9050938310 0.9445905751 0.5216989713
    [66] 0.6288786521 0.0779811712 0.8079791581 0.1460444080 0.6044540247
    [71] 0.3123244764 0.4338606363 0.7922435300 0.8487154534 0.9621009501
    [76] 0.1341666197 0.1460911678 0.6690474292 0.6994640650 0.1753575271
    [81] 0.7569685774 0.9635937205 0.9688451262 0.1407272732 0.1247575032
    [86] 0.4819995603 0.7691798331 0.2770829564 0.7398792403 0.9455288341
    [91] 0.0142598422 0.5757995462 0.3264586972 0.7765562674 0.7212093296
    [96] 0.3382495707 0.5386134582 0.1874346128 0.2770865532 0.9756078040
   [101] 0.3598678457 0.3787368378 0.0336104156 0.7930309540 0.4556919830
   [106] 0.5968733123 0.4800120266 0.5082123494 0.7291658567 0.8010839960
   [111] 0.2440112652 0.2543116277 0.3297683792 0.6898056259 0.1709183427
   [116] 0.8172594225 0.3652914511 0.8364139041 0.7308603646 0.8574370509
   [121] 0.5668790238 0.6843406847 0.0041328939 0.8657312011 0.1571911620
   [126] 0.1515661618 0.4946765127 0.6039636200 0.1716931062 0.5944860568
   [131] 0.0216783853 0.7796958005 0.4830693351 0.8811538061 0.9392988207
   [136] 0.7054086691 0.7787626283 0.3238977483 0.9345228346 0.6920572734
   [141] 0.7416381286 0.2213133117 0.6531367125 0.9270004695 0.6837724047
   [146] 0.7596302023 0.4038295718 0.5863056099 0.8833675133 0.0234545907
   [151] 0.1904199244 0.9314282304 0.8747900622 0.8658862999 0.3222072395
   [156] 0.1402667379 0.9417352723 0.3880109927 0.8524485053 0.6225920981
   [161] 0.0635160403 0.7777673593 0.0949597927 0.5698525226 0.5644770013
   [166] 0.2581893047 0.4688428165 0.4893531566 0.7165335910 0.8950231364
   [171] 0.9213364990 0.4728107667 0.5448361086 0.0137197515 0.1787935318
   [176] 0.9492834276 0.0865587788 0.9758018246 0.0757819386 0.6724752104
   [181] 0.3528138148 0.0646059951 0.2123116376 0.6924158698 0.2436904805
   [186] 0.2304000547 0.1891860040 0.0615155315 0.6664191531 0.4448751351
   [191] 0.6714784326 0.0249943798 0.1066603856 0.4150128951 0.5301339002
   [196] 0.4456873452 0.9029189693 0.4063277086 0.3116955282 0.2132237917
   [201] 0.4740829961 0.9254838517 0.2861561449 0.3875822043 0.7500879215
   [206] 0.0122876903 0.3229348483 0.8270198768 0.0557056265 0.8910548678
   [211] 0.8449785681 0.0503775990 0.6974584805 0.7313524922 0.1109886291
   [216] 0.0837593442 0.5036584036 0.2681163242 0.0757723125 0.0415869569
   [221] 0.5675709294 0.0311429640 0.0787194194 0.1920298403 0.4437042670
   [226] 0.9339570394 0.6104038330 0.2568096435 0.0472233640 0.9720533928
   [231] 0.4073100807 0.6953799492 0.5064889686 0.7805142691 0.1246848973
   [236] 0.7234809613 0.2187216925 0.8010015031 0.8375137867 0.8160691923
   [241] 0.3587910729 0.8081237068 0.6196225854 0.4446221506 0.0911296355
   [246] 0.5451784572 0.4509040238 0.7988180285 0.7347719567 0.2192694834
   [251] 0.7026692899 0.2425903892 0.1315187262 0.6057988545 0.4511245908
   [256] 0.2545578545 0.4672258096 0.5123341670 0.8689727159 0.6028287923
   [261] 0.7962183105 0.3518507322 0.9451395990 0.5041810041 0.5188296335
   [266] 0.5753487637 0.7826258810 0.5176164126 0.0620655464 0.7138455650
   [271] 0.7244834723 0.9222907489 0.0133932428 0.7797427168 0.5579171153
   [276] 0.3298182404 0.9576554047 0.7775682649 0.0465109469 0.2809512978
   [281] 0.2671092646 0.0740939071 0.0405800613 0.5766352033 0.0945906686
   [286] 0.3778271815 0.4156470718 0.0934377974 0.8198031383 0.0778786531
   [291] 0.0890436741 0.8333541667 0.1987319337 0.6922041019 0.3646372082
   [296] 0.9579863315 0.4661831157 0.1752217105 0.8556822212 0.5570979333
   [301] 0.6414476084 0.8348042509 0.2357970299 0.9015439218 0.2870902615
   [306] 0.6086462727 0.0680652829 0.9305752432 0.9708639132 0.4500062903
   [311] 0.9622625234 0.7235185271 0.6807484524 0.5728239706 0.3102077516
   [316] 0.7058307743 0.4431148814 0.3062123200 0.8492399873 0.3395290440
   [321] 0.3940143781 0.3083248725 0.3038198324 0.0479951417 0.5535923587
   [326] 0.8895978769 0.3552560331 0.1251553064 0.5536275404 0.1953674844
   [331] 0.1895570429 0.3790348982 0.5681960033 0.9978619358 0.8734075846
   [336] 0.2596880859 0.6974602537 0.8475687495 0.8082702132 0.2215025341
   [341] 0.0545832859 0.3339769086 0.5126118781 0.0698790913 0.8057676451
   [346] 0.2056940487 0.9822554858 0.0422864771 0.4134194897 0.0999386450
   [351] 0.8788564624 0.3736909693 0.3324376540 0.6314072004 0.7965043159
   [356] 0.0963610923 0.4096277105 0.5905164322 0.8564491989 0.8240473038
   [361] 0.2362410324 0.0010204604 0.8799534710 0.2705366826 0.7036388563
   [366] 0.7870029947 0.3892682614 0.2525826162 0.0120813446 0.7992445221
   [371] 0.6867350312 0.9272094881 0.3826416479 0.9509644946 0.2620121362
   [376] 0.0133923655 0.7222449677 0.2129385164 0.7774263890 0.7481116867
   [381] 0.4918326186 0.2179905316 0.8814496221 0.3267829483 0.0276510911
   [386] 0.2248600116 0.1003002496 0.5780613935 0.5656661140 0.1914441427
   [391] 0.0576698305 0.6230216986 0.2191017168 0.7074150136 0.2725746306
   [396] 0.2687126612 0.1591042923 0.5362118026 0.7853321852 0.8860868877
   [401] 0.2485316591 0.5164079657 0.8616628619 0.5223054802 0.3788671242
   [406] 0.5724534234 0.0249164226 0.9974177247 0.7602585452 0.5847917488
   [411] 0.6664235862 0.7354157781 0.4146823930 0.8693523547 0.4839725913
   [416] 0.0796643561 0.1222328143 0.0164176812 0.9984107586 0.8427054202
   [421] 0.0705356942 0.8388653835 0.3983710529 0.8404378658 0.4572877185
   [426] 0.1797855189 0.9631236466 0.1606722092 0.2959204363 0.3294727346
   [431] 0.3135524811 0.9160602754 0.6743893223 0.8017934551 0.7412568303
   [436] 0.2313998854 0.7170878397 0.2199280700 0.8657778623 0.2153145438
   [441] 0.4998865025 0.0614881208 0.8699502023 0.6663081264 0.1682969378
   [446] 0.0130084893 0.5633784952 0.2631945675 0.5087609487 0.6838145843
   [451] 0.5240389677 0.9899025476 0.2230645763 0.4292645296 0.5680059912
   [456] 0.5446551805 0.1558771627 0.0333663514 0.7973036440 0.4835247016
   [461] 0.7254154133 0.0007701656 0.4758822741 0.8483621543 0.8072324591
   [466] 0.2081353655 0.9837200614 0.0291020786 0.3211319195 0.6648728093
   [471] 0.0990495803 0.6104421979 0.7712069647 0.1332620075 0.8587093623
   [476] 0.9528981065 0.9890043782 0.3579433104 0.2466204586 0.2582329577
   [481] 0.3298136191 0.6547850957 0.9603880020 0.8692621505 0.5720808851
   [486] 0.6091259560 0.5060277702 0.5540330177 0.7699481742 0.6333918860
   [491] 0.8708177479 0.5243795132 0.3089173483 0.1340884706 0.0242746891
   [496] 0.9388518995 0.4146391945 0.0381680718 0.4972556802 0.6400214350
   [501] 0.3648274885 0.4287720164 0.2891847016 0.8761600619 0.6542980568
   [506] 0.0403477838 0.3534041913 0.7572950935 0.3631328391 0.3631411931
   [511] 0.9106516065 0.1956389016 0.9779689508 0.1070635905 0.8406609865
   [516] 0.0803936040 0.9164127456 0.7749340376 0.4018895151 0.4369307524
   [521] 0.0045788782 0.0950964978 0.5293690832 0.3203460192 0.1577543663
   [526] 0.0634120246 0.9606828513 0.1933888858 0.5141876535 0.3446259489
   [531] 0.4400668116 0.5387673294 0.2720026718 0.7831100663 0.2506363513
   [536] 0.4558275109 0.4792379038 0.7729798248 0.3247378143 0.9916084623
   [541] 0.0270104455 0.2375865122 0.1824250640 0.9562717741 0.0958050685
   [546] 0.9683844438 0.9480610462 0.9729062831 0.3048882829 0.0731731495
   [551] 0.6950443508 0.5117077595 0.8148473995 0.2837145617 0.3686607750
   [556] 0.6585335946 0.6332545923 0.8727252027 0.5370598854 0.3678324884
   [561] 0.3734559612 0.9302433720 0.2203565808 0.9499491369 0.7164855944
   [566] 0.7493713340 0.0478576543 0.5428039199 0.8261046307 0.0713925054
   [571] 0.9934133561 0.3179475879 0.9669653224 0.9402636429 0.9388939561
   [576] 0.1709909504 0.5759000974 0.9164820304 0.3157913061 0.6464095628
   [581] 0.0363356220 0.4003276667 0.0749454023 0.8467234140 0.4058318632
   [586] 0.8144804528 0.2343959482 0.0760516142 0.3467461513 0.3960123798
   [591] 0.2553589167 0.9680420822 0.5100222426 0.3477547159 0.4963281127
   [596] 0.8481762325 0.6221043812 0.0990401944 0.9953017356 0.0804486638
   [601] 0.5249763625 0.4258202007 0.8301339420 0.1484218454 0.4193255948
   [606] 0.1801569602 0.3070114078 0.2206558054 0.5608321624 0.3790907254
   [611] 0.2270548912 0.9505128181 0.6595828878 0.4028819641 0.4810457947
   [616] 0.2603370911 0.2326103942 0.0526285460 0.2222777279 0.8600094533
   [621] 0.1595571684 0.2172579309 0.8675330700 0.2498770421 0.6914646225
   [626] 0.8998943558 0.1761845322 0.9580579912 0.1626871573 0.3536010226
   [631] 0.6574217202 0.7621211177 0.6559312241 0.0764972856 0.5556026967
   [636] 0.6451406097 0.8704193877 0.4162508389 0.6637305440 0.2361257141
   [641] 0.4431793885 0.5339449039 0.2150549265 0.4848254239 0.9734582053
   [646] 0.4773204168 0.1027986528 0.3209081655 0.0002611177 0.1133932164
   [651] 0.6780092390 0.0475164866 0.1830157684 0.6704462310 0.3755354704
   [656] 0.2191967433 0.9353612261 0.6393966665 0.4181289645 0.7542037880
   [661] 0.7620620476 0.7845381936 0.8486707332 0.0311806565 0.5490473406
   [666] 0.0136581352 0.1405227454 0.7202132558 0.0565748261 0.8575296542
   [671] 0.6360044898 0.0982600516 0.8655199008 0.3087789407 0.0629945369
   [676] 0.5989567554 0.0267896997 0.7701273998 0.3796571018 0.3467960125
   [681] 0.6638621585 0.8620088389 0.1972936066 0.4256820893 0.7784500765
   [686] 0.8395616552 0.0313192429 0.6318605607 0.5092901783 0.3689960232
   [691] 0.6303645344 0.4612229979 0.0940571772 0.4133360824 0.6335018994
   [696] 0.0809866553 0.7844028370 0.9775371244 0.8055972131 0.0357491588
   [701] 0.9641200351 0.4629777810 0.1007863702 0.4379181927 0.7204318242
   [706] 0.3813272109 0.8040768476 0.3925161874 0.1184054958 0.1777872881
   [711] 0.0010742666 0.4063600069 0.4284916418 0.9137097886 0.6258339556
   [716] 0.5316156363 0.5571882175 0.5585885784 0.3368375124 0.9937278694
   [721] 0.9308296042 0.6414268008 0.4710943671 0.0537249958 0.0825006710
   [726] 0.0114790639 0.3263683440 0.8548984891 0.1920758383 0.4583686283
   [731] 0.0215292247 0.0038576936 0.8293831395 0.9415795943 0.1850293102
   [736] 0.6359595126 0.1504932838 0.1793240895 0.7215049835 0.7151130950
   [741] 0.7971337186 0.3467844566 0.9065032722 0.3179595238 0.7492276924
   [746] 0.6337304404 0.0593334110 0.6524265027 0.3805583166 0.4115113756
   [751] 0.0440434041 0.5606580442 0.9675576286 0.7594233742 0.8485215874
   [756] 0.2563191568 0.9012206541 0.1004515132 0.4917231919 0.0462755328
   [761] 0.8521444695 0.6963870218 0.5090219053 0.7866482353 0.1386922644
   [766] 0.7523194691 0.2656864347 0.8232433861 0.5482824044 0.8805039516
   [771] 0.3484820696 0.1663568532 0.8618024932 0.6736032804 0.2853972437
   [776] 0.6499539381 0.3311484354 0.1373051694 0.8434950979 0.8252240634
   [781] 0.3598884987 0.7323144218 0.1548900427 0.3385104602 0.6370523768
   [786] 0.7757201185 0.9208493205 0.5436148560 0.9740452515 0.9517378053
   [791] 0.9440195682 0.0984771615 0.0946868556 0.6818266800 0.2387783797
   [796] 0.2962301588 0.6283755349 0.1041817805 0.9697108688 0.8806291884
   [801] 0.5563773112 0.4126011720 0.4682112308 0.0958568370 0.3612399446
   [806] 0.3047281345 0.5772093059 0.7207026249 0.1293319957 0.2896683505
   [811] 0.5740221413 0.8371176934 0.8565068888 0.6049820920 0.9213305535
   [816] 0.0831444925 0.2068919735 0.4930514088 0.0962806912 0.1402214682
   [821] 0.9748025881 0.5868223151 0.7477105977 0.2048627501 0.4997811215
   [826] 0.1549219778 0.4315017732 0.1947128391 0.2847610759 0.9561509034
   [831] 0.1740557374 0.4389762944 0.0673561292 0.4533501258 0.1138955923
   [836] 0.6032224214 0.5942741977 0.1366433939 0.4713925840 0.5985649591
   [841] 0.3488564985 0.7060543587 0.0966176661 0.2252167678 0.4817416118
   [846] 0.8634987613 0.8453180613 0.3004195159 0.1946545439 0.4641516199
   [851] 0.0330188246 0.0207483685 0.8273207890 0.7771894177 0.2172494056
   [856] 0.3087916737 0.8975053923 0.6059072902 0.5578952106 0.8942056512
   [861] 0.3441770123 0.0172112128 0.0056741657 0.8791440791 0.2237969832
   [866] 0.4304851880 0.5687382789 0.5380629813 0.1097333776 0.8158334335
   [871] 0.9074002029 0.1019003158 0.4448000686 0.7516975692 0.5069847973
   [876] 0.2766306615 0.0969207929 0.0918488046 0.6788056912 0.2461949056
   [881] 0.3679182129 0.9917551270 0.6392668458 0.9098049318 0.7054279791
   [886] 0.0443234881 0.9170891168 0.1036233073 0.3679377930 0.2750169924
   [891] 0.3386618169 0.5568179702 0.2929514693 0.7463470837 0.8415192785
   [896] 0.3319919175 0.4182779985 0.5217407336 0.3659424158 0.4999878919
   [901] 0.7064456092 0.7387826284 0.0746852877 0.7990680700 0.1222408311
   [906] 0.5418323567 0.1508266600 0.0284687495 0.8133725571 0.6240165969
   [911] 0.4237465682 0.9263300365 0.7442611055 0.1285963049 0.0732278796
   [916] 0.2820005333 0.0329522835 0.6597089013 0.6616828563 0.0327170258
   [921] 0.2411564486 0.1524854591 0.7445047172 0.0946591711 0.8674125718
   [926] 0.3525428912 0.3085442008 0.6783791846 0.2933772998 0.6548511377
   [931] 0.2887111278 0.8386065280 0.4332390176 0.0519753536 0.4127009632
   [936] 0.0084275967 0.3362569110 0.9416930964 0.6238463791 0.2678404069
   [941] 0.9924250292 0.5439865133 0.3320938172 0.0966842836 0.5912613468
   [946] 0.6774095288 0.7431050511 0.3619445479 0.4837218272 0.6448300323
   [951] 0.5154837491 0.2894322043 0.0972394841 0.9785470655 0.9961270364
   [956] 0.1698386287 0.0538884448 0.7947830101 0.2837020578 0.5491652554
   [961] 0.7416730123 0.5075507751 0.3702475401 0.6535282647 0.5889417278
   [966] 0.6518959841 0.6109003546 0.7983382707 0.2919264333 0.5665141633
   [971] 0.7717470797 0.5318550086 0.2454063343 0.6857429286 0.9058005633
   [976] 0.2631170219 0.4264970617 0.1909291735 0.3071014853 0.1242463505
   [981] 0.9815647351 0.7711712560 0.7929449202 0.8171282178 0.7662650244
   [986] 0.2434361866 0.5642318996 0.1944657182 0.0887072133 0.7820518157
   [991] 0.8939459743 0.3252095049 0.9057432609 0.5075740209 0.8937547775
   [996] 0.7943624770 0.7223818647 0.1850288948 0.6087365868 0.9871594673
datos<-data.frame(u)

Calcular el promedio y varianza de

# 1 generar la media de la muestr a partir de la muestra generada
mean(u)
  [1] 0.491864
# 1 Genrar la varianza de la muestr a partir de la muestra generada
var(u)
  [1] 0.08748006

Generar un histograma de con la función ggplot()

 ggplot(datos, aes(u)) +
        geom_histogram(binwidth = 0.08, col='black', fill='orange', alpha=0.4)+
        theme_bw()+ 
        labs(y="Frecuencia", title = "Hisograma de la muestra u ")

Probar la muestra simulada tiene distribución uniforme.

Probar que provenga de una distribucion uniforme [0,1] DECISION DE HIPOTESIS A PROVAR \[H_0: U\sim Uniforme(0,1)\] \[H_1: U\nsim Uniforme(0,1)\]

ks.test(u,punif,0,1)
  
    Asymptotic one-sample Kolmogorov-Smirnov test
  
  data:  u
  D = 0.037818, p-value = 0.1145
  alternative hypothesis: two-sided

De acuerdo a los resultados, p-valor = 0.11, si alpha es mayor que p-valor se rechaza Ho. Note que p-valor es mayor que alpha Por lo tanto NO RECHAZA. es decir, la muestra simulada proviene de una distribución uniforme ########################################################################################### ##########################################################################################

  • Probar las observaciones de la muestra son estadísticamente independientes.
prueIND<- acf(u, type= "correlation");prueIND  # Ver si hay independencia en la muestra generada

  
  Autocorrelations of series 'u', by lag
  
       0      1      2      3      4      5      6      7      8      9     10 
   1.000 -0.079 -0.007 -0.004 -0.039 -0.031  0.002  0.006  0.002  0.023 -0.010 
      11     12     13     14     15     16     17     18     19     20     21 
  -0.021  0.015  0.067 -0.026 -0.035  0.025  0.009 -0.063  0.021  0.021 -0.009 
      22     23     24     25     26     27     28     29     30 
  -0.033 -0.034  0.017  0.051  0.012  0.025  0.008  0.045 -0.043

No hay correlación en los datos de la muestra generada de una uniforme Como en el gráfico se ve que las barras salen del rango, Por lo tanto se concluye , que la muesta es independiente.

  • Probar que la muestra es aleatoria (sugerencia: test de rachas, test de la mediana, test gap )

Con el estadístico Ljung-Box, se puede probar \(H_{0}\): Muestra aleatoria \(H_{1}\): Muestra no aleatoria

Se rechaza \(H_{0}\) si \(p-valor <\alpha\)

Box.test(u, lag = 10, type = "Ljung")
  
    Box-Ljung test
  
  data:  u
  X-squared = 9.5064, df = 10, p-value = 0.4848

De acuerdo a los resultados, p-valor = 0.4848, Note que \(p-valor > \alpha\), llegamos a conclusion ,que No se rechaza \(H0\) Por lo tanto La muestra generada es aleatoria

  • Comparar sus resultados con la función rlcg()

PROBLEMA 4:

  n <-1000# numeros aleatorios a generar (1000)
   c<- 0
   a<- 7^5#valor de operacion
   m<- 2^31-1
   seed<-45813#Fijar un valor a la semilla  asignamos el valor de muestr
   us <- numeric(n)

   for(i in 1:n) 
     {
       seed <- (a*seed+c)%%m
       us[i] <- seed/m 
   }
#A aprtir de la Asignacion de la mestr
#podemos calcular
#los siguientes
# 1 generar la media de la muestr a partir de la muestra generada
mean(us)
  [1] 0.4836723
# 1 Genrar la varianza de la muestr a partir de la muestra generada
var(us)
  [1] 0.08393356
View(us)
dat2<-data.frame(us)
library(ggplot2)
## generando graficos ...
ggplot(dat2, aes(us))+geom_histogram(binwidth = 0.1,col="black",fill="orange",alpha=0.4,) + labs(y="Conteo",x="Valor de datos muestr/m",title = "Valores generados con operador congrencial multiplicativo")+  theme_bw()

ks.test(us,punif,0,1)
  
    Asymptotic one-sample Kolmogorov-Smirnov test
  
  data:  us
  D = 0.041936, p-value = 0.05936
  alternative hypothesis: two-sided

De acuerdo a los resultados, p-valor = 0.059, si alpha es mayor que p-valor se rechaza Ho. Note que p-valor es mayor que alpha Por lo tanto NO RECHAZA. es decir, la muestra simulada proviene de una distribución uniforme ########################################################################################### ##########################################################################################

prueIND<- acf(us, type= "correlation");prueIND  # Ver si hay independencia en la muestra generada

  
  Autocorrelations of series 'us', by lag
  
       0      1      2      3      4      5      6      7      8      9     10 
   1.000  0.019  0.035  0.013 -0.010  0.015  0.034  0.007 -0.037 -0.095  0.006 
      11     12     13     14     15     16     17     18     19     20     21 
  -0.053 -0.017 -0.004 -0.024 -0.042 -0.016 -0.012 -0.009 -0.037  0.005  0.017 
      22     23     24     25     26     27     28     29     30 
   0.006 -0.030  0.007 -0.024 -0.049  0.019 -0.026  0.006  0.005

No hay correlación en los datos de la muestra generada de una uniforme Como en el gráfico se ve que las barras salen del rango, Por lo tanto se concluye , que la muesta es independiente.

  • Probar que la muestra es aleatoria (sugerencia: test de rachas, test de la mediana, test gap )

Con el estadístico Ljung-Box, se puede probar \(H_{0}\): Muestra aleatoria \(H_{1}\): Muestra no aleatoria

Se rechaza \(H_{0}\) si \(p-valor <\alpha\)

Box.test(us, lag = 10, type = "Ljung")
  
    Box-Ljung test
  
  data:  us
  X-squared = 13.904, df = 10, p-value = 0.1774

De acuerdo a los resultados, p-valor = 0.1774, Note que \(p-valor > \alpha\), llegamos a conclusion ,que No se rechaza \(H0\) Por lo tanto La muestra generada es aleatoria

  • Comparar sus resultados con la función rlcg()

Problema 5

Generación de números pseudo-aleatorios propuesta por Park, Miller y Stockmeyer (1993): \(c=0\), \(a=48271\) y \(m=2^{31}-1\)

  n<-1000# numeros aleatorios a generar (1000)
  c<-0
  a<-48271#valor de operacion
  m<-2^31-1
  seed<-45813 #Fijar un valor a la semilla  asignamos el valor de muestr

   ul <- numeric(n)
   for(i in 1:n) 
     {
       seed <- (a*seed+c)%%m########
       ul[i] <- seed/m 
   }
   View(ul)
   dat3<-data.frame(ul)
#A aprtir de la Asignacion de la mestr
#podemos calcular
#los siguientes
# 1 generar la media de la muestr a partir de la muestra generada
mean(ul)
  [1] 0.5072776
# 1 Genrar la varianza de la muestr a partir de la muestra generada
var(ul)
  [1] 0.08516119
library(ggplot2)
## generando graficos ...
ggplot(dat3, aes(ul))+geom_histogram(binwidth = 0.1,col="black",fill="red",alpha=0.4,) + labs(y="Conteo",x="Valor de datos muestr/m",title = "Valores generados con operador congrencial multiplicativo")+  theme_bw()

ks.test(ul,punif,0,1)
  
    Asymptotic one-sample Kolmogorov-Smirnov test
  
  data:  ul
  D = 0.028823, p-value = 0.3771
  alternative hypothesis: two-sided

De acuerdo a los resultados, p-valor = 0.3771, si alpha es mayor que p-valor se rechaza Ho. Note que p-valor es mayor que alpha Por lo tanto NO RECHAZA. es decir, la muestra simulada proviene de una distribución uniforme ########################################################################################### ########################################################################################## - Probar las observaciones de la muestra son estadísticamente independientes.

prueIND<- acf(ul, type= "correlation");prueIND  # Ver si hay independencia en la muestra generada

  
  Autocorrelations of series 'ul', by lag
  
       0      1      2      3      4      5      6      7      8      9     10 
   1.000  0.031 -0.007  0.022 -0.034 -0.017 -0.006 -0.018 -0.019  0.007  0.006 
      11     12     13     14     15     16     17     18     19     20     21 
   0.008 -0.021  0.003  0.031 -0.022  0.041  0.031 -0.038 -0.021  0.004  0.025 
      22     23     24     25     26     27     28     29     30 
   0.044  0.014 -0.009  0.029  0.013 -0.018 -0.050 -0.018  0.020

No hay correlación en los datos de la muestra generada de una uniforme Como en el gráfico se ve que las barras salen del rango, Por lo tanto se concluye , que la muesta es independiente.

  • Probar que la muestra es aleatoria (sugerencia: test de rachas, test de la mediana, test gap )

Con el estadístico Ljung-Box, se puede probar \(H_{0}\): Muestra aleatoria \(H_{1}\): Muestra no aleatoria

Se rechaza \(H_{0}\) si \(p-valor <\alpha\)

Box.test(ul, lag = 10, type = "Ljung")
  
    Box-Ljung test
  
  data:  ul
  X-squared = 3.7455, df = 10, p-value = 0.9581

De acuerdo a los resultados, p-valor = 0.9581, Note que \(p-valor > \alpha\), llegamos a conclusion ,que No se rechaza \(H0\) Por lo tanto La muestra generada es aleatoria

  • Comparar sus resultados con la función rlcg()