#install_github("rubenfcasal/simres")
library(simres)
##  simres: Simulation and resampling techniques,
##  version 0.1.3 (built on 2022-10-26).
##  Copyright (C) R. Fernandez-Casal 2022.
##  Type `help(simres)` for an overview of the package or
##  visit https://rubenfcasal.github.io/simres.
#install.packages("devtools")
library(devtools)
## Warning: package 'devtools' was built under R version 4.2.3
## Loading required package: usethis
## Warning: package 'usethis' was built under R version 4.2.3
library(remotes)
## Warning: package 'remotes' was built under R version 4.2.3
## 
## Attaching package: 'remotes'
## The following objects are masked from 'package:devtools':
## 
##     dev_package_deps, install_bioc, install_bitbucket, install_cran,
##     install_deps, install_dev, install_git, install_github,
##     install_gitlab, install_local, install_svn, install_url,
##     install_version, update_packages
## The following object is masked from 'package:usethis':
## 
##     git_credentials
library(simres)
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.2.3
#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}= (a^kx_n+c)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:

  • \(c=0\), \(a=7^5\) y \(m=2^{31}-1\)
  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 
   }
  • Calcular el promedio y varianza de \(u\)
#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
  • Generar un histograma de \(u\) con la función ggplot()
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()

  • 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(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 ########################################################################################### ##########################################################################################

  • Probar las observaciones de la muestra son estadísticamente independientes.
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()