Question

R Simulation:Write an R code that does the following:

(a) Generate n samples x from a random variable X that has a uniform density on [0,3].

(b) Now generate samples of Y using the equation: y = α x + β

(c) For starters, set α = 1, β = 1.

Answer 1

# Command

# a) Generate n samples x from a random variable X that has a
# uniform density on [0,3].

x=runif(n=1000, min = 0, max = 3)
x

# (b) Now generate samples of Y using the equation: y = α x + β
# by setting α = 1, β = 1.

y=1*x*1
y

## Result

> # a) Generate n samples x from a random variable X that has a
> # uniform density on [0,3].
>
> x=runif(n=1000, min = 0, max = 3)
> x
[1] 2.676802092 2.647266751 1.142663233 0.904854520 1.265866985 0.325854298
[7] 0.894797261 1.357160680 0.727363242 0.309877805 2.192792193 1.904744571
[13] 0.280010416 0.948966312 1.292643484 0.875926370 0.566390444 0.374292284
[19] 0.851245773 1.532447577 1.030802960 2.636592830 2.207418869 0.525802504
[25] 2.722728894 0.435688520 1.963871353 1.888447433 0.120782048 0.675865368
[31] 0.968241070 1.400812750 0.360185143 0.461715393 1.633237300 2.135191188
[37] 0.294834923 0.327644398 0.483224281 1.526024804 2.638749729 1.478006466
[43] 0.265844068 0.797292060 0.109200684 0.840625135 1.542920876 0.328529871
[49] 2.856175290 0.453143700 1.296332157 0.732526272 0.656189739 2.875638968
[55] 1.766858028 1.712032260 0.019516400 2.771668484 0.438898775 2.106072068
[61] 1.992352557 0.065972863 0.775083967 0.021375590 1.169381370 1.347733125
[67] 1.044571696 0.643344127 2.011164244 0.718011632 1.699650990 1.418938603
[73] 1.416186175 0.034627883 2.629104369 1.720379209 2.265726412 0.111018760
[79] 2.881741571 2.813602321 0.064045459 1.086367323 0.794659439 2.634215939
[85] 1.708043943 2.793018612 1.537711808 1.315281914 1.133780782 2.106808895
[91] 0.864058798 2.916466886 1.736698017 0.049541139 0.119374737 1.853509666
[97] 1.652494699 0.075735923 1.406412591 1.703051053 1.594400414 2.601438700
[103] 2.616584104 2.354931562 1.874945610 1.077878064 0.255736993 2.698710989
[109] 0.015617120 2.620048924 1.209274315 2.349230856 2.599505916 1.488369063
[115] 2.950331909 2.827827287 1.455684650 0.450013462 1.789377334 2.112566150
[121] 1.401627458 1.512398987 0.809010925 1.970042960 2.836847333 0.975467973
[127] 1.139072646 2.444285722 1.746892262 2.827174183 0.953314464 1.355272904
[133] 1.544772302 0.937084564 1.206254232 1.803770050 0.566893601 2.530928078
[139] 0.938602198 1.889064001 2.910973602 2.680789822 0.731735594 1.546938195
[145] 0.238630167 1.629107094 0.695593747 1.207905935 1.042162632 0.203724205
[151] 0.632919237 2.054258195 1.497964252 0.643118813 1.284844618 2.162731552
[157] 0.520032645 1.645939377 2.845823622 2.316532753 0.219192153 0.956918137
[163] 0.863959063 1.818817488 2.890135624 2.766651844 2.833007193 0.541888791
[169] 2.070374697 2.903043843 0.971133204 2.027751713 1.840560631 1.345358645
[175] 0.839752283 2.225680668 0.307684067 2.347342610 0.570524600 0.525931107
[181] 1.334087865 0.548665835 0.166540833 0.943286261 1.416852695 2.437104883
[187] 1.679533850 1.208425318 0.008908470 2.917420946 1.864684974 1.082178421
[193] 1.025405228 1.181058393 2.262259800 1.558967675 0.487566486 2.518634196
[199] 2.683709018 2.122739435 2.415099974 2.370522309 0.526187401 1.147688439
[205] 1.882783848 0.572579823 0.943623411 0.159254387 0.164722740 0.493067731
[211] 1.896636432 0.065627435 0.909611362 0.779255184 2.190603917 1.735775188
[217] 2.568100779 1.794239729 1.357593931 2.103323856 2.152039629 1.874829559
[223] 0.270037810 0.274548172 2.185426628 1.513420845 1.486891034 1.108369045
[229] 1.361194534 1.820851286 1.845158414 0.717087555 0.585436119 1.039223725
[235] 1.615378193 0.591485264 2.974604244 2.136148182 0.920994455 0.693502574
[241] 1.687219707 0.344380286 1.986613385 0.701016198 1.548732808 2.897491619
[247] 2.809449624 0.091892229 0.817399122 0.491817229 2.087130647 1.638454056
[253] 0.781764665 1.958740574 1.238035155 0.402622561 1.553263854 2.476417525
[259] 1.563846064 2.801636185 1.347773578 0.266262873 0.660696280 1.478670565
[265] 1.757935020 2.243785667 2.881021964 2.493581293 2.011086601 1.305899799
[271] 2.650949181 0.927301773 1.240218987 2.680989374 0.799822787 0.234812291
[277] 2.151554628 0.352742162 2.208097669 1.747685115 1.189757076 0.799678921
[283] 0.553603120 0.876132644 0.858621846 1.268829903 0.460254866 1.491486340
[289] 0.511196380 1.629076075 1.078909002 2.520935616 2.490863852 1.535810899
[295] 2.326194568 0.938564472 2.986176671 2.141291746 2.979448672 2.973308258
[301] 2.589439893 1.380489572 2.719310785 1.307115020 2.449838760 2.278591066
[307] 1.159107445 1.263092554 0.833318748 2.811367192 0.769104353 1.935307031
[313] 0.124821315 2.492760432 0.910199342 0.904291116 1.143728853 2.731029775
[319] 0.556950248 1.205866857 0.894270189 0.533762487 0.328521761 1.493319459
[325] 0.993925350 1.103897452 2.476268664 2.027373120 0.587475296 0.197113599
[331] 1.800156025 2.872851696 1.571103775 0.888057601 1.601705143 0.399493226
[337] 1.718234446 2.338849854 0.347306270 1.803484111 2.114343847 1.840105937
[343] 1.516471511 2.976729618 1.376613035 2.760030944 1.374084817 1.755401920
[349] 1.870346736 2.333880160 0.096258027 1.926458673 1.111396355 1.787529452
[355] 1.662747940 0.058420943 2.399623944 0.387071269 1.445820604 1.734271264
[361] 2.963552524 0.654075243 1.867877332 1.157613241 1.141600071 2.585877505
[367] 2.875310110 1.264296569 2.635827196 1.879010610 1.214081643 1.522207294
[373] 1.777240050 2.774328301 1.450969805 0.838394711 2.999960462 0.885667841
[379] 0.556020925 1.202312118 1.592085056 1.315280053 0.398515301 2.053895853
[385] 1.374793099 0.044952307 2.098011162 2.547326184 0.826419795 1.423389510
[391] 2.633057055 2.982308631 0.820249675 1.623682461 0.826390242 1.307686623
[397] 0.219340400 2.597372848 0.538275958 0.214039839 2.892392841 2.702985528
[403] 1.263740700 2.891550131 0.277858643 1.386701037 2.413661564 1.093548317
[409] 1.566984355 1.262001750 0.713726285 2.891250931 0.013119290 1.097069383
[415] 2.166478035 0.180916968 0.858717191 2.228131058 1.117739244 1.283515663
[421] 2.882668938 1.265347830 1.606090075 1.551382755 2.553674325 2.493271854
[427] 0.801173080 2.398023040 1.108120937 0.209957281 1.455545916 2.125412311
[433] 1.844793035 0.642294946 2.895708099 1.061425817 2.947552524 2.571776104
[439] 2.034827612 2.736889469 1.928857241 1.999859019 2.557839039 1.537654473
[445] 0.210301799 1.779655425 1.876558801 1.732081353 2.486056336 1.492660909
[451] 2.035036644 2.099993255 1.391467337 2.793796331 1.414763335 0.261239130
[457] 0.979563270 1.679364236 1.798135735 2.177388164 0.191806728 2.971540978
[463] 2.352038087 0.662788665 0.077260001 2.984600870 0.501800965 2.282185377
[469] 1.082891475 0.119997015 2.374057999 0.378100640 2.413587602 0.940722417
[475] 0.988678792 1.748621525 0.110095273 0.892559128 0.804474367 0.480486205
[481] 0.768176522 1.918965984 2.046678430 1.528228917 0.403173198 0.505911958
[487] 0.180515137 2.242188504 0.245482223 2.941010605 1.929560593 0.930993567
[493] 1.043893739 0.008120488 1.603958904 1.394178220 1.587791387 0.749835637
[499] 1.147182121 2.493283454 0.175073508 2.010901216 0.811632408 2.170075509
[505] 0.343706899 0.373357506 0.866600188 1.131940152 0.786469580 1.123410932
[511] 0.753313065 0.387015299 0.736521449 0.434035504 1.853079937 1.051070089
[517] 1.366217450 0.121740223 0.730741778 1.761940765 1.361690845 1.254293365
[523] 0.868425224 1.315852898 2.810424606 0.234080449 1.255732516 1.243656577
[529] 2.948490043 2.314766808 2.923328966 1.126326464 1.140850061 2.343640965
[535] 0.378718517 0.608485678 2.975297461 2.799542012 2.581840408 0.559951908
[541] 1.288209561 1.338027007 1.353994786 1.487170896 2.519385746 2.170981879
[547] 1.629825548 1.980487828 1.278012568 0.370287517 2.845085302 1.046344909
[553] 0.901026674 0.536236286 1.310885476 1.208352573 0.987095816 2.395847655
[559] 2.949781163 1.702588726 1.162315852 1.520945266 1.730786630 2.777827939
[565] 1.246571668 0.411753007 2.238370216 1.841595821 0.889438743 2.868993031
[571] 1.824373902 2.703819947 2.508461246 0.075342400 2.877531174 0.217118677
[577] 1.455378370 1.282544254 0.041145917 1.376377327 0.023849883 2.095277386
[583] 0.077475472 1.406784612 1.940246231 1.250818724 1.539158506 2.847951739
[589] 1.734721299 1.828714567 1.987944703 2.665985271 1.547963228 0.284522977
[595] 1.605389050 1.581501719 0.166564085 1.366428263 2.679443274 2.175215405
[601] 2.770628903 0.445961741 1.464759237 1.263358206 0.299488102 2.081664316
[607] 0.298914067 2.802991329 1.193385364 2.038833054 0.596322545 1.666535515
[613] 0.506859632 1.044796490 2.623166213 2.647858894 2.216643811 0.852435834
[619] 2.340621638 0.761346379 1.148077515 2.649329115 0.726563766 1.118275736
[625] 0.857698032 2.193755225 0.969061602 1.072396305 1.765542267 2.792453825
[631] 2.709633999 0.431035277 1.980526413 2.111790321 0.576525740 0.916906342
[637] 2.998403637 2.469324647 2.375635939 0.594457795 2.216809930 1.581765660
[643] 0.950612207 2.258478628 1.826993378 2.154989818 1.621331092 2.350716449
[649] 1.194374752 1.169968776 1.711673138 1.131793747 2.662144273 1.308473506
[655] 1.774351457 1.086471499 2.897480187 1.903999815 0.376742049 0.008849337
[661] 0.742757337 1.920297991 0.964314847 2.531681783 0.574159132 1.636374225
[667] 2.713975519 1.394547201 1.619194751 0.508388424 0.829940780 1.411039627
[673] 0.225203106 2.453193176 0.996267595 0.174878740 1.669997490 1.506066393
[679] 0.620681652 2.564817844 0.721691995 1.354753877 0.332894717 1.477819417
[685] 0.582811601 2.315756100 1.094756807 2.998259368 0.486095076 2.571141628
[691] 1.040963495 2.867842658 2.771559647 0.477956310 0.736177634 2.784443681
[697] 2.840800912 0.694399514 2.899027302 1.669307260 1.536367684 2.158492385
[703] 0.565437707 1.636142834 0.596779204 2.542176963 2.979296997 2.104508631
[709] 0.445636817 1.801847886 2.595369193 1.405879471 2.951325785 1.609670775
[715] 0.900235302 0.941198111 0.963548005 0.281636490 2.044759199 0.531367521
[721] 2.289250423 0.383434904 2.636865567 1.976923543 2.929048050 1.509989969
[727] 1.430741412 2.091164003 2.737919751 2.621091903 2.638037063 1.798433441
[733] 0.356278604 2.166462281 1.362444885 2.842279608 1.061248317 0.319539345
[739] 0.270907212 0.232214879 2.775776910 2.851253719 2.856140597 0.559870681
[745] 1.916470021 1.731711600 1.821771835 0.113905740 0.172391079 0.999022463
[751] 1.371089315 2.017777717 0.023879154 0.450930475 0.042052663 1.967815916
[757] 2.988283878 1.523767370 0.650876921 1.748941905 0.423612293 1.649035601
[763] 1.937604636 2.646937842 2.950167339 2.607838984 1.342940197 1.890220799
[769] 0.426096788 2.244312792 2.331070727 2.905399644 2.219416856 2.775482757
[775] 1.646428223 0.204037302 2.816041595 1.564781236 0.450506593 2.479341987
[781] 0.242711648 0.447008561 2.288215351 0.586899045 0.455664067 1.053952867
[787] 1.970310149 0.839073625 1.250480843 0.330354312 0.011150497 0.100755906
[793] 1.685669449 2.352617451 1.946448607 1.351369270 1.592186424 2.723212974
[799] 1.596676102 0.674400272 0.147289508 1.800336253 2.341951106 2.644274957
[805] 0.023012056 1.424707533 2.868350438 2.916999731 2.806863906 2.257690983
[811] 1.235041182 2.369721034 1.480285781 0.899806588 0.369491091 2.702258821
[817] 1.626866318 2.718381303 2.509623988 1.398341333 1.289999171 2.436285048
[823] 2.883421427 1.031854587 0.058766908 2.732582415 2.231158694 0.002495152
[829] 1.993744340 0.968663843 0.734666400 2.307582897 1.982631068 2.588799891
[835] 2.022288668 0.529422573 0.487225442 1.136649318 2.702275609 1.098223264
[841] 1.423751487 1.203916191 1.835942396 2.808767042 2.505487421 1.737009213
[847] 1.662988451 0.228693550 0.145984068 0.005255650 1.989478957 2.741423376
[853] 2.187459129 0.204183205 2.619503101 2.795129484 2.918881287 1.850403899
[859] 1.196053751 1.527479107 1.831868846 2.154579686 2.070811240 2.073407645
[865] 0.407310863 1.502387640 0.544734237 0.371907421 2.550872779 2.595659834
[871] 1.939854305 0.613602305 0.125783802 1.148778042 0.508312077 0.279525306
[877] 0.901069007 0.930060562 0.828461352 0.486810904 2.663349626 0.881613536
[883] 0.108073417 2.497289078 1.128973570 0.097860954 0.280546505 0.330521120
[889] 2.097386086 1.019116474 1.053232664 1.355687097 2.692510446 0.941572927
[895] 0.886999377 2.732032801 0.150847294 0.978044816 0.439010123 2.770103380
[901] 2.681294184 1.964232357 2.304386792 1.158550601 2.992884340 0.186991236
[907] 2.318852405 2.086792421 2.776832271 1.550079424 1.246130189 0.089101262
[913] 2.501498363 0.561057662 2.005483455 0.134935801 1.505943014 0.089284297
[919] 1.283231721 2.882945248 2.753389703 1.301047120 1.360613977 0.657229470
[925] 0.102027000 1.217374523 2.432020529 2.934684061 1.643214797 0.239989466
[931] 1.629189973 1.878237095 2.326322452 1.763521830 2.134006822 2.824763300
[937] 2.925473819 1.121776406 1.679903474 2.695782554 0.730419602 0.011386745
[943] 2.295451770 1.944345305 0.190466672 1.999953043 1.391519405 2.282360273
[949] 0.702124896 1.690033324 1.892915234 0.886074913 2.181487532 2.084184623
[955] 2.497656155 0.316707009 2.798734228 0.104359852 1.139695558 0.059417929
[961] 2.980093820 2.169005570 1.098800377 0.019232176 0.860744262 2.326691038
[967] 2.404097789 1.265805699 0.813857867 1.667354307 0.429475564 2.034847562
[973] 1.151630463 1.366885190 0.244827911 0.618031947 0.045592366 1.861640586
[979] 1.127660802 1.719224102 0.316498332 2.864433773 1.156899682 0.849168154
[985] 2.506816230 1.073521237 0.866037027 1.391112157 0.825435792 0.457736844
[991] 0.386978430 0.991842245 1.678112041 2.519593278 0.833953156 0.982420962
[997] 1.110738410 0.747109163 0.888145170 2.075462978
>
> # (b) Now generate samples of Y using the equation: y = α x + β
> # by setting α = 1, β = 1.
>
> y=1*x*1
> y
[1] 2.676802092 2.647266751 1.142663233 0.904854520 1.265866985 0.325854298
[7] 0.894797261 1.357160680 0.727363242 0.309877805 2.192792193 1.904744571
[13] 0.280010416 0.948966312 1.292643484 0.875926370 0.566390444 0.374292284
[19] 0.851245773 1.532447577 1.030802960 2.636592830 2.207418869 0.525802504
[25] 2.722728894 0.435688520 1.963871353 1.888447433 0.120782048 0.675865368
[31] 0.968241070 1.400812750 0.360185143 0.461715393 1.633237300 2.135191188
[37] 0.294834923 0.327644398 0.483224281 1.526024804 2.638749729 1.478006466
[43] 0.265844068 0.797292060 0.109200684 0.840625135 1.542920876 0.328529871
[49] 2.856175290 0.453143700 1.296332157 0.732526272 0.656189739 2.875638968
[55] 1.766858028 1.712032260 0.019516400 2.771668484 0.438898775 2.106072068
[61] 1.992352557 0.065972863 0.775083967 0.021375590 1.169381370 1.347733125
[67] 1.044571696 0.643344127 2.011164244 0.718011632 1.699650990 1.418938603
[73] 1.416186175 0.034627883 2.629104369 1.720379209 2.265726412 0.111018760
[79] 2.881741571 2.813602321 0.064045459 1.086367323 0.794659439 2.634215939
[85] 1.708043943 2.793018612 1.537711808 1.315281914 1.133780782 2.106808895
[91] 0.864058798 2.916466886 1.736698017 0.049541139 0.119374737 1.853509666
[97] 1.652494699 0.075735923 1.406412591 1.703051053 1.594400414 2.601438700
[103] 2.616584104 2.354931562 1.874945610 1.077878064 0.255736993 2.698710989
[109] 0.015617120 2.620048924 1.209274315 2.349230856 2.599505916 1.488369063
[115] 2.950331909 2.827827287 1.455684650 0.450013462 1.789377334 2.112566150
[121] 1.401627458 1.512398987 0.809010925 1.970042960 2.836847333 0.975467973
[127] 1.139072646 2.444285722 1.746892262 2.827174183 0.953314464 1.355272904
[133] 1.544772302 0.937084564 1.206254232 1.803770050 0.566893601 2.530928078
[139] 0.938602198 1.889064001 2.910973602 2.680789822 0.731735594 1.546938195
[145] 0.238630167 1.629107094 0.695593747 1.207905935 1.042162632 0.203724205
[151] 0.632919237 2.054258195 1.497964252 0.643118813 1.284844618 2.162731552
[157] 0.520032645 1.645939377 2.845823622 2.316532753 0.219192153 0.956918137
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>