Question

1. Generate 1000 random numbers from ??3? starting with standard normal random numbers in R.

Answer 1

The R code is given below

The histogram of the random numbers from t(3) is

From the histogram it is clear that the random numbers are from t distribution

1000 random numbers which are obtained from the given code are given below

   [1] -0.127599301  0.449350313  0.166071229 -2.832973151 -1.253517499 -1.841312069 -1.511593526  0.421056959 -0.549328337 -0.076734035  0.094467005  0.944483073
  [13] -0.866753991 -0.924904563  0.182555473 -0.196078231  0.852168793 -0.015942610 -0.103774486 -0.264645308 -1.010018088 -2.679696490  0.953369453  1.911646091
  [25] -0.288483958 -0.368437277 -0.910898977  0.596065272  1.225718783  2.121335571  0.280696946 -1.316331533 -1.158736551  1.046251818  2.687746896  3.365896540
  [37]  1.740192944  3.357734194 -0.884211886 -0.307966344  0.920421139 -0.618456119  1.521597362  0.135018051 -0.944373303  1.984214462  0.281260723  0.240720677
  [49] -0.859300828  1.075182948  2.117077809 -0.072513596  2.691443180 -2.299429457 -0.184996142  1.335418532 -6.547474719 -1.230841285  1.233998512  2.544976665
  [61] -0.389952269 -0.366575101  0.811042226  0.083387617  0.076851261  0.739733387  1.224746674  0.705048777 -2.268424960 -0.720407708 -0.724818730  1.914091660
  [73] -0.313645724 -0.408730566  0.200992781 -0.221031029 -0.529516870 -0.593958693 -0.196585493 -0.577102260 -1.528882040 -0.856018444  1.707912450 -0.857173603
  [85] -0.433834933  3.931151230  1.330334469  0.551818806  1.629937729  2.137719742 -1.729948135 -1.559947265 -0.138494551 -0.679833885 -1.635283720  0.535309253
  [97] -2.519939711 -0.991982528  0.006426805  0.829256781  4.031417981 -0.042091871  0.248483648 -1.745614141  1.096577880 -2.288288052  0.673247430  1.627069997
 [109] -2.712243562 -0.266297183  1.417796288 -0.322604688  0.193144021  1.245674237  3.169101137  0.168457012 -1.740447462  0.880617334  1.217308766  0.752770088
 [121]  0.347223962 -2.294488570 -0.095907424 -0.923349827 -0.422459737  1.245269891  0.462061434 -3.480096109  0.839687495 -0.801875527  0.460795333 -0.730396542
 [133] -0.530211271  0.255549073 -0.020687473 -0.973452358  0.538027732 -0.129231054 -0.386852939 -0.080483306 -0.136042826  0.715929348 -1.191247845 -0.913817886
 [145]  0.572060226 -1.150500372  0.406831942 -3.379796582 -0.985799706  0.659486222 -0.084421293 -1.522353005  0.523555159 -0.675080815  0.287249679  0.300402899
 [157] -0.002580491 -0.535064718 -0.001366722 -1.450050776 -2.937008885  0.758073425  0.570049970  0.681073727  0.335230194  0.213767490  1.128456047  0.409680626
 [169] -0.822027313 -0.597382021  0.282124591 -0.684116946  0.442577062  3.172186661  2.982872438 -1.003657923  0.649724856 -0.308448872  0.223628260 -0.305630767
 [181]  3.258905748 -0.049059853  0.677069074  1.535080182 -9.992204215 -0.698195511 -0.181369217 -0.407684333  0.118666200  0.677715336 -2.253249897  2.078663482
 [193] -0.501836585 -1.366094798 -0.420095692 -1.551347516  1.562456928  0.876393810 -5.664906634 -0.879125155 -0.069781265  0.372013848 -1.678446141  0.402317189
 [205]  0.871156206 -0.822979630  0.252403922 -1.756275398  0.561135015 -0.505022715 -1.829420303 -2.836324368 -1.509637726 -1.172011022 -0.902140967 -0.266131091
 [217] -1.345516968 -0.255526460 -0.414854029 -0.791059346  0.812033723 -0.932776423 -0.226974507  0.163974445 -1.081199905  0.273697504  0.573286561 -1.168117310
 [229] -0.727455741 -0.213712848 -0.962521742 -0.215376574  0.489748172  0.784786413 -0.687261280  1.519101494 -1.464847030 -0.497758981  1.309293379 -0.663090345
 [241]  0.571797537 -4.373343342  0.783514737 -1.700968850 -1.308226411  0.420912746 -0.741048645 -0.128084001  0.241983486 -0.492279406 -0.750234370  0.637115195
 [253] -1.039798733  0.679750239 -1.387549964  3.099500779 -1.272208729 -0.099116739 -0.017544178 -0.028398784  0.416545955 -0.254830058  1.026535606  0.870588048
 [265]  8.921270876 -0.815290324 -0.774701659 -0.708091136 -0.307463908 -0.512708559 -1.525830958  1.577120770 -0.632160424  1.562582957 -0.115207432 -1.219643257
 [277] -0.209925819  0.525062878 -1.335819303 -1.641962048  0.130635363  0.263012375 -0.179186651 -1.239978456  0.176887472  0.413041526 -8.924823788  0.215928355
 [289]  1.504561928  1.063702813  0.323749565 -1.031975613  8.851537057 -0.295567459  1.038110016 -0.587057262  0.250556428 -1.914693905  0.155454136 -1.692530683
 [301]  0.143926526  1.096564991 -0.101148355 -1.976543862 -0.219699934 -0.197281531  2.570775075  1.062424722  0.814556822  1.599794067 -0.829037744  1.018638730
 [313] -0.270409204  1.247405015  0.939451761 -2.384355187  0.457444917 -1.627979542  0.043151374  1.542742308 -0.349197207  1.751199353 -0.178622619 -0.033348508
 [325] -2.007835430  1.466662203  0.374668138  0.746035057 -0.044992213  0.779062734 -0.823002211  0.677020912  0.200891582 -2.648954329 -0.885744320 -1.736798005
 [337]  0.769421140 -0.099685597 -3.304385386 -1.008897925  0.844571371  1.377993858 -0.270765003 -0.201389728 -1.384306055  1.122857871  1.890461465 -1.735897927
 [349]  0.407325608  0.071046449  1.763399343  0.312939331  1.136301601 -0.835759678 -2.575128414  0.631406526  0.181223803 -1.944412563  0.046335689  0.890341508
 [361]  1.005705682 -0.840177757  0.846385494  0.743560693 -2.386929968 -0.466980889  5.788939853 -0.037986456  0.186455800 -0.114986922 -0.386084917 -0.571719042
 [373] -0.440632739  2.118914858 -0.647054084 -1.155225802  0.459261180  1.260023352 -0.083069225 -1.146251128  0.023409225 -1.081193774  0.260849423 -0.398054602
 [385] -1.744114253  1.271491973  1.180679672  0.952337014  1.855862559 -2.653173648  0.683108862  0.048132760  0.397266530  0.698631425  0.344136787 -0.508353094
 [397]  1.311798203 -3.744621291  0.020553736  0.741120335  0.271342682  0.491749989  1.323227606  0.389335317 -0.052146557 -1.159619544 -1.045206495 -0.609202541
 [409]  0.199326331  0.442966857  0.592707655 -0.991291935  0.141481031  1.529555086  0.647943592 -0.025382073 -0.453444998  0.887049679  0.432558327 -1.199254855
 [421]  0.019664070 -0.475843584  0.084211670  1.167686229 -1.370779467  1.156342447  0.323294674 -0.144061315 -0.745530564 -0.819128685 -0.117463879  0.850820516
 [433]  0.902520951  0.727020513  2.519207867  2.113258297  1.064720865 -1.038774464 -0.346360866 -0.808087065  2.329681332 -0.496233066  0.534684581 -0.167097821
 [445]  1.168946536 -1.889921486  0.252575446  2.644360871 -0.446525060  1.856033981 -0.136650198 -1.588745486 -1.057300999  3.527631921 -0.287431150 -0.335132136
 [457] -0.539861310 -0.246021615  1.534889614 -0.110406700 -0.731319884  0.728485161 -0.483251046  1.546063842  0.830736966 -0.409397065  0.425134851 -2.402287027
 [469]  2.223137378  0.574451328  2.860119458  1.266569952 -0.375138205 -0.285639811  6.790383458  1.244539426 -0.569996199 -0.388234560 -7.104341794 -0.519551838
 [481]  0.610728616  2.046318199 -0.100523775  0.757549359 -0.669735534 -0.079510780  0.670921280 -0.867692733  0.274086343  3.285265616 -0.517301784  2.176908475
 [493] -0.284969968 -0.602971070  0.115634770  0.658276608  1.512188573  3.140211368  0.523430610 -0.114608408 -2.238724056 -8.222529266  1.470289067 -0.300615706
 [505] -2.946728116 -0.072376658  0.292701302  0.916671789  0.649011295  0.570381793 -1.611621218  2.185157369 -1.647211089  0.776250261 -0.408053775  0.141291939
 [517]  1.039011858  0.890250250 -0.211180277  1.802918222  0.139346334  1.980173527  1.087574048  2.750661902  0.049125910 -0.312174576  2.958510164  0.985550344
 [529] -0.399275193  0.798530464 -2.161957372 -0.661819124  0.937012392 -0.559198826 -0.986679655  1.492843757  1.372830956 -2.307362096 -0.996440308  0.511706331
 [541]  0.056108128 -0.995396268 -0.421330335 -1.904006956  0.028523387 -1.254476765  0.640287365 -0.374241510  0.372959791  1.220261037  6.745769001 -0.280924298
 [553] -0.285874574  1.125197967 -0.107185369 -5.204182854 -1.744160893  0.373590605 -1.213371969  2.540495956 -1.445534545 -0.731352446 -1.523438920  0.312495113
 [565] -0.090414703 -1.030152377  0.213140895  0.005098283  1.422428405 -0.785518237 -2.176300660  0.332966803  1.272395132  0.227693405  0.479987104  0.406492065
 [577]  0.363189839  0.398039589  0.460117600 -0.358408501  0.795890900  1.995956471 -0.681562530 -4.030022346  0.203738476  0.903614858  0.821649121 -1.500130376
 [589]  0.015239108  0.658754148  1.059966212 -0.008765340  3.991316816 -0.809727109  0.780685885 -0.068602668 -1.468155236 -0.476479679 -0.566427791  0.512904178
 [601]  0.315733409 -1.210536644  0.127807610  1.125013865  1.520615857  6.311135501  0.485403538  0.392112918 -1.037624185 -0.584532120 -1.172839113 -0.196413100
 [613] -0.736950057 -0.260526535  0.205065287 -0.556502246 -1.981806904 -7.566353535 -0.953333845  1.200809825 -0.512510770 -1.223821184 -0.835672926 -1.239550767
 [625] -2.198519883 -1.250641375 -0.729345795  0.968431401  0.704200135 -1.821879824 -0.820074885  0.947406519  0.340127083 -2.096722425  0.221103154 -0.439509446
 [637] -1.104448421  0.057363251  1.991889438  0.965077346  0.612874888 -0.047327073  1.080774273  0.195321185 -1.144863746  0.169594125 -1.460420962 -0.465902007
 [649]  0.680647429 -1.734321485 -0.990956716 -2.246178057  0.028153959  0.046512108  0.579608317  0.657213522  0.770868943 -0.738681494  2.240590160  0.570644664
 [661]  1.321272773 -0.057852096 -0.509457248  0.311091060  0.441106835  0.644823986 -0.246242097  1.468242629 -1.236952589  0.608503190 -1.026844141  0.061825970
 [673]  0.274724203  0.448679700  0.090065287  0.355819407 -0.173311063 -0.469238821 -3.309878764  0.175246824 -1.303745796  1.026779000  0.003377233  1.352391841
 [685]  2.255082247 -1.136277892 -0.698813822 -2.254093586  0.119897729 -0.012408217  3.811769349 -0.355516970  0.494992935  0.009370813 -1.614882233 -0.350392253
 [697]  0.749985870 -1.764312835  4.790790453  0.051799244 -1.764536820 -1.925565047 -0.497098058  0.539437293 -1.070463290  1.165233302  1.701287078 -1.550659570
 [709] -0.999899363 -0.560777733 -2.040850966  0.695306830 -0.414269350  0.276425995 -0.392949325  0.681821562 -0.088309406  0.364790541 -0.816783683  0.536294025
 [721]  1.145884396  1.204015191 -0.452903243  1.857095566 -0.178709805 -0.520670719  0.051425936 -2.080277764 -1.058543728  0.663796586  0.360866166 -1.010269728
 [733] -0.342076039 -0.355739583 -0.689641570 -0.096019881  0.718922723  1.615787467  1.083174923  0.474658527 -0.014191068 -1.463483523  0.388271645 -2.807849469
 [745]  2.715471789  1.200056645 -1.569320627  0.182129359 -0.220533660  0.050416752 -0.467901077 -0.354473792  0.314561778 -0.469210326 -0.863396175 -0.686905546
 [757]  0.562142872 -0.312914399  2.002348454  4.265727533 -4.932938649  2.183730384 -0.752931382  0.389432168  1.709592967 -0.016929037  0.450314725  0.738086883
 [769] -2.709446752 -0.701057270 -3.086346134  0.678816653 -0.183587688  0.919316051  0.083714294 -0.292260469 -0.458393297 -0.159976648  0.018276012 -2.322537166
 [781] -0.587343324  1.243502538  0.565150371  0.776327910  1.617503507  0.740134942  0.158994437  1.022584270 -0.158326161 -0.388342589  0.686975437 -0.261785026
 [793]  0.971716305 -1.739026272  0.075468740  0.274707110 -4.421499277  0.691433158  1.703818510 -1.486869418  0.791279095 -1.320855009  2.813473924  0.161544619
 [805]  0.903026522  0.176449015  0.026210408  1.904926075  0.168535071  0.327720480  1.441919652 -2.480976115  0.735141670 -0.825317082  0.248487999 -0.342238954
 [817] -0.213382038 -0.210940842  0.854036330  0.487660048 -0.065082666  1.037680468  0.493700250  0.305693607 -0.236432865  2.591302399 -0.962440854  1.529031879
 [829] -3.499259405  1.336203495 -0.550030379  0.618601014  0.026316215 -0.558898836  2.619023190 -0.429228142  3.075546687 -2.041450759  0.311806339 -0.968075385
 [841] -0.920771103 -0.636056412  1.236658523  0.415371102  3.715673802  2.566708053 -1.794439439  2.371152823  0.214681523 -0.449613671 -0.321609212 -0.105552303
 [853]  0.118839676  3.935128394  1.088200532 -0.161164989  0.762692678 -0.265231936  0.408698224  0.294674907  1.165070388 -2.136631555 -2.082583681  0.654444761
 [865] -0.146868794 -3.580824454  0.760417568 -1.040110701 -0.253217499 -0.197001020  0.738497983 -0.641417530 -0.696508744  1.723174756  0.157450659  0.142288888
 [877] -0.445756779  0.085543343 -2.159556581  1.508278307 -0.377151487  0.399942719 -2.997706910 -0.088989944 -1.623127291  0.819763937 -1.006453766 -8.357211351
 [889] -1.191057853 -1.710889638  0.670559582 -0.773334453 -3.216522774  3.130601798 -1.035065475 -0.204904758  0.345201180  0.149431772  0.125089419  0.194658451
 [901]  0.604063579 -0.370619146 -0.349305031  0.499253359 -2.435152832  0.316855494  0.268502170  0.354052860  0.556724435 -6.668264039 -0.670644011 -2.747661480
 [913]  0.457845037  0.855961580 -0.024127718 -0.147038022 -0.058877456  0.233876594  0.152935856 -0.221898748 -0.053068317  0.405484486  1.951573590  0.003972543
 [925]  0.236996766  1.065948435  0.577058577  0.004176317 -3.210613986 -0.190412053  0.824997507  0.446596692 -1.697183310 -1.700432522 -0.631428467 -1.757101922
 [937]  1.413680687 -2.098423948  1.515882455 -1.152029531  4.785021703 -0.831475167  3.805208612  0.281842382  0.278996811  0.746308693 -2.841571161 -1.628822708
 [949]  0.427818193 -0.075431309  0.432890414 -0.195183026  2.249294033 -1.832570331 -2.151024832  1.248091588 -0.407651559 -1.051875640  0.341823365  1.019104074
 [961]  0.786337352 -1.370180669 -1.059174652  0.213449807  0.499250835  0.119739153 -0.713643831 -1.053529207  0.316425339 -0.546757014  0.652897632  0.234227127
 [973]  0.227172987 -0.331621578 -0.027908335  2.339396052  1.083397681  2.449909078 -1.130590015 -0.754213036 -0.254147124 -2.195565429  0.551510881 -0.354206293
 [985]  0.362791610  1.120947204 -1.979115121 -0.177633750 -2.695584353  0.429985778 -0.388190025  3.867318575 -1.251150108  1.282846436 -0.048782241 -1.465473160
 [997] -0.153469743 -1.516518746 -1.546400395 -0.225832638