Question

Use R to complete the following questions. You should include your R code, output and plots in your answer.

Two methods of generating a standard normal random variable are:

a. Take the sum of 5 uniform (0,1) random numbers and scale to have mean 0 and standard deviation 1. (Use the properties of the uniform distribution to determine the required transformation).

b. Generate a standard uniform and then apply inverse cdf function to obtain a normal random variate (Hint: use qnorm).

QUESTION 1.

For each method generate 10,000 random numbers and check the distribution using

a. Normal probability plot

b. Mean and standard deviation

c. The proportion of the data lying within the theoretical 2.5 and 97.5 percentiles and the 0.5 and 99.5 percentiles. (Hint: The ifelse function will be useful)

Answer 1

Algorithm:
Step 1: Generate uniform random number of size n (say).
Step 2: Use the inversion formula for particular distribution to generate the random from that distribution.

> ######generate Unifrom random number (sample size = 5)
> u1=runif(n=5,0,1)
> u1
[1] 0.3807355 0.6838301 0.5793533 0.7001886 0.1476591
> sum(u1) ###sum of 5 uniform (0,1) random numbers
[1] 2.491767
> ####### Generate a standard uniform number use qnorm.
> u2=runif(5);u1
[1] 0.3807355 0.6838301 0.5793533 0.7001886 0.1476591
> u=qnorm(u2);u
[1] 1.9863522 -0.4100890 0.3649655 -1.2394764 0.2776269
> ####### Q.1
> u3=runif(1000);u3
[1] 0.178273742 0.338654670 0.502983252 0.574192463
[5] 0.938588518 0.164298457 0.568822881 0.277261303
[9] 0.587949903 0.822968805 0.486584451 0.308141776
[13] 0.057207227 0.523044843 0.481863485 0.357590487
[17] 0.278691400 0.547701614 0.912094467 0.255963780
[21] 0.017637369 0.827254662 0.049479483 0.595249967
[25] 0.981266572 0.569439175 0.727461077 0.151835970
[29] 0.318237809 0.699689800 0.854843975 0.378279480
[33] 0.275432114 0.988443919 0.134025161 0.977952887
[37] 0.642202243 0.600516791 0.162106824 0.143829044
[41] 0.762430230 0.781958816 0.217104930 0.714697188
[45] 0.293083606 0.271073145 0.903164100 0.934551410
[49] 0.452999341 0.329169206 0.398795732 0.169447463
[53] 0.209013989 0.827561936 0.347139587 0.599187326
[57] 0.452403407 0.093152807 0.896113889 0.262902533
[61] 0.711164872 0.249534035 0.896619297 0.174416814
[65] 0.468116865 0.795492926 0.341465854 0.478540409
[69] 0.260783592 0.985835146 0.438381592 0.730666561
[73] 0.680872272 0.907649460 0.760866767 0.853651691
[77] 0.254686764 0.453843033 0.784928615 0.591713174
[81] 0.988043745 0.927597355 0.885562700 0.569294132
[85] 0.919895268 0.290173398 0.256231378 0.720954146
[89] 0.162788707 0.482136352 0.980341022 0.714786028
[93] 0.603208002 0.756598214 0.042582439 0.892577048
[97] 0.766692444 0.472288112 0.897595979 0.407147537
[101] 0.767170512 0.941631244 0.765189774 0.650673144
[105] 0.923338404 0.270001958 0.960051585 0.441324859
[109] 0.262421818 0.362591780 0.902642826 0.294795499
[113] 0.214329107 0.052150619 0.936144301 0.840517639
[117] 0.612928761 0.048513069 0.269719144 0.760647244
[121] 0.917457117 0.042594744 0.623126212 0.025791519
[125] 0.996860833 0.177718724 0.654859019 0.661239052
[129] 0.820549976 0.876687104 0.082439308 0.830690825
[133] 0.417017611 0.066647755 0.104336319 0.731123660
[137] 0.594064977 0.534583113 0.048505401 0.615337142
[141] 0.738851126 0.117799712 0.366601524 0.135035213
[145] 0.297117565 0.908073284 0.112918757 0.564549285
[149] 0.008583148 0.787502195 0.807160685 0.509219521
[153] 0.478424136 0.403077338 0.543652771 0.007867210
[157] 0.173895296 0.896861917 0.519487014 0.973213919
[161] 0.805484351 0.949031616 0.285357425 0.619058772
[165] 0.795161493 0.935566761 0.526652905 0.831893430
[169] 0.800116418 0.658076967 0.995597235 0.845622191
[173] 0.879890635 0.482919763 0.094897597 0.028267485
[177] 0.009439711 0.823091849 0.043040400 0.823170943
[181] 0.091859132 0.864137657 0.022668031 0.050639909
[185] 0.949141232 0.228148169 0.626769278 0.439620057
[189] 0.599773124 0.968633908 0.994204796 0.273533738
[193] 0.287168470 0.199977183 0.196244740 0.991003036
[197] 0.955346841 0.506245054 0.236431236 0.517316789
[201] 0.577782649 0.218334426 0.222556222 0.825124176
[205] 0.363241312 0.924038738 0.320962447 0.120048779
[209] 0.210499389 0.559180991 0.403326527 0.789544941
[213] 0.471045552 0.873281119 0.176097968 0.305159749
[217] 0.395722471 0.004994244 0.898505884 0.424280648
[221] 0.020754369 0.632583446 0.253635833 0.789984525
[225] 0.114601608 0.069106708 0.200499010 0.293540743
[229] 0.391404357 0.353940719 0.003051260 0.304292710
[233] 0.531214076 0.863909848 0.259208594 0.093974944
[237] 0.083889605 0.257744178 0.086540011 0.117788461
[241] 0.186304734 0.069494832 0.338623188 0.860353916
[245] 0.781542517 0.277202218 0.723161361 0.633321233
[249] 0.156334724 0.152739369 0.158851962 0.895088423
[253] 0.289939353 0.018619489 0.049620316 0.593186052
[257] 0.241353801 0.304142122 0.794493628 0.750661148
[261] 0.589345373 0.136211386 0.627481826 0.864842833
[265] 0.597692031 0.197969936 0.982050263 0.131674617
[269] 0.181336075 0.788943350 0.236138895 0.860342829
[273] 0.419425327 0.651857140 0.579699595 0.323007263
[277] 0.431335230 0.417773142 0.522972820 0.432197149
[281] 0.182775282 0.070289564 0.964844320 0.542852027
[285] 0.748663097 0.554676080 0.680594160 0.250223322
[289] 0.284464552 0.126662511 0.693831057 0.880094765
[293] 0.968801859 0.317698028 0.045397989 0.949943755
[297] 0.400208505 0.371284345 0.324154023 0.560770155
[301] 0.091314258 0.774883735 0.413061508 0.823627154
[305] 0.365933198 0.883123858 0.245355511 0.594524894
[309] 0.365887217 0.745536448 0.011344925 0.925059291
[313] 0.869512550 0.638616879 0.295088985 0.935575329
[317] 0.663063156 0.618219852 0.561209138 0.176157437
[321] 0.520921070 0.711753411 0.925892704 0.067051924
[325] 0.236690307 0.695405485 0.332264071 0.113306195
[329] 0.939782017 0.372087179 0.706058034 0.751002471
[333] 0.403755571 0.227361978 0.273286580 0.072920854
[337] 0.910408618 0.459842255 0.600472450 0.275654292
[341] 0.973215841 0.873161961 0.751179213 0.136125233
[345] 0.655204212 0.080378616 0.265310630 0.223805349
[349] 0.498866989 0.924211003 0.800580584 0.806108595
[353] 0.664286047 0.229822823 0.286321921 0.061472379
[357] 0.949580774 0.649938368 0.617928571 0.072906912
[361] 0.494374668 0.440830963 0.013917039 0.248682106
[365] 0.171046166 0.345438675 0.456405981 0.654901610
[369] 0.953590865 0.726935534 0.247356471 0.613288462
[373] 0.128787193 0.052366293 0.567321367 0.740011448
[377] 0.750754249 0.352847454 0.125696890 0.866560883
[381] 0.069579976 0.650474829 0.244731351 0.329884567
[385] 0.274328860 0.979834995 0.913374244 0.674176028
[389] 0.326536032 0.148618585 0.579086140 0.477506300
[393] 0.960701694 0.157627866 0.002057758 0.086027329
[397] 0.238918506 0.590526449 0.435348495 0.597216841
[401] 0.762081118 0.789204862 0.470843896 0.934225282
[405] 0.645869571 0.665627897 0.270417793 0.502791649
[409] 0.638165310 0.534609866 0.806940141 0.806592199
[413] 0.497163989 0.166165425 0.547377223 0.185009908
[417] 0.847412064 0.154381476 0.697753233 0.821393932
[421] 0.387256954 0.753594385 0.799000961 0.051941172
[425] 0.888509980 0.764760763 0.341600794 0.123064455
[429] 0.173528212 0.565935960 0.346349916 0.853956492
[433] 0.203340870 0.216345853 0.590784603 0.713518189
[437] 0.339589359 0.375325617 0.243176455 0.887737171
[441] 0.112777991 0.742362105 0.044284578 0.720649027
[445] 0.863230416 0.039876486 0.721131758 0.326834837
[449] 0.726444875 0.032174349 0.372234785 0.415885718
[453] 0.121231657 0.407248453 0.424385557 0.357479038
[457] 0.830057639 0.365814243 0.293699471 0.374506313
[461] 0.719548796 0.100653077 0.691715453 0.235637697
[465] 0.779742779 0.282059433 0.706762615 0.619718383
[469] 0.544563180 0.924641102 0.531242341 0.271171188
[473] 0.905813898 0.020223106 0.779149749 0.364991618
[477] 0.307303057 0.160867940 0.176943196 0.693851550
[481] 0.621652109 0.468918368 0.869892344 0.693520504
[485] 0.506954286 0.913280662 0.812330135 0.404537447
[489] 0.794142246 0.017007901 0.548779433 0.254125526
[493] 0.253319016 0.624762121 0.090745371 0.460815903
[497] 0.913422323 0.485198421 0.388692948 0.171777135
[501] 0.677473963 0.292836735 0.075758994 0.719629838
[505] 0.132899838 0.991176752 0.098000679 0.837644470
[509] 0.548980407 0.983141540 0.703873824 0.281914421
[513] 0.253390006 0.177424038 0.002899268 0.454129540
[517] 0.992385163 0.816874481 0.066406031 0.898935678
[521] 0.829591758 0.024087609 0.472519187 0.272334988
[525] 0.336495083 0.057294943 0.205245066 0.869877560
[529] 0.940513984 0.960492567 0.123542711 0.881292025
[533] 0.565366857 0.532137827 0.599130224 0.057422890
[537] 0.152380878 0.322147847 0.617132141 0.080236230
[541] 0.957498794 0.928580394 0.950919238 0.106630008
[545] 0.557867326 0.742990294 0.342305454 0.370324479
[549] 0.661520795 0.077816255 0.189265553 0.408123146
[553] 0.212645051 0.980271553 0.455587409 0.955461214
[557] 0.027056524 0.040871849 0.516093694 0.102335132
[561] 0.762620967 0.161880870 0.168592367 0.857959896
[565] 0.060331176 0.329365385 0.821863838 0.446692215
[569] 0.958970293 0.770069090 0.251505435 0.219794163
[573] 0.975352971 0.647225778 0.646266310 0.638372786
[577] 0.096874715 0.459759643 0.963887846 0.921417943
[581] 0.200894469 0.305496338 0.087948175 0.489524599
[585] 0.390109701 0.178165076 0.226199844 0.341700639
[589] 0.126379829 0.065163645 0.277436177 0.600410688
[593] 0.273144482 0.941489053 0.799276451 0.089914952
[597] 0.392770776 0.174849976 0.456753087 0.481765486
[601] 0.142861928 0.034419505 0.202051959 0.264258587
[605] 0.744757670 0.353446902 0.877193653 0.135480624
[609] 0.570409286 0.412244422 0.225636233 0.797575301
[613] 0.398676400 0.591562685 0.906964443 0.384809852
[617] 0.371891612 0.374291895 0.182152950 0.041934537
[621] 0.261202744 0.304963026 0.178527893 0.558146396
[625] 0.371059665 0.642810347 0.924115872 0.259350448
[629] 0.239350285 0.848965338 0.699011598 0.884429610
[633] 0.209242901 0.528550479 0.228069716 0.102115365
[637] 0.102061887 0.150018109 0.116709932 0.626661309
[641] 0.515648632 0.670773006 0.861331310 0.636687285
[645] 0.570006530 0.894844278 0.275015817 0.490434975
[649] 0.946826611 0.142462477 0.174120710 0.886910460
[653] 0.507694072 0.190409639 0.562036618 0.547679755
[657] 0.893743749 0.704780649 0.561151840 0.447658210
[661] 0.105691165 0.658082008 0.168128955 0.486573109
[665] 0.167834845 0.129795962 0.416056700 0.255846069
[669] 0.693765950 0.249083919 0.958213799 0.764405156
[673] 0.841369813 0.875609934 0.009615549 0.036116867
[677] 0.285924633 0.080531561 0.968794668 0.564562469
[681] 0.984704898 0.836638826 0.880193851 0.244424738
[685] 0.732067207 0.498606006 0.503376291 0.568116239
[689] 0.652677042 0.064712493 0.676612402 0.735371693
[693] 0.111299667 0.046654618 0.130910312 0.880956365
[697] 0.839725489 0.868142720 0.033382926 0.107906996
[701] 0.549296889 0.109317743 0.722590516 0.725010276
[705] 0.789521863 0.617478245 0.503007366 0.456149423
[709] 0.034665802 0.209638843 0.342787279 0.545594801
[713] 0.769484416 0.428764684 0.686461671 0.521868988
[717] 0.796536368 0.903052028 0.432457164 0.331940473
[721] 0.917114556 0.209279713 0.812734745 0.300712818
[725] 0.572182454 0.386782175 0.279481675 0.260042868
[729] 0.959383609 0.634132585 0.081877257 0.926206241
[733] 0.890942530 0.912085011 0.718498562 0.682527713
[737] 0.555186460 0.401709436 0.863837790 0.074413949
[741] 0.283284486 0.162637435 0.763896585 0.116375717
[745] 0.167471821 0.475883521 0.005788442 0.974413290
[749] 0.114147759 0.080774309 0.322919441 0.118785095
[753] 0.985772469 0.404147644 0.241781906 0.354878908
[757] 0.153240084 0.287547626 0.256672147 0.157690251
[761] 0.029771646 0.811628372 0.579172815 0.576731940
[765] 0.701486865 0.063766200 0.502482899 0.383949951
[769] 0.426416348 0.544390162 0.643638207 0.791192387
[773] 0.846853398 0.072626310 0.860110642 0.956164637
[777] 0.926419157 0.053963660 0.246409464 0.487369670
[781] 0.100063382 0.195159257 0.213571525 0.207682233
[785] 0.028771427 0.234566428 0.107185220 0.277315790
[789] 0.696021955 0.266257392 0.660476814 0.212416518
[793] 0.005266102 0.102902506 0.287400001 0.444156097
[797] 0.692992780 0.020884316 0.955751769 0.804110624
[801] 0.911739890 0.309466796 0.323927877 0.138359778
[805] 0.957560620 0.232669176 0.023714907 0.052111627
[809] 0.932719734 0.038751502 0.807113662 0.696882099
[813] 0.388050787 0.390583366 0.982332567 0.216765057
[817] 0.566959663 0.980335618 0.574558311 0.632866844
[821] 0.782992921 0.856875613 0.041593013 0.895225023
[825] 0.188516586 0.387341884 0.494502024 0.370316362
[829] 0.693625107 0.733155085 0.308549734 0.972441896
[833] 0.243334341 0.835028139 0.390575896 0.908863697
[837] 0.607566295 0.607495053 0.894327145 0.975288389
[841] 0.676351208 0.091138610 0.777949302 0.517544394
[845] 0.652370169 0.339297696 0.292120431 0.783712422
[849] 0.319965783 0.489277602 0.134141912 0.653294066
[853] 0.783924127 0.921809286 0.258463774 0.513232899
[857] 0.858583322 0.151413594 0.641468017 0.825477348
[861] 0.972295296 0.173125720 0.710021871 0.110715289
[865] 0.416779255 0.284621345 0.341154850 0.168895211
[869] 0.547096898 0.797205828 0.858630747 0.332270424
[873] 0.515725486 0.371684692 0.436701348 0.742428995
[877] 0.011101299 0.046900366 0.371817852 0.658156762
[881] 0.529650157 0.588913933 0.785948208 0.504572585
[885] 0.593546614 0.872144154 0.891242010 0.939548917
[889] 0.848769896 0.811404374 0.824689138 0.916008334
[893] 0.723415147 0.910254459 0.785772749 0.224662109
[897] 0.025505915 0.382444482 0.920264228 0.842048720
[901] 0.879420725 0.715013948 0.945104592 0.377839050
[905] 0.475368502 0.764876201 0.762309468 0.276107021
[909] 0.013929509 0.191360186 0.508207597 0.357133010
[913] 0.064195928 0.694495361 0.911169011 0.449594197
[917] 0.854834523 0.751918294 0.374545260 0.609409488
[921] 0.792774655 0.386679893 0.124152965 0.267549657
[925] 0.521273961 0.590104377 0.886542718 0.328781780
[929] 0.404275393 0.047014354 0.480532115 0.748229080
[933] 0.443648359 0.468462170 0.731393147 0.293246638
[937] 0.163580966 0.503860285 0.192165966 0.321273932
[941] 0.029732935 0.869980632 0.647166965 0.318806882
[945] 0.293482209 0.669097224 0.407680992 0.663763635
[949] 0.049678547 0.495533216 0.785123125 0.906233663
[953] 0.020996784 0.162863294 0.654907126 0.887484326
[957] 0.925433173 0.448821391 0.039802389 0.661046990
[961] 0.174042276 0.026689433 0.626593738 0.031572743
[965] 0.114810244 0.951679254 0.955425197 0.609623392
[969] 0.104700660 0.692146658 0.891687072 0.737418877
[973] 0.331607591 0.622847309 0.765353283 0.077647318
[977] 0.650668845 0.395230689 0.781125475 0.626846384
[981] 0.015295418 0.916748053 0.133362752 0.079918742
[985] 0.219676089 0.007125481 0.821645687 0.113951277
[989] 0.312969944 0.090880581 0.972712883 0.852674136
[993] 0.498993299 0.274403698 0.291713902 0.053610240
[997] 0.970454900 0.577010068 0.498106406 0.682937474
> mean(u3)
[1] 0.4906725
> var(u3)
[1] 0.08887691
> sd=sqrt(var(u3));sd
[1] 0.2981223
> qqnorm(u3)
> ########### part C
if(u3>0)
{
(u3>2.5)
else
(u3<97.5)
}
if(u3>0)
{
(u3>0.5)
else
(u3<99.5)
}