Question

(a) Write an R function rnormmax that has three arguments, n, mu, and sigm,
# and returns the maximum of a vector of n random numbers from the normal distribution
# with mean mu and standard deviation sigm. Make the arguments mu and sigm optional
# with default values of 0 and 1, respectively.

# (b) Write an R code that replicates rnormmax(n=1000) hundred thousand times and creates
# a histogram of the resulting vector via standard hist function.

Answer 1

> t=rnorm(1000,0,1)
> t
   [1] -0.2025522364 -0.5628747394 0.3008293670 -0.0109936054 -1.3858062077
   [6] -1.0962769079 -0.4107434903 -0.9213600548 1.0035003042 -0.2287652089
[11] -0.1893576987 -0.3368195312 1.1492968266 0.2464101887 0.1275073552
[16] 0.0212622805 -1.1876985396 1.1493823781 -0.5495349805 1.2107117707
[21] 0.1476249844 -0.5137755952 0.7154159488 -0.9079026442 0.9561299448
[26] 0.9152880058 0.0531798150 0.7805725446 -0.4689627635 1.2295162792
[31] -1.5463529114 1.3277500555 -0.6857960485 0.5753645844 1.2946730943
[36] 1.5413879495 2.4402947504 0.8391489027 -0.4098228396 0.5674069851
[41] 1.0035146891 -0.0486254537 0.0215929972 -0.1105703858 0.4394433651
[46] 1.2026577962 -0.8336760830 -0.5495587990 0.1331179584 0.6856567892
[51] -0.3440381303 0.1631663831 -0.0949072779 1.1075549157 0.4194171146
[56] 0.6229146401 0.0716801413 -0.7113681939 -0.3922119388 0.3187030466
[61] -0.1981484884 0.1527989123 -0.2197427597 -0.3125554996 -0.0058168765
[66] -0.3333864452 0.8002497373 -0.9048584392 0.6686276084 -0.3769060917
[71] 0.3714546746 1.0371545897 0.6211163640 1.0348307519 0.9547224732
[76] 0.5280907971 -1.8446492578 1.3593743376 -0.8847838502 -1.6531572284
[81] -0.4015734208 0.2643377498 -0.5787477472 1.4719815707 0.3594939962
[86] 0.3282870768 -0.5745701271 -0.1798735695 -0.4119014426 1.2932262841
[91] 0.7229004919 -0.4589662382 -0.3662283877 -0.5225528414 -0.1378938617
[96] -0.1145632847 0.3932730051 -2.0932515616 0.3966931100 0.8927617739
[101] -1.6354471324 -0.1093359641 0.2675152310 0.5453859298 0.5501264148
[106] 0.5293513730 -0.1408506042 -1.6892246992 0.7675472476 -0.2610344597
[111] -0.0378232906 -1.3798153285 0.3157370288 -0.3803098311 -0.2124364966
[116] -0.2849530665 -1.7276265139 -0.3046483811 -0.7738003718 -0.2248115092
[121] -0.2748815879 0.0361064430 0.6220237545 0.5596878638 0.2614530359
[126] -0.4972331978 1.2920001635 -0.4705439438 -0.1166023995 -1.4252332256
[131] 0.7089964238 1.0474221429 0.6067528331 -0.9403456063 1.3385762767
[136] 0.5771250379 0.6278930018 1.0876027066 0.4944085221 0.2091250630
[141] -1.0352714293 -1.4980403551 -0.3706285436 1.7335901319 0.6673666077
[146] -2.2803426750 -0.0056788980 0.2184565331 -1.2256772101 0.4720021056
[151] -1.8838095431 -0.4832002909 -0.3460169051 -1.3122355349 -2.3052494413
[156] 0.2385604711 1.8094952747 0.4648103488 0.5727124343 0.4854108249
[161] 0.0909614677 0.9241014109 0.5492545344 -1.7165811306 -1.4557613700
[166] -0.1546512044 -0.2759981970 1.0201058149 -0.2776869610 -0.8527108929
[171] -0.3418175450 -0.0693307020 -0.3225739425 1.6739460969 -0.2265769340
[176] -0.2802373394 0.0673886795 -0.2604722815 -0.7654919623 0.0497559056
[181] 0.0387649155 1.5625207710 -0.4624629431 -0.3799278336 -0.9715111231
[186] 1.2477611681 0.3748317460 0.2773003468 0.8189999985 -0.2405951842
[191] -2.9023942032 0.4289912594 0.8790687109 -0.1225902032 1.0734815627
[196] -0.8889225905 1.1138304904 0.4893193643 -0.3519402895 -0.0532397498
[201] 0.0681823932 1.9106881252 0.5303131066 -1.0482230633 -1.5455368745
[206] 0.3927506928 -0.5973904842 0.4607816957 -0.1184833355 0.5013185369
[211] -0.9257726257 0.4036309429 -0.1048784607 0.0962906019 1.3508240386
[216] -0.5284700256 1.2251854935 -0.4524261686 -0.6877876995 1.8435222891
[221] -0.8981120073 -0.5329528079 -0.1199703087 -0.6945806316 0.0173226127
[226] -0.5252919367 0.1690543594 -0.9733771358 -0.4381525474 -0.0534270074
[231] -0.1298956717 1.6721701851 0.7005755692 1.2566961939 -0.7486124099
[236] -0.0347105508 -0.7074079574 2.3355110143 -0.2456218950 -0.2290159140
[241] -1.2461296948 1.9902288020 1.2248503730 0.2903711810 -0.7355894686
[246] 0.5261391475 1.3923401939 -0.6358666526 1.4424399936 1.8306958836
[251] 2.4213035383 0.0837812302 1.6425769993 0.7746677664 -0.0497985577
[256] -1.9570295902 -0.6319616357 -1.1310456029 0.1066941146 -1.5054429991
[261] 0.4696454255 0.2263526487 0.3905677629 -0.7445831666 0.2253111180
[266] 0.0860805830 0.0985692203 -1.6823983828 1.3297486793 -0.5502847221
[271] 0.5549378471 -1.3025372254 0.1165715299 1.8348790169 0.6559939187
[276] 0.0442428148 -1.6504688524 0.2468442698 0.4567844533 -0.3169586269
[281] -0.2252031159 0.5519666751 0.6579709070 -1.5195197139 0.7018229172
[286] 1.8976091251 -0.4675660837 -0.9865530451 0.5673402079 -1.3704862272
[291] -0.2130310577 -0.4034004471 -1.3799468390 -0.3694589173 -0.1462071054
[296] -0.6069931918 0.7261927725 1.3081854450 -0.3746482453 1.2451163625
[301] -0.9125328761 0.9289978665 2.6891318928 -0.2586707326 -1.0070125502
[306] 0.3603336147 -1.0583139642 -0.0841236383 1.0215098273 1.0337456989
[311] -0.3266162307 -0.6234850822 -0.1517971175 -0.1126043581 1.0096082536
[316] -1.6520311288 -0.5499717296 0.0429793409 -0.0752242226 0.6688033515
[321] -0.6885596083 0.4653900893 -1.5244869072 -1.1171355401 -0.4199355616
[326] 0.3631389084 0.1591553201 0.0237735846 2.1823580672 2.3367839389
[331] -0.6562386040 0.1430467205 -0.8611223386 0.5803456594 0.1968340588
[336] -0.3482853396 0.3955895806 0.6257726279 -0.1638736557 1.3918778205
[341] 0.2918034860 -1.5905217299 1.1631371069 0.4759245338 0.5751176725
[346] 0.0150661102 -0.8906693662 0.0972057298 0.4037993881 1.6068118375
[351] -1.6004795492 0.1882771921 0.1848302009 0.4150622129 -2.1537787052
[356] 0.6673806943 -0.2541352003 -0.2179916424 0.9195637575 -0.3308583099
[361] -0.2258694649 0.6257730583 0.2785488052 0.7635807053 1.3227192226
[366] 0.9739855938 -0.4870525713 1.6225563758 -0.1462025782 -1.2628124613
[371] 1.8895307356 0.7691235818 -0.5476250834 0.4645921413 -0.5464979680
[376] 0.8736952160 -1.5829098206 -0.9482162218 -0.2228455835 -0.3880619261
[381] -0.2141438067 0.6949777714 2.0830916173 1.2848939381 1.2794629002
[386] -0.2910378892 1.0768057444 -0.2026091824 -1.0908672330 0.0179094787
[391] -0.2383015052 0.6060615429 1.1884873587 0.2443356144 1.5881090284
[396] 1.4540601009 0.3311912352 1.0291899196 0.2412533253 0.9193650717
[401] -0.0654841823 -0.4741903806 0.8555858843 0.0280003309 0.8169500141
[406] 1.3549886007 -0.2478626954 2.1208050706 1.2261948898 1.0929750617
[411] -0.6289900900 -1.1932378657 -0.5214520393 -0.8655393555 1.4502974787
[416] 2.2923100172 -0.5582308091 1.2484139100 0.2590657379 -2.5241867219
[421] -0.1910783869 -0.9446639654 0.1428637860 -0.2725308100 -0.5627612462
[426] -0.6580382764 -1.5773343426 1.3906270799 0.1131595136 -0.8602791459
[431] -1.1302796253 -0.1047571987 0.4879611412 1.3701551546 0.3098070859
[436] -0.0831686569 0.9222211151 -1.4407204861 0.5284084718 0.0597960994
[441] 0.9036362772 -2.2536380053 -0.2113691496 0.9187724246 0.4670220266
[446] 0.1646240428 0.7597226237 -1.7588754831 -0.7403506108 -1.9511788069
[451] -0.4880138273 -0.3048329160 -0.7225816599 0.8481880727 -0.7530278870
[456] 0.7321343650 -1.1482381580 1.2271838361 -1.2004626373 0.5813847374
[461] 1.5033792898 0.2589796849 -1.2329232751 1.5764566921 -0.0483379271
[466] 0.1054956620 -0.3600173275 -0.1897516109 0.0220610662 0.0513737103
[471] -0.7213467797 0.5429561532 -0.2664934972 2.1995922957 0.5423069511
[476] 0.6465778025 1.0694886605 0.4745449712 -0.1184987694 -0.4964397429
[481] 0.0241505338 -1.8137380461 -0.4413217248 1.7276270519 -2.1521126447
[486] -0.8045814682 0.7782749523 -0.3821389208 0.4123923539 0.2372050732
[491] 0.2847092770 -0.8125047101 -0.3177382266 0.1286939771 -1.3705712406
[496] 0.1290176206 2.2511802661 -1.0565634905 0.2979908197 -0.8519048973
[501] -0.9583495140 -0.0839868743 0.3570596725 0.2738153128 0.8235904047
[506] -0.8192518817 0.2003253370 -0.4759068901 -1.1490941684 -0.2271246907
[511] 0.1477184440 1.8149550172 0.1338182546 0.9225289409 -0.2288736627
[516] 0.0281198056 -0.0015688186 0.1176072514 -0.5216006271 1.3486204248
[521] 0.0042147465 0.4077771608 1.3180463568 1.8686562111 -0.8830259975
[526] -1.5558168267 0.2243064102 -1.0240114130 0.4061856272 0.6450547696
[531] -1.1244732146 -0.4089068168 0.2702221483 1.1043985537 0.3833591805
[536] 0.5023906351 -0.2782964847 1.5553542498 -0.1581569029 1.7999941434
[541] -0.4562172867 -0.9735227507 -2.7918661272 -0.4885742379 -0.1442111559
[546] 0.9306161431 0.8262168593 -0.8675169697 0.3549181138 0.0383216395
[551] -0.6539441697 0.3533190190 -0.9888945504 -0.7273937994 -1.0018624035
[556] 0.9959127661 0.1796787852 0.2794359836 0.0337487391 0.2169360497
[561] 0.9233530205 -0.9706109671 -0.9142521953 1.4310574974 -1.2025934468
[566] 0.4539987524 -0.3313439667 0.5474132001 2.4584080514 -1.5734152526
[571] 0.1585364028 -0.2031045472 -1.0694911271 -0.3413409296 -1.5404381972
[576] 0.2115576330 -0.0400667516 1.5467148524 -0.4010646259 -0.2109143404
[581] -1.2043219696 0.0889806338 0.4012528597 -0.4995257052 0.0721152257
[586] 1.0809195779 0.8674474489 1.5079709555 0.1058030966 0.0617148380
[591] 0.0376292328 -0.3930555385 -0.3343346127 -1.5732488985 -0.0717255239
[596] -0.6015760461 1.3603736137 -0.0752886720 -0.1292807843 2.6279209830
[601] 0.0455296615 -0.6212435238 -0.0762190283 0.8643313243 0.6478249188
[606] 1.6862427578 -0.2336247048 -0.1492314386 -0.2947612216 -1.3469762447
[611] 0.2095358956 -0.5998770223 1.0366479537 -2.2634089492 -0.3300725712
[616] -0.7656621677 1.0260070282 -1.2892935698 1.2343083247 0.4095433685
[621] 0.3151639688 1.4063281845 -0.0664833564 0.8597618308 0.3636957891
[626] -0.5314789145 -0.0490763935 1.7584796002 -1.5137473584 0.2244716755
[631] -1.4208371968 -0.9032490317 -1.0064585233 -0.7739049383 -0.1230299392
[636] -0.5369971828 0.7763913215 0.0725337998 1.3420969707 0.7506599384
[641] 0.8081633562 1.2949831000 1.1543429506 -1.0579547168 0.1469428261
[646] 1.2041718894 -0.5031059892 -1.4370073518 -0.2961384980 0.5866857532
[651] 1.7237128395 0.1289828167 -0.3974472810 -0.1717501963 0.1840754202
[656] -2.0488973975 -0.9040558378 -0.1887247464 0.4189533910 -0.2940189003
[661] -1.0647253813 1.0675945396 -1.7001960251 -0.8213046659 -0.4292963576
[666] -0.1419867395 -0.2523163861 -0.5034848517 1.6205813577 -0.9198642984
[671] -2.2815878561 2.0069212673 1.2851827738 -0.1081651585 -0.7888897090
[676] -0.3422770624 0.6009313369 0.6179863375 0.5917973490 1.0805804145
[681] 0.4802099041 -1.3927499953 0.0881291867 0.1024914542 -0.0069081810
[686] -0.3319110002 1.7474602145 1.1213831376 -0.4850091102 -0.2482988401
[691] 0.1500464637 -1.0566799940 0.4437819984 0.7410912345 -0.3446143199
[696] -1.1766983534 1.4314953740 -0.0906640444 1.6160234295 1.4126012670
[701] 0.0526620907 -0.6238274618 1.5513567481 -0.0499453835 -0.0877633596
[706] -0.2822211892 0.0126528912 0.9566042125 -0.6480352312 1.0753458010
[711] -1.5353609706 -0.4527125954 -0.0226842697 0.2180156363 0.3941926967
[716] 0.3355424995 0.8620386665 -0.7008064992 -1.5239781204 0.6605754957
[721] 1.3320421496 -0.4581911619 -0.7530934936 0.8089266495 0.1833977991
[726] 0.2518361672 -0.0208305644 -0.7743529255 -0.5470275968 0.5608024360
[731] -1.2927490080 -1.2592401084 -1.1216186043 1.8744969681 -0.4662144739
[736] -0.7799670090 0.2853911069 -0.5874754601 0.8596082195 0.6941070214
[741] 1.1444028021 1.2941795739 0.2763331015 1.3732324786 0.7858863137
[746] 2.4819165409 -0.6049604536 -0.4760087252 0.4645846561 -0.1285365344
[751] 1.6112714903 -0.0822862867 -1.0177379158 -1.0645297566 -1.0664905562
[756] -1.3192808108 -0.4403430908 1.6494052300 0.2163710859 -1.0256137403
[761] 1.1043271903 0.3895521134 0.8210252534 0.2403786529 1.6833057021
[766] -0.6368842652 -0.3532137735 0.3141642029 -0.1807452018 0.1304195619
[771] -0.0321070307 -0.0279149320 -2.4968672451 0.3869069960 -0.8412269233
[776] 0.3934260423 0.8613270625 0.9782614554 -0.5025541329 0.5605497685
[781] -1.1820689627 -0.2925104982 -0.7302979023 -2.1881624439 -0.7831335319
[786] 0.1325351214 1.4387742089 1.0893274658 -0.2681151926 -1.2653356247
[791] 2.0591460915 0.8660551336 -0.2414855172 1.2341549536 0.9787053748
[796] 0.1676958877 0.0002627212 -0.3436245738 0.3355070821 -1.0961507503
[801] 0.9096701387 0.9343824669 -0.7292963676 0.0935680795 -0.9214714620
[806] -1.2169640097 -0.0004269139 3.3232361257 0.4565090582 -0.9933600850
[811] 2.2115017111 0.3296333965 0.9121033943 0.4605937155 -1.5097314232
[816] -0.2792864031 -0.0780261053 0.9293770198 1.9793188106 0.7946419635
[821] 0.9964196650 0.4683186098 -0.6405469838 0.3390313210 0.5907919632
[826] 0.5591667108 -0.8367462055 0.7166699362 -0.4601088877 -1.1374632502
[831] 1.5140403839 1.4578163137 -0.9691587517 -1.2967940816 -1.2730160568
[836] -0.2348633234 0.3372999748 2.3536463206 -0.2579395111 1.1694337793
[841] 0.9689264477 -1.4074859837 2.0076428301 0.6059970961 0.4454152741
[846] 0.6601473190 1.1852762469 1.0476314751 -2.0052641658 -0.4152799083
[851] 0.6768107811 0.7883943984 -0.2534256709 -1.2713486914 -0.9016666693
[856] -0.9094724290 0.9472326735 -0.7657601596 0.6621015323 0.8837176282
[861] 0.6160946025 0.9724531843 -0.3947072440 0.0911752472 0.1538235362
[866] -0.0760096639 0.7894791585 0.1982334740 -1.5294479106 1.4653507324
[871] -1.0511147531 1.5074226763 0.5124224372 0.6815433922 0.5351481503
[876] 0.8984217141 1.1756350304 0.2563228332 0.4182695243 -0.4856156007
[881] 0.6138598060 2.0438838161 -0.0553659216 0.7085852343 -0.6921416458
[886] -0.0031843341 0.2757759790 1.3061246018 1.8925396249 -1.0233122247
[891] 0.9226713096 -0.1431867236 0.4176144558 0.9641666005 0.3126338809
[896] -0.1602993676 0.5592487129 -0.8712203120 1.7828855604 0.9580406423
[901] 1.7787747426 0.5622376974 -0.0546073628 -0.3298531806 0.7126948071
[906] -0.1904715929 -0.0304130447 0.4859526168 0.2183964166 -2.8138725719
[911] -0.5146259790 -0.0028629758 -2.2429701506 -1.0343403313 -0.1726532944
[916] 0.9382250427 2.0023552270 1.4577895139 1.6551450583 -0.4303598034
[921] -0.5970542750 -1.0989064725 -0.4677631386 0.2887607673 -1.2022388250
[926] -0.9089280343 0.0297518274 -0.4937845154 1.7398197302 0.0933517932
[931] -0.2314156103 0.4891114180 0.5352807415 0.7056495389 -1.2987299152
[936] 0.0377396263 0.8930964965 0.9749332113 0.6498870308 -0.1425854823
[941] -0.5512767320 1.3404459214 -0.5178598029 -0.1221323915 0.2635844005
[946] 1.5303607934 1.4323874394 -0.0285541637 -0.3224577745 0.0426622302
[951] 1.0010185024 0.3349582252 1.4395941896 0.6995824455 -0.7403406882
[956] -0.2560683186 0.1363601997 -0.1498333014 -0.4572043915 -0.3165032880
[961] -0.8590543665 0.9892851110 -0.1196541699 0.8411344036 0.0138365932
[966] 0.0485593163 -0.3081305927 -1.0641868817 -0.3778310990 0.5628966410
[971] -0.8917892261 -0.1487478563 -0.0228908345 1.1538592605 0.9726759817
[976] -1.3952401339 0.1609346410 -0.2248287568 1.8758701039 0.6716424041
[981] -0.4542886703 0.9908003553 0.5669183114 -1.5805776525 1.1266489295
[986] -1.0703544398 -0.6409674196 -0.6764726007 -1.4095410834 -0.8150808612
[991] 0.7511286628 2.2711968103 0.6441377107 -0.2725438063 2.4297589286
[996] 0.4252838814 1.2545811617 -0.0469414671 0.6087547070 0.7241390714
>

> max(t)
[1] 3.018436

# (b)

t=rnorm(1000,0,1)
t
hist(t)