HomeworkStats Fall Project
mean <- 50
lambda <- 1/mean
z <- rexp(1000,lambda)
print(z)## [1] 31.07853674 15.20050375 163.70364166 13.78178473 15.79945087
## [6] 0.67776702 72.81195694 38.67950037 5.69712289 83.07631583
## [11] 6.23219532 217.87193360 33.05935755 11.68631901 27.41203266
## [16] 32.71744023 31.50431905 0.19668180 48.99107269 29.19385799
## [21] 45.66502166 34.45754200 56.84156585 18.92522837 122.73670965
## [26] 16.12960622 62.62389705 1.52764970 49.65345252 91.49009157
## [31] 112.62185247 69.91578195 95.88578837 17.20480155 5.58128201
## [36] 22.39151092 2.30419382 26.15458879 41.51132277 329.69773737
## [41] 1.40852563 34.08265619 102.93677123 33.10392208 166.10719114
## [46] 52.41673805 47.86135019 53.06615989 51.89894857 24.84331136
## [51] 9.07592401 119.28456053 44.01545170 6.11749880 67.01380288
## [56] 61.87738543 28.72955520 4.08471329 51.79503704 26.39602579
## [61] 53.31788809 67.01673972 5.85798565 117.18327822 52.53071878
## [66] 85.32338235 72.75898439 74.88688174 170.57712785 2.20204832
## [71] 28.19517190 65.19986796 51.61853852 102.09404183 107.64921977
## [76] 5.71944914 54.01678677 58.18833983 14.42747060 5.67818813
## [81] 54.73312035 67.24967756 1.98816233 27.57586173 57.84207522
## [86] 25.23406299 3.38392336 21.41229594 25.55293438 18.85183703
## [91] 138.54270083 56.60759662 63.26103317 98.63701025 120.32121893
## [96] 33.17733225 13.08614700 16.57002172 30.13661611 62.99114530
## [101] 33.11197611 61.51953027 59.56900259 9.06824104 10.84488917
## [106] 11.46575610 13.17863190 49.40196778 42.07018909 41.91669831
## [111] 55.89752567 21.11519924 18.32980479 15.70824170 118.65581692
## [116] 24.49432970 37.68425119 12.21850074 12.32455283 138.32070724
## [121] 28.51753947 135.20988365 155.62603845 31.85257020 71.55977878
## [126] 89.87198425 0.75667573 53.44607951 79.83900290 58.30413173
## [131] 67.09810104 27.77343760 51.41875646 32.85310818 114.82172207
## [136] 49.94560056 416.87517052 80.49175749 61.73917595 42.11326940
## [141] 11.11389399 5.06655299 40.15985076 1.64643486 84.60405198
## [146] 42.03767478 9.55241838 153.94293851 38.46434298 22.38312890
## [151] 87.58451780 114.72141863 90.33322567 39.94978382 300.52617712
## [156] 33.24546739 42.21397061 85.61566798 34.68255313 4.29002734
## [161] 156.85288868 49.54425911 18.31540933 72.99338845 3.31096763
## [166] 20.62121216 121.95327759 276.98599928 2.79481169 8.88044699
## [171] 10.91647905 4.94808394 83.42263130 54.08160284 51.64709687
## [176] 2.64518247 8.20183663 49.23961631 50.19695256 36.13743193
## [181] 150.57897865 75.48532148 119.99799227 16.37609552 28.75865765
## [186] 68.27772162 5.90461559 71.30479542 72.24031940 22.12551213
## [191] 30.28595878 26.27169904 1.19224896 4.71703645 5.15761951
## [196] 35.03234065 0.09454433 39.17986441 18.82657669 10.29084427
## [201] 11.50309075 20.38339451 42.42473366 13.96901384 9.39521263
## [206] 32.81940694 20.04537000 67.20350762 43.73570025 4.91737972
## [211] 10.94318582 123.13824118 55.20911799 60.29346213 50.82465929
## [216] 8.06421480 178.62488566 13.30391178 5.94642019 75.37896884
## [221] 40.11854594 56.18308766 83.00106115 219.13196983 300.13207717
## [226] 19.37697160 18.44321059 59.17549138 54.79787979 24.84307892
## [231] 4.72411981 77.82753713 65.13532284 242.93839865 0.94002730
## [236] 143.43592114 30.24137905 29.72847361 69.92317597 101.01757757
## [241] 1.56537101 19.31419270 42.36848647 16.06191917 21.98336599
## [246] 7.32202025 62.38768691 35.19727755 9.27161886 79.62436843
## [251] 113.83283464 11.69263482 31.82357855 10.26902518 40.10071166
## [256] 57.16774645 9.21881406 42.41401826 3.60073878 15.02338619
## [261] 18.98617295 0.38776987 101.02926101 106.64653396 19.20909025
## [266] 143.21086331 43.15568292 7.39214607 24.95721364 25.10146089
## [271] 12.52230017 22.62254140 179.45776907 21.57305339 40.22403206
## [276] 65.36208512 139.79232926 50.72920262 33.23825763 7.04317894
## [281] 79.48435861 200.75191491 108.15161984 25.95147833 82.81986108
## [286] 55.23600401 37.68584723 50.07000394 94.63378405 41.03531097
## [291] 65.61973803 29.42083247 9.35542346 85.80910936 8.04479041
## [296] 4.29936899 17.48140517 62.74714842 36.21802828 93.28336641
## [301] 8.82131939 1.98864045 0.01831751 74.55962258 20.70784906
## [306] 26.38149799 21.48133779 102.98818536 4.43883650 4.40280866
## [311] 4.06572735 19.98443571 308.57944412 45.61168710 43.68835306
## [316] 38.70410188 6.21765438 23.86869292 18.14582217 1.01655298
## [321] 41.35834514 44.13617714 29.79659985 36.83350188 11.44297124
## [326] 22.18980824 119.75562255 3.53450330 50.68347906 135.44028578
## [331] 13.14026832 83.92247650 52.92620659 26.11266242 142.01942424
## [336] 14.01010344 100.12061074 1.42666250 60.18754542 32.70709380
## [341] 60.01300518 64.17591749 69.82266819 8.96068674 66.33315803
## [346] 28.29347469 15.67902182 98.85013904 91.34321111 9.23447127
## [351] 44.93378941 32.75343995 5.73306068 38.84844803 57.16040707
## [356] 88.58451759 10.27591191 41.97088983 9.85758352 0.78813258
## [361] 60.05877338 55.68548254 28.09662542 91.05669756 15.00521705
## [366] 75.98562159 88.01973313 11.56791842 147.34257757 38.68991999
## [371] 324.26891241 79.13068113 30.41921477 33.86735590 17.17645512
## [376] 45.64547199 12.07692656 87.36066613 99.36163669 35.69873608
## [381] 54.04000944 85.23753378 12.09976687 2.41049964 96.13127922
## [386] 50.22063860 48.08936744 112.17277763 96.89108412 90.37523382
## [391] 0.14243102 38.84723102 16.72031019 23.30357328 123.68916245
## [396] 2.37812535 111.00415860 75.02458744 92.39130979 17.41844204
## [401] 54.24615410 29.11680487 33.70622336 3.57750683 10.35857829
## [406] 236.01503613 38.66537949 14.25496226 16.89258555 16.91290947
## [411] 66.40547821 3.11265211 142.08349622 23.89234791 14.90102028
## [416] 108.75631300 15.00437346 32.89728593 11.03774088 22.46689215
## [421] 118.30159043 61.28407507 9.62371654 98.73291393 5.75713418
## [426] 142.58794895 1.63893799 99.38783291 18.34011581 7.01509497
## [431] 62.40181592 7.88901919 63.92933386 38.10117411 3.99720791
## [436] 20.20636131 55.36275273 30.13249296 26.45365287 21.55086184
## [441] 4.27414637 7.80693604 23.87405632 123.74620771 95.95855744
## [446] 35.76625478 113.17623229 0.57102819 42.69715435 74.36338365
## [451] 14.25040589 12.50543573 80.71999838 11.76585653 68.56335541
## [456] 115.38353070 35.82950174 77.46820012 118.89292271 76.84691611
## [461] 11.02898851 36.60618919 51.93850635 36.02730701 230.81319871
## [466] 119.99929776 15.79454957 53.82273658 9.56901640 30.31784801
## [471] 11.15511062 10.35315315 68.37444398 78.04014839 23.89908181
## [476] 47.69704933 8.99597807 26.46902229 14.88496177 81.10848814
## [481] 146.85435518 0.50680900 39.94959327 14.64198418 18.50691788
## [486] 61.51691063 53.32303578 3.38527205 5.65450124 45.08922505
## [491] 67.49752816 51.07477873 12.77749271 40.47309840 50.97943046
## [496] 128.67161938 28.98250865 19.12836200 10.07471054 11.95185469
## [501] 0.99204956 9.75437609 14.16952652 37.89965158 46.51038905
## [506] 32.12802645 39.69327812 7.22602585 96.88247945 32.13338580
## [511] 16.76195380 9.15761432 82.60368239 6.93266958 83.31240611
## [516] 58.38110684 10.47943942 82.94279855 4.94183307 23.64611179
## [521] 6.76412069 20.47120205 60.85734674 45.38981016 11.58851238
## [526] 15.12213622 126.20510207 65.31948629 165.53869470 145.57311199
## [531] 32.21793564 8.33361994 4.81307646 201.86704424 53.92941739
## [536] 41.09991477 37.99983845 50.12274110 7.12744951 25.25036195
## [541] 43.95307144 110.72459937 15.33196398 37.44597458 14.07304109
## [546] 18.03836878 73.85493977 119.73038184 3.29535359 45.98714746
## [551] 74.73853966 184.69083336 42.19315220 51.88950803 27.02574059
## [556] 12.10819385 113.72715430 0.07441917 155.54684996 4.88321362
## [561] 11.43065967 46.17691864 30.55919544 6.36291873 31.95827194
## [566] 19.51544518 53.86814819 17.15763148 119.25031551 32.59205462
## [571] 0.01719419 47.26985400 37.40853094 48.95274401 11.69492056
## [576] 209.22036641 158.13597328 22.05578727 21.55097956 34.54042496
## [581] 17.96602716 278.27059550 13.79644666 4.10687947 8.57938366
## [586] 72.40007045 37.82409000 33.92065149 0.99637492 43.58612543
## [591] 71.95546021 79.77865701 27.46282602 86.71636646 49.01885297
## [596] 106.03839998 46.18006488 51.96654089 32.53458024 119.03872573
## [601] 0.40220686 37.48008394 8.71413435 44.91586860 53.23550165
## [606] 89.19708198 6.97758464 9.33923230 0.67092928 73.95142465
## [611] 45.76199851 103.06501211 3.83829726 39.01829924 8.99026534
## [616] 9.48207239 12.09961735 15.92123429 46.47606909 127.83561562
## [621] 182.69579313 232.15795847 14.43170763 0.56003095 54.72921566
## [626] 23.73477682 37.84602520 25.50643054 2.10356673 28.25920216
## [631] 28.51435458 57.64208040 72.95451453 14.01163163 31.09757861
## [636] 5.14819529 40.45482383 66.81379047 34.47321251 12.55068781
## [641] 13.38203619 164.35059681 35.87166793 26.03434029 21.93864794
## [646] 7.82165870 71.55252788 65.10848985 79.77195605 35.98430324
## [651] 19.53972646 161.39853857 62.73886738 1.16376991 0.02448659
## [656] 29.42095783 54.96863667 83.71339608 22.62207072 51.02866431
## [661] 0.55373525 37.53902411 44.62831374 13.62733101 50.95138856
## [666] 80.64847076 54.53707902 10.17119028 53.66171859 61.00357757
## [671] 43.81222522 39.68982631 53.76364849 146.39030908 191.84622526
## [676] 1.50165902 27.36445495 25.75317591 29.81619865 22.31272801
## [681] 16.48647767 6.04030292 9.73739173 4.43728887 67.97922374
## [686] 30.99937248 11.53810648 97.81854993 4.79813190 4.25279245
## [691] 26.96963353 27.90989601 57.10442057 43.59968735 15.40309598
## [696] 4.00046261 62.17675931 11.20104140 32.80792364 25.19214731
## [701] 0.98041890 105.71590857 5.29060573 63.93285673 16.65430942
## [706] 23.24076765 2.61729498 13.45307655 60.36995975 0.10786026
## [711] 121.75990426 54.16725967 61.32045178 19.03821107 53.61464433
## [716] 45.48840701 23.72594359 11.59796233 113.74227413 8.94855335
## [721] 56.36476963 22.07743027 16.40556329 17.72877167 9.99698456
## [726] 3.09082121 9.31556664 65.79222376 32.14008994 6.42028796
## [731] 94.58494121 26.15022338 22.28248511 2.67602215 50.29948759
## [736] 9.59608932 11.13718916 11.07948855 94.32602245 46.67279523
## [741] 36.00356672 6.76290917 11.45844623 153.21210585 111.63038490
## [746] 61.68283512 17.00358246 18.73006830 2.94041830 117.94964270
## [751] 68.69605616 50.27948036 17.05506030 18.42307181 7.32628428
## [756] 3.80061031 48.26547976 106.47868158 45.44742092 195.58689770
## [761] 53.56290061 56.15330940 1.02222010 134.52577898 3.75515476
## [766] 52.32463619 4.36898294 22.09418183 1.27630620 26.77742660
## [771] 3.22224586 36.43590370 64.81461669 37.11776463 42.90308389
## [776] 18.73519557 120.05188091 138.84852787 1.35771628 13.95452174
## [781] 47.41913792 118.54354934 67.15257508 24.49482395 300.93693657
## [786] 100.77730687 66.85831956 14.70743783 74.77214830 39.74422565
## [791] 47.91017162 59.73570817 120.56593010 96.47300448 61.65476502
## [796] 83.04722076 23.45006187 48.91105853 76.48058664 32.15418353
## [801] 31.93031759 37.45078821 27.89645114 20.24228356 70.95856042
## [806] 22.40075783 42.57176428 76.97791016 74.94520094 22.66743302
## [811] 49.38376918 7.63465078 94.17557953 28.61872728 13.68665628
## [816] 5.41696318 0.24840588 9.76233361 162.30096444 16.87900187
## [821] 106.42072306 91.80285229 34.35419409 25.32088633 13.36326553
## [826] 148.69723024 113.92801245 0.78533676 10.68069229 21.84723315
## [831] 33.91693945 18.22486003 208.91507719 61.67117781 43.50687903
## [836] 23.55901555 1.55583207 11.32928277 97.09992939 59.78612276
## [841] 16.20321004 76.21285626 8.39530541 10.92593395 56.60354122
## [846] 109.55776115 152.69948140 68.40269021 33.41857388 97.46550889
## [851] 65.55942292 46.03259857 71.18986786 119.45696836 163.15652057
## [856] 238.93336746 4.72033818 59.37055237 50.63409237 5.70298329
## [861] 5.99974749 5.05384254 12.53352392 213.03829494 19.13471380
## [866] 104.32227848 13.33170867 16.62964555 32.21860928 8.68610444
## [871] 14.01446490 12.46631473 12.81795253 3.81507807 3.25644687
## [876] 31.08592171 1.40998033 106.12993510 11.86260103 2.64287014
## [881] 13.06033260 73.60071979 27.37944499 19.05043754 44.59328367
## [886] 72.66723937 101.99667774 21.30383778 246.17611103 9.67164480
## [891] 23.03705120 1.61972439 28.54601080 41.46456322 27.19588552
## [896] 121.43999477 48.50014248 62.91022533 7.71893122 49.62055888
## [901] 22.03187791 12.01603187 0.77616843 9.50818811 46.50170496
## [906] 164.69598970 21.03157891 12.94849841 40.31580854 91.37465776
## [911] 42.16535110 116.21074406 50.80600963 31.52636758 5.97172324
## [916] 41.84652916 1.94849337 20.92215106 91.60019761 127.04167244
## [921] 115.79445144 4.65277738 44.41609758 33.13263238 16.34211952
## [926] 62.05756096 24.63066638 38.69718960 17.78919732 22.74890554
## [931] 10.32245646 101.74661791 122.35541019 25.37162609 19.41257778
## [936] 41.28559264 102.16836556 16.08261040 88.13460889 46.46561281
## [941] 88.23170531 8.76333858 7.03817627 14.42288805 94.05569038
## [946] 3.61885540 63.53717791 49.42983296 6.46134840 67.82519137
## [951] 21.87952220 42.79172855 1.48442998 65.68233352 24.71330892
## [956] 12.03694390 9.68749670 46.17084237 133.46401913 35.22952413
## [961] 54.93991338 78.72527185 34.64509391 68.67343192 87.29457315
## [966] 177.16379992 28.75776265 0.33451572 27.27753525 54.62597473
## [971] 12.15700477 35.39036638 117.37316390 4.30673399 75.93021830
## [976] 71.01700706 93.32829570 81.79838848 22.19288670 82.50465329
## [981] 68.52492690 101.61916185 5.61055459 25.79453140 5.98166164
## [986] 40.03269677 7.32653063 116.66521234 25.58403886 10.10945800
## [991] 44.34135915 6.28993079 58.95545217 27.21009080 23.30383623
## [996] 128.93179645 46.27245199 0.89495673 46.15514404 27.92758425t <- seq(5,50,by=5)
# Estimating P(X>t) the probability of the area at the right of t, where t = 5,10,15,...,50
a <- 1-pexp(t,lambda)
# Estimating P(X>t+10|x>10)
b <- (1-pexp(t+10,lambda))/(1-pexp(10,lambda))
plot(b,a, xlab = "estimated probabilities in b", ylab ="estimated probabilities in a", main ="estimated probabilities comparison", type ="o")
## lines 11 and 15 give exactly the same answers, that being said we have proved the memoryless property of exponential.
## From the graph we can also see that the line y=x meaning the two probabilities are the same.
c <- (1-pexp(t+20,lambda))/(1-pexp(20,lambda))
plot(c,b, xlab ="estimated in c", ylab="estimated probabilities in b", main="estimated probabilities comparison", type="o")
## Question 2
ceiling(z)## [1] 32 16 164 14 16 1 73 39 6 84 7 218 34 12 28 33 32 1
## [19] 49 30 46 35 57 19 123 17 63 2 50 92 113 70 96 18 6 23
## [37] 3 27 42 330 2 35 103 34 167 53 48 54 52 25 10 120 45 7
## [55] 68 62 29 5 52 27 54 68 6 118 53 86 73 75 171 3 29 66
## [73] 52 103 108 6 55 59 15 6 55 68 2 28 58 26 4 22 26 19
## [91] 139 57 64 99 121 34 14 17 31 63 34 62 60 10 11 12 14 50
## [109] 43 42 56 22 19 16 119 25 38 13 13 139 29 136 156 32 72 90
## [127] 1 54 80 59 68 28 52 33 115 50 417 81 62 43 12 6 41 2
## [145] 85 43 10 154 39 23 88 115 91 40 301 34 43 86 35 5 157 50
## [163] 19 73 4 21 122 277 3 9 11 5 84 55 52 3 9 50 51 37
## [181] 151 76 120 17 29 69 6 72 73 23 31 27 2 5 6 36 1 40
## [199] 19 11 12 21 43 14 10 33 21 68 44 5 11 124 56 61 51 9
## [217] 179 14 6 76 41 57 84 220 301 20 19 60 55 25 5 78 66 243
## [235] 1 144 31 30 70 102 2 20 43 17 22 8 63 36 10 80 114 12
## [253] 32 11 41 58 10 43 4 16 19 1 102 107 20 144 44 8 25 26
## [271] 13 23 180 22 41 66 140 51 34 8 80 201 109 26 83 56 38 51
## [289] 95 42 66 30 10 86 9 5 18 63 37 94 9 2 1 75 21 27
## [307] 22 103 5 5 5 20 309 46 44 39 7 24 19 2 42 45 30 37
## [325] 12 23 120 4 51 136 14 84 53 27 143 15 101 2 61 33 61 65
## [343] 70 9 67 29 16 99 92 10 45 33 6 39 58 89 11 42 10 1
## [361] 61 56 29 92 16 76 89 12 148 39 325 80 31 34 18 46 13 88
## [379] 100 36 55 86 13 3 97 51 49 113 97 91 1 39 17 24 124 3
## [397] 112 76 93 18 55 30 34 4 11 237 39 15 17 17 67 4 143 24
## [415] 15 109 16 33 12 23 119 62 10 99 6 143 2 100 19 8 63 8
## [433] 64 39 4 21 56 31 27 22 5 8 24 124 96 36 114 1 43 75
## [451] 15 13 81 12 69 116 36 78 119 77 12 37 52 37 231 120 16 54
## [469] 10 31 12 11 69 79 24 48 9 27 15 82 147 1 40 15 19 62
## [487] 54 4 6 46 68 52 13 41 51 129 29 20 11 12 1 10 15 38
## [505] 47 33 40 8 97 33 17 10 83 7 84 59 11 83 5 24 7 21
## [523] 61 46 12 16 127 66 166 146 33 9 5 202 54 42 38 51 8 26
## [541] 44 111 16 38 15 19 74 120 4 46 75 185 43 52 28 13 114 1
## [559] 156 5 12 47 31 7 32 20 54 18 120 33 1 48 38 49 12 210
## [577] 159 23 22 35 18 279 14 5 9 73 38 34 1 44 72 80 28 87
## [595] 50 107 47 52 33 120 1 38 9 45 54 90 7 10 1 74 46 104
## [613] 4 40 9 10 13 16 47 128 183 233 15 1 55 24 38 26 3 29
## [631] 29 58 73 15 32 6 41 67 35 13 14 165 36 27 22 8 72 66
## [649] 80 36 20 162 63 2 1 30 55 84 23 52 1 38 45 14 51 81
## [667] 55 11 54 62 44 40 54 147 192 2 28 26 30 23 17 7 10 5
## [685] 68 31 12 98 5 5 27 28 58 44 16 5 63 12 33 26 1 106
## [703] 6 64 17 24 3 14 61 1 122 55 62 20 54 46 24 12 114 9
## [721] 57 23 17 18 10 4 10 66 33 7 95 27 23 3 51 10 12 12
## [739] 95 47 37 7 12 154 112 62 18 19 3 118 69 51 18 19 8 4
## [757] 49 107 46 196 54 57 2 135 4 53 5 23 2 27 4 37 65 38
## [775] 43 19 121 139 2 14 48 119 68 25 301 101 67 15 75 40 48 60
## [793] 121 97 62 84 24 49 77 33 32 38 28 21 71 23 43 77 75 23
## [811] 50 8 95 29 14 6 1 10 163 17 107 92 35 26 14 149 114 1
## [829] 11 22 34 19 209 62 44 24 2 12 98 60 17 77 9 11 57 110
## [847] 153 69 34 98 66 47 72 120 164 239 5 60 51 6 6 6 13 214
## [865] 20 105 14 17 33 9 15 13 13 4 4 32 2 107 12 3 14 74
## [883] 28 20 45 73 102 22 247 10 24 2 29 42 28 122 49 63 8 50
## [901] 23 13 1 10 47 165 22 13 41 92 43 117 51 32 6 42 2 21
## [919] 92 128 116 5 45 34 17 63 25 39 18 23 11 102 123 26 20 42
## [937] 103 17 89 47 89 9 8 15 95 4 64 50 7 68 22 43 2 66
## [955] 25 13 10 47 134 36 55 79 35 69 88 178 29 1 28 55 13 36
## [973] 118 5 76 72 94 82 23 83 69 102 6 26 6 41 8 117 26 11
## [991] 45 7 59 28 24 129 47 1 47 28a1 <- 1-pexp(t,lambda)
b1 <- (1-pexp(t+10,lambda))/(1-pexp(10,lambda))
c1 <- (1-pexp(t+20,lambda))/(1-pexp(20,lambda))
plot(b1,a1, xlab = "estimated probabilities in b1", ylab ="estimated probabilities in a1", main ="estimated probabilities comparison", type ="o")
plot(c1,b1, xlab ="estimated in c1", ylab="estimated probabilities in b1", main="estimated probabilities comparison", type="o")
* In conclusion, we have found that the 3 results lead to the same probability distribution. Hence the memoryless property of the exponential has been proven. * Regardless of the fact that all the numbers of the distribution have been rounded up, we still get the same probability density. This is mainly because the probability is dependent of the inverse of the mean and the values within the sequence t(5,10,…,50).
*** Note, for some of the above functions to work, one may need to activate some packages on the user library, those include dyplr, dslabs, stats, graphics, datasets, grDevices. >>> This is the End of the Project<<<