x<-0:20 # generate an integer sequence from 0 to 20
p<-dbinom(x,size=20,prob=1/4) # p is a prob vector, p=Pr(X=x)
round(p,digits=4) # display the vector p
## [1] 0.0032 0.0211 0.0669 0.1339 0.1897 0.2023 0.1686 0.1124 0.0609 0.0271
## [11] 0.0099 0.0030 0.0008 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## [21] 0.0000
1-pbinom(8,size=20,prob=1/4) # Pr{X>8} = 1 – Pr{X<=8}
## [1] 0.04092517
1-sum(p[1:9]) # Pr{X>8} = 1- Pr{X<=8}
## [1] 0.04092517
(q<-seq(0,1,0.1)) # create and display q
## [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
qbinom(q,size=20,prob=1/4) # quantiles from bin(n,p) corresponds to q
## [1] 0 3 3 4 4 5 5 6 7 8 20
round(cumsum(p),digits=4)
## [1] 0.0032 0.0243 0.0913 0.2252 0.4148 0.6172 0.7858 0.8982 0.9591 0.9861
## [11] 0.9961 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
## [21] 1.0000
set.seed(123) # set initial seed
x<-runif(500,0,20)
x
## [1] 5.751550402 15.766102709 8.179538436 17.660348080 18.809345686
## [6] 0.911129988 10.562109761 17.848380888 11.028700289 9.132294706
## [11] 19.136666907 9.066683124 13.551412709 11.452668039 2.058493653
## [16] 17.996499408 4.921754687 0.841190671 6.558414386 19.090072983
## [21] 17.790786321 13.856068123 12.810136275 19.885395532 13.114115982
## [26] 14.170609363 10.881320494 11.882840409 5.783194746 2.942272946
## [31] 19.260484651 18.045980902 13.814105568 15.909348354 0.492273690
## [36] 9.555919422 15.169190750 4.328158717 6.363620153 4.632515707
## [41] 2.856000448 8.290926716 8.274486526 7.376909018 3.048894955
## [46] 2.776121269 4.660681989 9.319249005 5.319452807 17.156554307
## [51] 0.916623333 8.844001484 15.978496913 2.437985200 11.218959675
## [56] 4.130627792 2.550633005 15.066157286 17.900907183 7.489255518
## [61] 13.302303893 1.896813218 7.679392756 5.487672891 16.292800778
## [66] 8.970326828 16.201287061 16.247790190 15.886846422 8.796633752
## [71] 15.089503173 12.584422631 14.203648027 0.012495467 9.506331482
## [76] 4.402377703 7.596330754 12.255420065 7.035958185 2.222708487
## [81] 4.872389454 13.361111749 8.352935594 15.763916681 2.057292885
## [86] 8.697854830 19.699139600 17.861022288 17.729381216 3.501053005
## [91] 2.613913831 13.062038501 6.870329445 13.135162559 6.407464850
## [96] 3.753822385 15.645886026 1.871899734 9.335580831 10.230109198
## [101] 11.999779185 6.656470806 9.772260673 19.089476550 9.658047943
## [106] 17.807004442 18.288763738 12.174699646 8.213795531 2.941893819
## [111] 18.705996065 6.024577999 1.214411431 18.954538801 14.411925469
## [116] 2.845885912 10.985693121 19.081824771 11.709667062 8.090205635
## [121] 12.957869587 6.396412342 6.154400217 4.395352625 7.389777317
## [126] 19.684384069 3.084046016 1.820879998 2.838138156 13.800142030
## [131] 12.385129668 17.827882343 13.459981852 14.741554759 10.422714516
## [136] 13.196768994 16.436109203 15.725631034 19.596438347 8.788630725
## [141] 6.234044041 8.189499052 0.209342237 3.676990480 16.854586378
## [146] 4.623235641 4.781999113 1.533823307 4.914473561 14.642704111
## [151] 16.949063302 9.950545342 7.758180597 4.928979883 2.221929226
## [156] 7.799888705 11.438706280 4.337855256 8.895360040 4.359813374
## [161] 10.045991265 7.078091437 12.999703176 7.494279132 7.108907616
## [166] 10.673758909 14.806687208 4.422058756 8.254922372 5.313733737
## [171] 12.599461069 3.676569816 17.272882229 14.931360083 13.365692995
## [176] 12.360357465 7.444761205 10.596713717 17.493646857 11.635001996
## [181] 16.795355296 6.248963298 14.165806444 5.300356122 11.886863881
## [186] 9.625796010 5.300654629 11.291808696 18.263764461 18.037487790
## [191] 5.483332430 6.429655128 19.712817688 12.399866204 18.746281783
## [196] 9.330654046 8.136651865 13.184606484 3.046932337 11.457341164
## [201] 4.774520537 19.247178729 12.027314520 10.300594545 8.051466844
## [206] 17.604930825 7.281837296 5.764785614 3.412904702 3.443434928
## [211] 9.640852110 5.059298570 4.325095792 13.487527762 0.953272549
## [216] 14.017061749 7.037772769 8.178879959 16.419026479 18.377146963
## [221] 5.650566602 19.222095874 14.567888565 13.727501640 1.056878855
## [226] 7.904402693 9.556907597 11.205065274 13.965231897 18.313670768
## [231] 12.367024547 8.568430175 10.841607344 1.169569772 5.217137141
## [236] 7.943039066 3.954894734 16.638551257 3.057744456 16.068370841
## [241] 10.936523131 13.246352840 3.433969878 12.661107196 6.237394935
## [246] 14.491086928 7.978796493 19.387128223 19.347967412 14.534050780
## [251] 5.144334929 4.435758693 11.860913034 5.350428643 10.621407977
## [256] 15.705833426 3.361216211 8.087983620 9.431525553 17.362136138
## [261] 18.514159117 17.639551186 13.483736850 19.003339587 10.328897885
## [266] 11.530380426 6.726624118 6.946492628 0.400486011 10.056260913
## [271] 17.420868273 0.126015679 1.441142480 3.284224500 15.406681485
## [276] 14.703686116 19.437512713 9.329447537 1.487690266 12.976362486
## [281] 15.171863390 2.742121620 7.931691892 4.499706584 1.159171211
## [286] 7.917853766 1.298566000 4.517728658 1.092582182 13.405640791
## [291] 5.954835652 2.014431646 1.438081944 17.608811376 15.084948041
## [296] 16.332117752 19.642807473 2.071992899 1.980836587 15.976632223
## [301] 15.691505331 0.188598102 15.581317656 14.587813038 12.602637056
## [306] 9.618216599 3.132737027 0.164310397 9.049167885 9.845866575
## [311] 7.791742231 9.293318838 14.265580014 1.106038527 7.095661950
## [316] 16.056245458 16.714176740 4.754988099 7.079722052 17.137708403
## [321] 17.075267476 5.917909108 2.940966492 14.079841226 2.076133764
## [326] 0.674555460 19.988090559 0.697496082 6.767825675 18.301275233
## [331] 12.344705393 5.725706974 14.755948060 16.681086184 6.285415608
## [336] 9.851330956 13.947475324 12.829247080 12.878458300 19.557068096
## [341] 8.294706689 2.388095600 10.520593203 4.501467017 9.728235281
## [346] 7.404295970 19.667003616 7.766382303 4.584896807 12.465950926
## [351] 2.730803941 19.349389886 10.301436158 3.261406585 12.438045903
## [356] 19.719083305 13.375430349 8.378317938 6.466899863 16.705106399
## [361] 2.876340873 3.856318938 17.934773653 6.162391086 7.266010870
## [366] 15.678929575 3.867573598 0.355316238 8.132157326 9.663353385
## [371] 8.436899004 6.856176034 17.329666293 9.102161019 10.675297473
## [376] 19.276866638 15.491830842 4.177526971 6.175736655 19.426849005
## [381] 11.698001870 15.216472507 7.454187889 15.383878225 10.753543661
## [386] 18.279908993 3.705928838 5.644368348 1.899248264 4.209741582
## [391] 19.541979926 5.926043508 14.519660538 15.713756685 2.108354922
## [396] 4.791892581 5.410897448 2.021169881 2.358276825 19.824731117
## [401] 19.721085946 2.741349428 18.106191633 11.526036751 7.908977182
## [406] 8.996049683 14.130038023 1.650054916 6.786251604 13.615751024
## [411] 6.338984962 16.631371961 4.303441653 9.958978724 5.520993453
## [416] 3.840466375 19.012425281 6.434510755 9.569127676 0.559851439
## [421] 10.949189360 12.884804411 11.925270893 6.438747514 17.822286249
## [426] 12.525138948 6.058098306 7.764093275 3.209501849 17.251037951
## [431] 19.062024276 11.272893669 6.590948268 19.932344374 4.696393516
## [436] 12.253439305 2.163570588 9.740651399 1.988916467 3.223315254
## [441] 5.659857416 11.677446719 14.634153177 3.310418222 17.329355264
## [446] 14.171482744 15.207990725 2.941682171 7.161139268 13.466649628
## [451] 10.476452033 6.996035855 4.810614032 1.163835926 4.732394842
## [456] 17.801558264 16.236548321 14.950326444 3.098234516 2.494841772
## [461] 19.494516170 8.722599922 9.280332536 3.305961625 11.698731147
## [466] 5.415560389 4.601938408 13.824156630 5.657047993 16.207961324
## [471] 1.878332142 16.440601423 8.548565614 15.117745195 13.247710131
## [476] 8.890547939 12.542923689 0.009306981 4.344870280 14.097444913
## [481] 4.303329764 16.278674426 6.155277486 13.754853713 18.653621580
## [486] 2.315593297 2.554113758 13.564477148 8.578971918 16.688019559
## [491] 19.428761336 1.409779554 9.195703864 14.031702369 1.738803359
## [496] 19.858880322 5.061979881 0.990768713 13.726498936 15.738547053
y<-matrix(x,ncol=10)
y
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 5.7515504 0.91662333 11.9997792 16.949063 4.7745205 5.1443349
## [2,] 15.7661027 8.84400148 6.6564708 9.950545 19.2471787 4.4357587
## [3,] 8.1795384 15.97849691 9.7722607 7.758181 12.0273145 11.8609130
## [4,] 17.6603481 2.43798520 19.0894765 4.928980 10.3005945 5.3504286
## [5,] 18.8093457 11.21895968 9.6580479 2.221929 8.0514668 10.6214080
## [6,] 0.9111300 4.13062779 17.8070044 7.799889 17.6049308 15.7058334
## [7,] 10.5621098 2.55063300 18.2887637 11.438706 7.2818373 3.3612162
## [8,] 17.8483809 15.06615729 12.1746996 4.337855 5.7647856 8.0879836
## [9,] 11.0287003 17.90090718 8.2137955 8.895360 3.4129047 9.4315256
## [10,] 9.1322947 7.48925552 2.9418938 4.359813 3.4434349 17.3621361
## [11,] 19.1366669 13.30230389 18.7059961 10.045991 9.6408521 18.5141591
## [12,] 9.0666831 1.89681322 6.0245780 7.078091 5.0592986 17.6395512
## [13,] 13.5514127 7.67939276 1.2144114 12.999703 4.3250958 13.4837369
## [14,] 11.4526680 5.48767289 18.9545388 7.494279 13.4875278 19.0033396
## [15,] 2.0584937 16.29280078 14.4119255 7.108908 0.9532725 10.3288979
## [16,] 17.9964994 8.97032683 2.8458859 10.673759 14.0170617 11.5303804
## [17,] 4.9217547 16.20128706 10.9856931 14.806687 7.0377728 6.7266241
## [18,] 0.8411907 16.24779019 19.0818248 4.422059 8.1788800 6.9464926
## [19,] 6.5584144 15.88684642 11.7096671 8.254922 16.4190265 0.4004860
## [20,] 19.0900730 8.79663375 8.0902056 5.313734 18.3771470 10.0562609
## [21,] 17.7907863 15.08950317 12.9578696 12.599461 5.6505666 17.4208683
## [22,] 13.8560681 12.58442263 6.3964123 3.676570 19.2220959 0.1260157
## [23,] 12.8101363 14.20364803 6.1544002 17.272882 14.5678886 1.4411425
## [24,] 19.8853955 0.01249547 4.3953526 14.931360 13.7275016 3.2842245
## [25,] 13.1141160 9.50633148 7.3897773 13.365693 1.0568789 15.4066815
## [26,] 14.1706094 4.40237770 19.6843841 12.360357 7.9044027 14.7036861
## [27,] 10.8813205 7.59633075 3.0840460 7.444761 9.5569076 19.4375127
## [28,] 11.8828404 12.25542007 1.8208800 10.596714 11.2050653 9.3294475
## [29,] 5.7831947 7.03595818 2.8381382 17.493647 13.9652319 1.4876903
## [30,] 2.9422729 2.22270849 13.8001420 11.635002 18.3136708 12.9763625
## [31,] 19.2604847 4.87238945 12.3851297 16.795355 12.3670245 15.1718634
## [32,] 18.0459809 13.36111175 17.8278823 6.248963 8.5684302 2.7421216
## [33,] 13.8141056 8.35293559 13.4599819 14.165806 10.8416073 7.9316919
## [34,] 15.9093484 15.76391668 14.7415548 5.300356 1.1695698 4.4997066
## [35,] 0.4922737 2.05729289 10.4227145 11.886864 5.2171371 1.1591712
## [36,] 9.5559194 8.69785483 13.1967690 9.625796 7.9430391 7.9178538
## [37,] 15.1691908 19.69913960 16.4361092 5.300655 3.9548947 1.2985660
## [38,] 4.3281587 17.86102229 15.7256310 11.291809 16.6385513 4.5177287
## [39,] 6.3636202 17.72938122 19.5964383 18.263764 3.0577445 1.0925822
## [40,] 4.6325157 3.50105301 8.7886307 18.037488 16.0683708 13.4056408
## [41,] 2.8560004 2.61391383 6.2340440 5.483332 10.9365231 5.9548357
## [42,] 8.2909267 13.06203850 8.1894991 6.429655 13.2463528 2.0144316
## [43,] 8.2744865 6.87032945 0.2093422 19.712818 3.4339699 1.4380819
## [44,] 7.3769090 13.13516256 3.6769905 12.399866 12.6611072 17.6088114
## [45,] 3.0488950 6.40746485 16.8545864 18.746282 6.2373949 15.0849480
## [46,] 2.7761213 3.75382239 4.6232356 9.330654 14.4910869 16.3321178
## [47,] 4.6606820 15.64588603 4.7819991 8.136652 7.9787965 19.6428075
## [48,] 9.3192490 1.87189973 1.5338233 13.184606 19.3871282 2.0719929
## [49,] 5.3194528 9.33558083 4.9144736 3.046932 19.3479674 1.9808366
## [50,] 17.1565543 10.23010920 14.6427041 11.457341 14.5340508 15.9766322
## [,7] [,8] [,9] [,10]
## [1,] 15.6915053 2.7308039 19.7210859 10.476452033
## [2,] 0.1885981 19.3493899 2.7413494 6.996035855
## [3,] 15.5813177 10.3014362 18.1061916 4.810614032
## [4,] 14.5878130 3.2614066 11.5260368 1.163835926
## [5,] 12.6026371 12.4380459 7.9089772 4.732394842
## [6,] 9.6182166 19.7190833 8.9960497 17.801558264
## [7,] 3.1327370 13.3754303 14.1300380 16.236548321
## [8,] 0.1643104 8.3783179 1.6500549 14.950326444
## [9,] 9.0491679 6.4668999 6.7862516 3.098234516
## [10,] 9.8458666 16.7051064 13.6157510 2.494841772
## [11,] 7.7917422 2.8763409 6.3389850 19.494516170
## [12,] 9.2933188 3.8563189 16.6313720 8.722599922
## [13,] 14.2655800 17.9347737 4.3034417 9.280332536
## [14,] 1.1060385 6.1623911 9.9589787 3.305961625
## [15,] 7.0956620 7.2660109 5.5209935 11.698731147
## [16,] 16.0562455 15.6789296 3.8404664 5.415560389
## [17,] 16.7141767 3.8675736 19.0124253 4.601938408
## [18,] 4.7549881 0.3553162 6.4345108 13.824156630
## [19,] 7.0797221 8.1321573 9.5691277 5.657047993
## [20,] 17.1377084 9.6633534 0.5598514 16.207961324
## [21,] 17.0752675 8.4368990 10.9491894 1.878332142
## [22,] 5.9179091 6.8561760 12.8848044 16.440601423
## [23,] 2.9409665 17.3296663 11.9252709 8.548565614
## [24,] 14.0798412 9.1021610 6.4387475 15.117745195
## [25,] 2.0761338 10.6752975 17.8222862 13.247710131
## [26,] 0.6745555 19.2768666 12.5251389 8.890547939
## [27,] 19.9880906 15.4918308 6.0580983 12.542923689
## [28,] 0.6974961 4.1775270 7.7640933 0.009306981
## [29,] 6.7678257 6.1757367 3.2095018 4.344870280
## [30,] 18.3012752 19.4268490 17.2510380 14.097444913
## [31,] 12.3447054 11.6980019 19.0620243 4.303329764
## [32,] 5.7257070 15.2164725 11.2728937 16.278674426
## [33,] 14.7559481 7.4541879 6.5909483 6.155277486
## [34,] 16.6810862 15.3838782 19.9323444 13.754853713
## [35,] 6.2854156 10.7535437 4.6963935 18.653621580
## [36,] 9.8513310 18.2799090 12.2534393 2.315593297
## [37,] 13.9474753 3.7059288 2.1635706 2.554113758
## [38,] 12.8292471 5.6443683 9.7406514 13.564477148
## [39,] 12.8784583 1.8992483 1.9889165 8.578971918
## [40,] 19.5570681 4.2097416 3.2233153 16.688019559
## [41,] 8.2947067 19.5419799 5.6598574 19.428761336
## [42,] 2.3880956 5.9260435 11.6774467 1.409779554
## [43,] 10.5205932 14.5196605 14.6341532 9.195703864
## [44,] 4.5014670 15.7137567 3.3104182 14.031702369
## [45,] 9.7282353 2.1083549 17.3293553 1.738803359
## [46,] 7.4042960 4.7918926 14.1714827 19.858880322
## [47,] 19.6670036 5.4108974 15.2079907 5.061979881
## [48,] 7.7663823 2.0211699 2.9416822 0.990768713
## [49,] 4.5848968 2.3582768 7.1611393 13.726498936
## [50,] 12.4659509 19.8247311 13.4666496 15.738547053
m<-apply(y,2,mean)
m
## [1] 10.401819 9.540540 10.307797 10.261198 10.253117 9.188950 9.849096
## [8] 9.838603 9.813296 9.602321
s<-apply(y,2,sd)
s
## [1] 5.885765 5.533551 5.880352 4.652955 5.462771 6.292274 5.749865 6.060184
## [9] 5.583015 6.017083
x<-0:20 # generate an integer sequence from 0 to 20
p<-dbinom(x,size=20,prob=1/4) # p is a prob vector, p=Pr(X=x)
p
## [1] 3.171212e-03 2.114141e-02 6.694781e-02 1.338956e-01 1.896855e-01
## [6] 2.023312e-01 1.686093e-01 1.124062e-01 6.088669e-02 2.706075e-02
## [11] 9.922275e-03 3.006750e-03 7.516875e-04 1.541923e-04 2.569872e-05
## [16] 3.426496e-06 3.569266e-07 2.799425e-08 1.555236e-09 5.456968e-11
## [21] 9.094947e-13