Bernoulli-Binomial Distribution Analysis

Suppose we are giving two students a multiple-choice exam with 40 questions, where each question has four choices. We don’t know how much the students have studied for this exam, but we think that they will do better than just guessing randomly.

1. What are the parameters of interest?

• Parameters of interest are \(\theta_1\)=true probability the first student will answer a question correctly, and \(\theta_2\)=true probability the second student will answer a question correctly.

2. What is our likelihood?

• Likelihood is \(Binomial(n=40, \theta)\), if we assume that each question is independent and that the probability a student gets each question right is the same for all questions for that student.

3. What prior should we use?

• For this prior for \(\theta\), we can ask what’s the prior probability that \(\theta > 0.25\) which would be guessing randomly? Then, look at \(\theta > 0.5\) and \(\theta > 0.8\).

• The conjugate prior is a beta prior. We can plot the density with dbeta.

theta=seq(from=0,to=1,by=.01)
plot(theta,dbeta(theta,1,1),type="l")  # uniform distribution, no improvement to guessing randomly

plot(theta,dbeta(theta,4,2),type="l")  # getting better, but still has a non-zero distribution at theta=0.5

plot(theta,dbeta(theta,8,4),type="l")  # the best scenario, dbeta = 0 for theta = 0.25.

4. What is the prior probability \(P(\theta >.25)\)? \(P(\theta>.5)\)? \(P(\theta>.8)\)?

• Find probabilities using the pbeta function.

1-pbeta(.25,8,4)

## [1] 0.9988117

1-pbeta(.5,8,4)

## [1] 0.8867188

1-pbeta(.8,8,4)

## [1] 0.1611392

5. Suppose the first student gets 33 questions right. What is the posterior distribution for \(\theta_1\)? \(P(\theta_1>.25)\)? \(P(\theta_1>.5)\)? \(P(\theta_1>.8)\)? What is a 95% posterior credible interval for \(\theta_1\)?

• Posterior is \(Beta(\alpha, \beta) = Beta(8+33,4+40-33) = Beta(41,11)\)

41/(41+11)  # posterior mean

## [1] 0.7884615

33/40       # MLE

## [1] 0.825

plot(theta,dbeta(theta,8,4),type="l")  # the best scenario, dbeta = 0 for theta = 0.25.
lines(theta,dbeta(theta,41,11))

Plot posterior first to get the right scale on the y-axis

plot(theta,dbeta(theta,41,11),type="l")
lines(theta,dbeta(theta,8,4),lty=2)
# plot likelihood
lines(theta,dbinom(33,size=40,p=theta),lty=3)
# plot scaled likelihood
lines(theta,44*dbinom(33,size=40,p=theta),lty=3)

Posterior probabilities

1-pbeta(.25,41,11)

## [1] 1

1-pbeta(.5,41,11)

## [1] 0.9999926

1-pbeta(.8,41,11)

## [1] 0.4444044

Equal-tailed 95% credible interval

qbeta(.025,41,11)

## [1] 0.6688426

qbeta(.975,41,11)

## [1] 0.8871094

6. Suppose the second student gets 24 questions right. What is the posterior distribution for \(\theta_2\)? \(P(\theta_2>.25)\)? \(P(\theta_2>.5)\)? \(P(\theta_2>.8)\)? What is a 95% posterior credible interval for \(\theta_2\)?

• Posterior is \(Beta(\alpha, \beta) = Beta(8+24,4+40-24) = Beta(32,20)\)

32/(32+20)  # posterior mean

## [1] 0.6153846

24/40       # MLE

## [1] 0.6

plot(theta,dbeta(theta,8,4),type="l")  # the best scenario, dbeta = 0 for theta = 0.25.
lines(theta,dbeta(theta,32,20))

Plot posterior first to get the right scale on the y-axis

plot(theta,dbeta(theta,32,20),type="l")
lines(theta,dbeta(theta,8,4),lty=2)
# plot likelihood
lines(theta,dbinom(24,size=40,p=theta),lty=3)
# plot scaled likelihood
lines(theta,44*dbinom(24,size=40,p=theta),lty=3)

Posterior probabilities

1-pbeta(.25,32,20)

## [1] 1

1-pbeta(.5,32,20)

## [1] 0.9540427

1-pbeta(.8,32,20)

## [1] 0.00124819

Equal-tailed 95% credible interval

qbeta(.025,32,20)

## [1] 0.4808022

qbeta(.975,32,20)

## [1] 0.7415564

7. What is the posterior probability that \(\theta_1>\theta_2\), i.e., that the first student has a better chance of getting a question right than the second student?

• We can estimate this by simulation: draw 1000 samples from each and see how often we observe \(\theta_1>\theta_2\)

theta1=rbeta(1000,41,11)
theta1

##    [1] 0.8846733 0.7217401 0.7074772 0.8100486 0.7727152 0.7664562 0.6783994
##    [8] 0.8266556 0.8346818 0.6904084 0.8247541 0.7582045 0.8350147 0.8606706
##   [15] 0.8330235 0.8166974 0.7175752 0.8046015 0.7526805 0.8381240 0.7243950
##   [22] 0.8509890 0.7778484 0.8182907 0.7444489 0.7928636 0.8753292 0.8449599
##   [29] 0.8447863 0.8649688 0.7425976 0.7600132 0.8350736 0.7782671 0.8034023
##   [36] 0.7542170 0.7810615 0.7153583 0.7824402 0.6884926 0.8027202 0.8190226
##   [43] 0.7572770 0.8342380 0.8225062 0.7360662 0.8628760 0.7294408 0.7857521
##   [50] 0.7554900 0.6816933 0.8015035 0.8170246 0.7346541 0.7896005 0.8518071
##   [57] 0.8635107 0.7729666 0.7901977 0.6961213 0.7063250 0.7594760 0.7294982
##   [64] 0.7892438 0.7880958 0.8295358 0.7853591 0.8172633 0.8235958 0.7062304
##   [71] 0.7879117 0.7960982 0.7276823 0.8350590 0.8127331 0.6637598 0.7403810
##   [78] 0.7970355 0.7394023 0.7974479 0.7532947 0.6957986 0.8291042 0.7028780
##   [85] 0.8474633 0.7569852 0.8585600 0.7668294 0.7054847 0.8822993 0.7481890
##   [92] 0.7459187 0.6852877 0.7645744 0.8700286 0.8412942 0.7957395 0.8658786
##   [99] 0.8704745 0.9007785 0.7787623 0.7347227 0.8480899 0.8223761 0.8310645
##  [106] 0.7676485 0.8331954 0.8226627 0.7822198 0.8402511 0.7602039 0.8137823
##  [113] 0.7411929 0.8607977 0.7139958 0.7001201 0.8481400 0.8661721 0.8366524
##  [120] 0.6546694 0.8452538 0.7643273 0.7894396 0.8903646 0.7536271 0.8080659
##  [127] 0.7582289 0.8082530 0.7749981 0.7909532 0.8061932 0.8181506 0.7678420
##  [134] 0.7598342 0.7441794 0.7909228 0.8078812 0.7461126 0.7821685 0.7340816
##  [141] 0.8018226 0.7082020 0.6888588 0.7018630 0.7205710 0.8057531 0.8699237
##  [148] 0.7172504 0.7718206 0.7049051 0.7898130 0.7204951 0.8262177 0.8133023
##  [155] 0.6827026 0.8805894 0.7251395 0.8478082 0.8002169 0.7463336 0.7429369
##  [162] 0.8903366 0.9225391 0.8431932 0.7992961 0.8205880 0.7302511 0.8363004
##  [169] 0.7820112 0.7203340 0.8418375 0.8450217 0.8253821 0.7074247 0.7132020
##  [176] 0.6977921 0.8275332 0.7146592 0.7813596 0.8401824 0.8368324 0.7796835
##  [183] 0.8456231 0.7974279 0.6605946 0.8605695 0.8357800 0.7270934 0.7222695
##  [190] 0.8094781 0.8192010 0.7566897 0.7910872 0.8408540 0.8055544 0.8289196
##  [197] 0.7900672 0.8355969 0.8032445 0.8398645 0.8135821 0.6948760 0.6281994
##  [204] 0.8158350 0.8671833 0.8585791 0.7562217 0.8232802 0.8808737 0.8188010
##  [211] 0.8168666 0.8099303 0.8037259 0.7985627 0.7352763 0.8297759 0.8288138
##  [218] 0.8168747 0.8525803 0.8640767 0.7911621 0.7685518 0.8117277 0.7938350
##  [225] 0.7925223 0.7561561 0.7904126 0.7500029 0.8772673 0.8113536 0.7170665
##  [232] 0.7620454 0.7468795 0.7735996 0.8338566 0.8999429 0.8175931 0.8334485
##  [239] 0.8257766 0.7415127 0.6646220 0.6853694 0.7676950 0.8431618 0.8165972
##  [246] 0.8386801 0.8144986 0.8590062 0.8504599 0.7708346 0.8674582 0.7661328
##  [253] 0.8340336 0.8084731 0.7129558 0.7642253 0.7731853 0.7097694 0.6923205
##  [260] 0.7991350 0.7779225 0.6826248 0.8482167 0.7927851 0.7310443 0.8242842
##  [267] 0.7503421 0.8351086 0.7411488 0.7453425 0.7437373 0.7377306 0.7360513
##  [274] 0.7090526 0.7883881 0.8056487 0.8288438 0.8271662 0.8483900 0.7902169
##  [281] 0.6598295 0.8217397 0.8311929 0.7596451 0.8123486 0.8400887 0.8194085
##  [288] 0.8153232 0.8793559 0.7574044 0.7217146 0.6953600 0.8608807 0.8057625
##  [295] 0.8514730 0.8346715 0.7249542 0.7409345 0.7850270 0.7933531 0.7279819
##  [302] 0.8462000 0.7012579 0.7532403 0.8545247 0.7912783 0.7952925 0.8205339
##  [309] 0.8140552 0.7137895 0.8775966 0.8171413 0.6723719 0.8399152 0.8669165
##  [316] 0.8512629 0.8659834 0.7848296 0.8255666 0.8691893 0.7748834 0.7328413
##  [323] 0.7431583 0.8445987 0.8162983 0.7521661 0.8091753 0.7298974 0.7602842
##  [330] 0.8725461 0.7087700 0.7733938 0.8671069 0.8037717 0.8423240 0.7004366
##  [337] 0.7687193 0.8800096 0.8535366 0.7180818 0.8242308 0.8041383 0.8311301
##  [344] 0.7839833 0.8415570 0.7813222 0.8553733 0.7793872 0.7310051 0.7671214
##  [351] 0.8264306 0.8332293 0.7913923 0.7796772 0.7346337 0.8221151 0.7364324
##  [358] 0.7848848 0.8013482 0.7679114 0.7705608 0.8986130 0.7421145 0.7993539
##  [365] 0.7970194 0.6926196 0.7850175 0.7677628 0.8460703 0.7965130 0.8261026
##  [372] 0.8150768 0.7006725 0.8673399 0.7378226 0.8534308 0.8414715 0.8235733
##  [379] 0.7877190 0.7980154 0.7998275 0.8397661 0.7567180 0.7486971 0.7190635
##  [386] 0.7515342 0.7327953 0.7315078 0.7319349 0.7886442 0.6477552 0.6970162
##  [393] 0.7783944 0.8470498 0.8099437 0.7346298 0.7548374 0.8056563 0.7069205
##  [400] 0.7862184 0.7821079 0.7255760 0.8284857 0.7862538 0.8876569 0.7886328
##  [407] 0.7375460 0.6777641 0.6876912 0.7746952 0.6901969 0.7945108 0.7742886
##  [414] 0.7085441 0.8280503 0.7480682 0.7821027 0.8071329 0.7634631 0.7332563
##  [421] 0.8059134 0.8646597 0.8429084 0.7517293 0.7607753 0.6828613 0.8273973
##  [428] 0.8075867 0.8165299 0.8241866 0.8076126 0.7533996 0.8279733 0.8194251
##  [435] 0.7798333 0.8617126 0.8500434 0.7854757 0.7277045 0.7576446 0.7297017
##  [442] 0.7057110 0.7428036 0.8003764 0.7909515 0.8419494 0.7081352 0.7762413
##  [449] 0.7114148 0.8766686 0.7495843 0.8160592 0.8373533 0.7514646 0.8102477
##  [456] 0.8003945 0.8101515 0.7156660 0.8142070 0.8055705 0.7409614 0.8256909
##  [463] 0.7684222 0.7702442 0.8894360 0.6515723 0.8210020 0.6931778 0.8219351
##  [470] 0.7890512 0.7848584 0.8558610 0.8034580 0.7620870 0.8130092 0.8718365
##  [477] 0.8683861 0.8061212 0.7815239 0.8320224 0.7919318 0.8276822 0.7456336
##  [484] 0.7834530 0.7795763 0.8037984 0.6995669 0.8817600 0.7594459 0.6876511
##  [491] 0.7677825 0.7814610 0.8488859 0.7057643 0.7200309 0.7449680 0.8538312
##  [498] 0.7964637 0.7856382 0.8404159 0.7317046 0.8266270 0.7098202 0.8171470
##  [505] 0.8264200 0.7876573 0.8523283 0.7834064 0.8402305 0.7802939 0.7278985
##  [512] 0.8177053 0.8348210 0.7992998 0.7666948 0.8467332 0.7262787 0.7357929
##  [519] 0.8393375 0.6638297 0.7290833 0.8094941 0.8291615 0.7974126 0.5249076
##  [526] 0.8095073 0.7657038 0.7641165 0.7127524 0.8150903 0.8253414 0.8650136
##  [533] 0.8105715 0.8319646 0.7641811 0.8484717 0.7483816 0.8249086 0.7847930
##  [540] 0.6669724 0.8029505 0.7918183 0.8157395 0.7654130 0.8406061 0.7181641
##  [547] 0.8059819 0.7896059 0.8498591 0.8413732 0.8529313 0.7834090 0.7819818
##  [554] 0.7301262 0.7421309 0.7755565 0.6797773 0.8806201 0.8607263 0.7900417
##  [561] 0.7512819 0.8862460 0.7885408 0.8061592 0.9185856 0.8812779 0.7049167
##  [568] 0.7690154 0.8115846 0.7823810 0.7812888 0.7704469 0.8057478 0.7777918
##  [575] 0.8455055 0.7523438 0.7523317 0.8329731 0.7593709 0.7821871 0.8548802
##  [582] 0.7141463 0.7599941 0.8701216 0.7939075 0.7531303 0.7926863 0.7681958
##  [589] 0.6812957 0.7900636 0.8608935 0.8664241 0.7227964 0.8079380 0.7928904
##  [596] 0.8044223 0.7307620 0.7192923 0.8063806 0.7626488 0.7607496 0.8659965
##  [603] 0.8198289 0.8500074 0.7543378 0.8135884 0.7811923 0.7310187 0.8768451
##  [610] 0.7815577 0.8395034 0.7461694 0.7361329 0.8102000 0.7948887 0.8068596
##  [617] 0.8876235 0.7703835 0.7926738 0.8539096 0.7953680 0.6579911 0.8348157
##  [624] 0.6740228 0.7076322 0.7536770 0.6093973 0.8406601 0.8189344 0.7260512
##  [631] 0.8138497 0.8207205 0.7797791 0.7341592 0.7156034 0.7536386 0.7186769
##  [638] 0.7912573 0.7284140 0.7680458 0.6975391 0.8203458 0.7260899 0.7768159
##  [645] 0.8026176 0.7318027 0.8153782 0.7781551 0.8279049 0.8150188 0.7210677
##  [652] 0.8131851 0.8053048 0.7201713 0.8012087 0.7878587 0.8586435 0.8091693
##  [659] 0.7490306 0.8308974 0.7440840 0.7654573 0.8276180 0.6772958 0.6587713
##  [666] 0.7717658 0.8270965 0.7065622 0.7822039 0.7215031 0.8176842 0.6644656
##  [673] 0.7164040 0.7692818 0.7873370 0.6381922 0.7984175 0.8542135 0.6739246
##  [680] 0.8065569 0.7636860 0.7386244 0.8224752 0.7577778 0.7975497 0.8232826
##  [687] 0.7559364 0.8036763 0.7657098 0.8004635 0.8255566 0.8153333 0.7494641
##  [694] 0.8843286 0.8517481 0.7924759 0.8001353 0.8148205 0.7938221 0.7403147
##  [701] 0.8384991 0.6791303 0.7976607 0.8527196 0.8006004 0.8498086 0.7356046
##  [708] 0.7513996 0.8180634 0.8043775 0.8188274 0.7063737 0.7627911 0.8109598
##  [715] 0.7721635 0.7899117 0.7920064 0.7478036 0.7451321 0.8070155 0.8763553
##  [722] 0.8290911 0.8378519 0.8526932 0.8056475 0.8306661 0.7729247 0.8434281
##  [729] 0.8519164 0.8335720 0.6961475 0.7950153 0.8110392 0.8149030 0.6632877
##  [736] 0.8800636 0.7766713 0.7560198 0.8535212 0.6961085 0.8583493 0.8014555
##  [743] 0.8168584 0.7564135 0.8495124 0.7915731 0.8585315 0.7990201 0.9143686
##  [750] 0.8227522 0.7584836 0.7401888 0.7598875 0.8189048 0.7490818 0.7689104
##  [757] 0.8224648 0.7780091 0.7673327 0.7988198 0.8825730 0.8284380 0.8061015
##  [764] 0.7744721 0.7434644 0.8509500 0.7969636 0.7062229 0.8152858 0.7892998
##  [771] 0.7648243 0.7795860 0.7158926 0.8046428 0.8432405 0.7285802 0.7077829
##  [778] 0.8306302 0.7486983 0.7458663 0.7573505 0.7979608 0.7410862 0.7541251
##  [785] 0.8178157 0.8445947 0.8160142 0.6984400 0.8318381 0.7109249 0.7474209
##  [792] 0.6914638 0.8273479 0.9019873 0.7395081 0.8392180 0.8381892 0.7385027
##  [799] 0.8011742 0.8328271 0.7557698 0.8907225 0.7923185 0.7660002 0.8182833
##  [806] 0.8577759 0.7906391 0.7356079 0.8149981 0.8348757 0.8054151 0.7893336
##  [813] 0.7530269 0.8244324 0.8619107 0.7917845 0.8656908 0.8160132 0.7514628
##  [820] 0.7405342 0.8533835 0.8181846 0.7956671 0.7287285 0.7505355 0.7992931
##  [827] 0.7829497 0.8243815 0.8604357 0.7303663 0.7896868 0.6144641 0.7505163
##  [834] 0.6966442 0.7210987 0.8584063 0.7678155 0.7770799 0.7758262 0.7747871
##  [841] 0.7375307 0.8269666 0.8440602 0.6347415 0.7494191 0.8410114 0.7647480
##  [848] 0.7945300 0.8504633 0.7179956 0.7565163 0.8678767 0.8585552 0.8014419
##  [855] 0.8451294 0.8523013 0.7475728 0.7327941 0.7939904 0.6784687 0.7721427
##  [862] 0.7193059 0.7309737 0.7576569 0.7995889 0.8431589 0.7958256 0.8767255
##  [869] 0.6726215 0.8164453 0.8242568 0.8399800 0.8369123 0.7796373 0.8465969
##  [876] 0.7861354 0.7588991 0.7285073 0.7025349 0.7665106 0.9099330 0.8502907
##  [883] 0.8309342 0.8332026 0.8779222 0.8873960 0.7592336 0.8019798 0.7651713
##  [890] 0.8563508 0.7311601 0.7550513 0.8116603 0.8383190 0.7869824 0.7907432
##  [897] 0.8320261 0.8814200 0.8406412 0.8054984 0.7321260 0.7790785 0.7712020
##  [904] 0.8608476 0.8125517 0.7920148 0.6719661 0.7719429 0.7627646 0.8768348
##  [911] 0.8224561 0.7663800 0.7729761 0.8364335 0.8237794 0.7533740 0.7740950
##  [918] 0.8273750 0.8861395 0.7879110 0.7647831 0.7698610 0.8667801 0.8063934
##  [925] 0.8270007 0.9044755 0.8140121 0.6665636 0.7901457 0.8503084 0.8026225
##  [932] 0.8206811 0.8438595 0.7654821 0.7836999 0.8131644 0.8252290 0.7790659
##  [939] 0.7650122 0.7543560 0.8564002 0.7951094 0.8898895 0.8163409 0.8338416
##  [946] 0.8497551 0.7244178 0.8393663 0.8525822 0.7139073 0.7729701 0.8312848
##  [953] 0.8346462 0.7874030 0.7912847 0.8623996 0.7721396 0.8469815 0.7618378
##  [960] 0.7054967 0.7876079 0.8005125 0.7736464 0.8080763 0.8374753 0.7891131
##  [967] 0.7821096 0.7893804 0.7260634 0.8333176 0.7894485 0.8675195 0.7118148
##  [974] 0.6877023 0.7527907 0.8793341 0.7902649 0.7679896 0.7847300 0.7842160
##  [981] 0.8323445 0.7444435 0.6966226 0.8225530 0.8771327 0.8310801 0.7848163
##  [988] 0.7902474 0.6767020 0.7771311 0.8786780 0.8110116 0.6609307 0.7983672
##  [995] 0.6746447 0.8790831 0.8208674 0.8669751 0.8294809 0.7888427

theta2=rbeta(1000,32,20)
theta2

##    [1] 0.6693666 0.5381663 0.6774750 0.6103431 0.5979286 0.6228797 0.5266123
##    [8] 0.6854664 0.4051132 0.5430502 0.6416328 0.7334167 0.5132187 0.7520971
##   [15] 0.5797036 0.6002107 0.5542022 0.7026043 0.6099425 0.7378764 0.6918885
##   [22] 0.5484368 0.6153943 0.6672938 0.6332345 0.5988188 0.6288923 0.6047525
##   [29] 0.5184679 0.5052921 0.6170702 0.6264363 0.7292428 0.6761850 0.7131386
##   [36] 0.6672954 0.6739630 0.5148102 0.5986965 0.5990536 0.6459450 0.6418154
##   [43] 0.5177176 0.6575418 0.4662086 0.5951708 0.5698599 0.5670072 0.6879270
##   [50] 0.6619184 0.6254539 0.6900463 0.5603833 0.6760862 0.7274504 0.5654798
##   [57] 0.6446813 0.6288391 0.6518314 0.7332666 0.5751071 0.7454491 0.6258621
##   [64] 0.6341230 0.5740419 0.5387569 0.5573828 0.6662167 0.6286147 0.6427977
##   [71] 0.6225741 0.6801754 0.4532705 0.6368280 0.6405154 0.6708372 0.6491356
##   [78] 0.6235994 0.5221002 0.5313392 0.5669817 0.5235560 0.6600457 0.6617032
##   [85] 0.7109460 0.5962384 0.6847641 0.6588489 0.7165810 0.5693491 0.7762137
##   [92] 0.6498740 0.6489787 0.6866779 0.5866620 0.6039109 0.6349549 0.6192341
##   [99] 0.7196048 0.5692299 0.6691994 0.6191409 0.5390349 0.6252682 0.6992057
##  [106] 0.6673435 0.6368509 0.7644378 0.5550783 0.5497832 0.6750528 0.6004082
##  [113] 0.4815063 0.6008535 0.6479255 0.7407153 0.6711869 0.6025436 0.6831674
##  [120] 0.6516864 0.6751929 0.6678519 0.5694851 0.6513238 0.6565910 0.5446842
##  [127] 0.4488082 0.5545920 0.5884652 0.6554662 0.6438792 0.5500178 0.7347463
##  [134] 0.4543830 0.5800600 0.5657119 0.5506190 0.6003788 0.7274603 0.6944248
##  [141] 0.5822224 0.5674917 0.5863931 0.5644032 0.7110760 0.7095261 0.4855515
##  [148] 0.6829482 0.6435806 0.7199756 0.5667579 0.5804222 0.6085130 0.6470461
##  [155] 0.5851887 0.6258452 0.5742184 0.5584822 0.5707011 0.4878632 0.6349082
##  [162] 0.5755250 0.6699109 0.5973313 0.5171957 0.6366920 0.6180613 0.5597709
##  [169] 0.6385771 0.6023323 0.6374819 0.5637656 0.6709203 0.5571144 0.5021658
##  [176] 0.6538125 0.5994290 0.5321318 0.6347444 0.7647220 0.6324827 0.6931932
##  [183] 0.6142185 0.5659455 0.6298532 0.5906557 0.6856581 0.4672683 0.6011641
##  [190] 0.5799706 0.5947508 0.4964107 0.6960937 0.5931808 0.6279235 0.6118266
##  [197] 0.5700017 0.6533690 0.4952154 0.5881637 0.6955950 0.5903361 0.6290145
##  [204] 0.5718677 0.5321148 0.5549382 0.5784452 0.5489419 0.5738673 0.6380425
##  [211] 0.4550259 0.4970860 0.5553806 0.6500841 0.6645086 0.6901697 0.6031355
##  [218] 0.6813456 0.4897152 0.6607993 0.6300951 0.5700425 0.6055359 0.5119386
##  [225] 0.5513707 0.4818625 0.6352568 0.6239467 0.6591635 0.6333844 0.6033109
##  [232] 0.6551307 0.6367759 0.6747551 0.5956503 0.5135521 0.7286260 0.6650795
##  [239] 0.6628310 0.6594858 0.5412270 0.5551035 0.6369146 0.5796250 0.7032025
##  [246] 0.5604735 0.6572501 0.5954594 0.5948506 0.6251645 0.5740274 0.5302017
##  [253] 0.6077045 0.7289821 0.5706194 0.5067700 0.5978512 0.7772620 0.5925313
##  [260] 0.6023103 0.5492994 0.6610854 0.6185179 0.6928889 0.5693189 0.5794596
##  [267] 0.5241534 0.5680458 0.5330342 0.5894289 0.6793213 0.7278451 0.5379303
##  [274] 0.6755182 0.6688655 0.6287002 0.6519012 0.5686694 0.5425975 0.5696589
##  [281] 0.6638481 0.5938812 0.6327966 0.6181656 0.5569073 0.5321444 0.6747018
##  [288] 0.6488159 0.7106355 0.5893666 0.6684270 0.5942361 0.7287554 0.5829348
##  [295] 0.5512071 0.6849260 0.6149324 0.6391075 0.5591499 0.6620766 0.5617110
##  [302] 0.6860379 0.5619820 0.6314810 0.5715060 0.6926690 0.6291794 0.6598372
##  [309] 0.6251519 0.6930911 0.5211603 0.5116604 0.6052129 0.6454190 0.7738205
##  [316] 0.6360146 0.6650246 0.5391385 0.6496800 0.7290726 0.6698282 0.6505611
##  [323] 0.6089775 0.7415910 0.5895747 0.6836398 0.5307996 0.7443592 0.7193571
##  [330] 0.6732021 0.6154237 0.5943650 0.5855898 0.4576111 0.6101208 0.5767025
##  [337] 0.6521235 0.5799009 0.5935411 0.5381304 0.7307907 0.6470795 0.4464808
##  [344] 0.7226846 0.5766680 0.5678074 0.5875826 0.6885066 0.6499826 0.5413900
##  [351] 0.6639269 0.6084018 0.5099699 0.6067764 0.6342350 0.6345645 0.5849542
##  [358] 0.4899342 0.6915963 0.6250137 0.6500811 0.6230220 0.7209224 0.6542816
##  [365] 0.6367259 0.5916040 0.7134901 0.5732757 0.7238965 0.6501776 0.6398226
##  [372] 0.6643727 0.6745545 0.6624408 0.5338985 0.6608737 0.5768644 0.6438657
##  [379] 0.6050746 0.6522702 0.6091400 0.6906262 0.5954609 0.6899805 0.5562158
##  [386] 0.6707009 0.5612372 0.6043454 0.5196878 0.4912994 0.6865298 0.6457546
##  [393] 0.6886715 0.5331274 0.5523346 0.6230601 0.8130622 0.5284867 0.5464231
##  [400] 0.6410227 0.6219657 0.5272896 0.6050813 0.5937654 0.4736244 0.5352044
##  [407] 0.6481274 0.6249425 0.5933807 0.4293872 0.6784777 0.5498209 0.6563430
##  [414] 0.6357270 0.6257843 0.5873544 0.7126692 0.6303153 0.5719171 0.6766698
##  [421] 0.6221296 0.5227017 0.6011069 0.4917934 0.5621077 0.6921149 0.6318375
##  [428] 0.5666724 0.6218039 0.5286400 0.5423230 0.6203873 0.5282766 0.5694643
##  [435] 0.5977345 0.5912338 0.6663779 0.6686350 0.5856401 0.6510447 0.5887071
##  [442] 0.5285773 0.5242402 0.7130163 0.6161205 0.7348450 0.7271724 0.6291462
##  [449] 0.5225717 0.6574641 0.5678945 0.6227157 0.6970698 0.5300796 0.5561365
##  [456] 0.6127554 0.4550709 0.6734622 0.7102050 0.5872064 0.5268953 0.5604778
##  [463] 0.6709406 0.5008837 0.6081954 0.7255131 0.5075775 0.6165455 0.6071298
##  [470] 0.6776771 0.4829884 0.6057957 0.6949515 0.5944346 0.6321324 0.6236541
##  [477] 0.6830569 0.6265511 0.6053059 0.7039487 0.6554503 0.5528657 0.5972634
##  [484] 0.6074042 0.7287974 0.5886895 0.5916494 0.6670789 0.4571227 0.5001109
##  [491] 0.6021526 0.6536995 0.5666194 0.7829552 0.5656670 0.6658174 0.6958401
##  [498] 0.6653953 0.6322282 0.6083822 0.5312202 0.5984280 0.4709072 0.6621322
##  [505] 0.5978085 0.5642160 0.6431835 0.5472090 0.7369450 0.6674789 0.6089692
##  [512] 0.6161620 0.4957559 0.6701454 0.5895427 0.6403001 0.6285034 0.5395673
##  [519] 0.6294196 0.6584485 0.5536041 0.6299846 0.5341160 0.6341337 0.6916955
##  [526] 0.7358603 0.5984904 0.7025304 0.6195483 0.5608476 0.5314143 0.5734246
##  [533] 0.5243906 0.6094395 0.4892060 0.6640149 0.6678693 0.5875450 0.6696926
##  [540] 0.6760109 0.6175463 0.5736992 0.5983470 0.6773675 0.5586198 0.7045444
##  [547] 0.5688076 0.6314880 0.6414102 0.6376563 0.5594364 0.6616753 0.5575914
##  [554] 0.6489292 0.6170373 0.6138564 0.6559980 0.5415890 0.6149336 0.6851857
##  [561] 0.5417744 0.7341743 0.7230177 0.6913848 0.6309950 0.7008545 0.5436878
##  [568] 0.7050004 0.5806413 0.5643034 0.6312287 0.5826379 0.6430942 0.6221849
##  [575] 0.5811463 0.5565909 0.5900750 0.6302972 0.6377786 0.6258689 0.5280460
##  [582] 0.5429793 0.7225650 0.5658988 0.6442058 0.6569823 0.5871890 0.6590480
##  [589] 0.6074943 0.5727659 0.5659476 0.5027837 0.6548334 0.5691527 0.5877413
##  [596] 0.4824481 0.6555845 0.5463058 0.6221628 0.6647114 0.5988987 0.5617084
##  [603] 0.5418505 0.6323463 0.5757486 0.7045501 0.6059816 0.6566064 0.5385813
##  [610] 0.5935910 0.5440844 0.7068717 0.6391592 0.6492613 0.6311335 0.7143808
##  [617] 0.4983675 0.6152385 0.5741519 0.5257198 0.5814939 0.6060730 0.6100025
##  [624] 0.5924436 0.7044260 0.6059803 0.6663270 0.6271617 0.4416624 0.5773002
##  [631] 0.6540769 0.4758553 0.7088950 0.4793864 0.5603772 0.5729106 0.5800163
##  [638] 0.6127316 0.5413364 0.6446111 0.6861931 0.6816244 0.6289896 0.6232948
##  [645] 0.5302517 0.5194662 0.6087542 0.6672417 0.4756103 0.6539853 0.7291336
##  [652] 0.6717090 0.6238014 0.6453885 0.5968856 0.4757359 0.6851803 0.6555846
##  [659] 0.6888724 0.6733033 0.6327955 0.6667165 0.6598710 0.5010227 0.6511148
##  [666] 0.6500438 0.5679584 0.5879916 0.6453386 0.6052763 0.6440080 0.7365144
##  [673] 0.6441224 0.6687826 0.5295954 0.7557611 0.6221426 0.6204120 0.5446680
##  [680] 0.6281550 0.5958495 0.4173216 0.5752029 0.5411240 0.6295659 0.6471556
##  [687] 0.5527193 0.5127366 0.5463141 0.5227394 0.5444795 0.6372822 0.6607815
##  [694] 0.5299646 0.5632824 0.5325555 0.6768347 0.5059506 0.6387416 0.6485711
##  [701] 0.6223721 0.5817634 0.7448042 0.6702435 0.6086604 0.5718148 0.5420099
##  [708] 0.6034746 0.5437199 0.6043012 0.6457496 0.6515462 0.7755679 0.6205588
##  [715] 0.6665552 0.6545492 0.6367266 0.5634826 0.5925274 0.6701613 0.6944939
##  [722] 0.6794975 0.6044275 0.6447248 0.6672177 0.6776380 0.5853464 0.6582291
##  [729] 0.5854416 0.5513091 0.5532668 0.4597094 0.6694867 0.6595524 0.5915039
##  [736] 0.6681709 0.5376309 0.6834415 0.5777037 0.5196334 0.5228661 0.4579903
##  [743] 0.5798739 0.5217713 0.6630816 0.5779871 0.5776743 0.5640943 0.5527090
##  [750] 0.6491767 0.7091135 0.5476967 0.6531405 0.6461314 0.6360693 0.6543273
##  [757] 0.6004211 0.7147010 0.5767958 0.7309200 0.6178079 0.6717152 0.5779235
##  [764] 0.5016514 0.6983460 0.5141445 0.5527610 0.5614660 0.5639041 0.5096935
##  [771] 0.6356964 0.5467650 0.4769941 0.4891160 0.6530873 0.7095505 0.7322876
##  [778] 0.4780968 0.6247971 0.6500674 0.6351301 0.6397581 0.5870274 0.7308309
##  [785] 0.6247118 0.6147927 0.6944534 0.5543721 0.5459672 0.6249110 0.5870717
##  [792] 0.5833743 0.6079619 0.6032678 0.6386388 0.6004721 0.5793887 0.6644469
##  [799] 0.5079686 0.6054812 0.6742061 0.5246992 0.6783594 0.5569540 0.5593974
##  [806] 0.5960028 0.5814832 0.5924001 0.6174573 0.4717176 0.5827798 0.7113468
##  [813] 0.7593596 0.5514918 0.7032058 0.6549840 0.6035165 0.5842008 0.6599217
##  [820] 0.6234918 0.6348524 0.4483339 0.5982531 0.6797543 0.5151775 0.7116508
##  [827] 0.6653247 0.5877765 0.6269115 0.7279242 0.5815926 0.5642292 0.5974489
##  [834] 0.6489748 0.6753052 0.5735715 0.6319258 0.4784567 0.6920145 0.6592442
##  [841] 0.4451242 0.5725599 0.7065517 0.7115405 0.6557096 0.5686597 0.5756954
##  [848] 0.6716451 0.6240093 0.6685960 0.4705600 0.6578628 0.6215676 0.6150071
##  [855] 0.6395585 0.6681637 0.6502934 0.6133745 0.7384002 0.6562161 0.6594382
##  [862] 0.6779440 0.5873646 0.6197349 0.6370132 0.5215932 0.5553993 0.7106818
##  [869] 0.7790537 0.5906139 0.6013836 0.7136107 0.6189789 0.6540099 0.6341936
##  [876] 0.7768735 0.6859749 0.5585495 0.7092618 0.6019323 0.4197190 0.6209696
##  [883] 0.4938703 0.6567386 0.5755455 0.6002061 0.6846253 0.6013421 0.6416165
##  [890] 0.5834305 0.7417625 0.6141635 0.6615178 0.6241506 0.6204885 0.6169527
##  [897] 0.6320839 0.6314074 0.6249776 0.6911351 0.5665048 0.5489173 0.6638340
##  [904] 0.7288768 0.6355429 0.5622839 0.7024494 0.6196395 0.4385311 0.5726520
##  [911] 0.6786184 0.6452795 0.5924332 0.5763578 0.5955685 0.7174488 0.8220610
##  [918] 0.6594398 0.5778311 0.6072277 0.6261682 0.6527707 0.6833337 0.5748028
##  [925] 0.6337050 0.6870601 0.6836941 0.4965024 0.5392026 0.6188081 0.6040021
##  [932] 0.4949613 0.6687377 0.6021406 0.5524734 0.5020027 0.7263085 0.5630452
##  [939] 0.6180902 0.6941203 0.6092083 0.6857939 0.6229209 0.5585555 0.5325266
##  [946] 0.6258184 0.5607611 0.7122695 0.6402562 0.6184269 0.6878886 0.6279478
##  [953] 0.6219030 0.7304332 0.7162146 0.6191707 0.6328776 0.5123996 0.5674513
##  [960] 0.6810268 0.4658909 0.5417657 0.2923296 0.5604424 0.6926089 0.4814127
##  [967] 0.6618942 0.6838017 0.6077242 0.6649439 0.6310998 0.6771057 0.5688614
##  [974] 0.5981540 0.6163708 0.5557066 0.6334449 0.5832472 0.6185583 0.5892214
##  [981] 0.5739884 0.6189260 0.5765527 0.6211775 0.6814136 0.6113361 0.6494130
##  [988] 0.7152113 0.6225867 0.5451237 0.6916945 0.5644787 0.6129952 0.6786246
##  [995] 0.6256261 0.4874117 0.5393667 0.6538065 0.6271814 0.6926112

mean(theta1>theta2)

## [1] 0.969

8. Note - for other distributions:

• dgamma,pgamma,qgamma,rgamma
• dnorm,pnorm,qnorm,rnorm