How long should it take until you visit the integers with property, "n -> type(sqrt(n),integer)", for the k-th time if you roll a fair die with, 6, faces for k from 1 to, 10 By Shalosh B. Ekhad If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 1, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99999999575297955338 Conditioned that you are succesful the expected number of rounds for that go\ al is, 9.0191987632483880687, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 14.227689917614188369 The skewness is, 5.3583301053316850786 The kurtosis is, 58.602529055171293791 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 2, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99999989832235369813 Conditioned that you are succesful the expected number of rounds for that go\ al is, 24.882609900701003299, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 28.574282434248348777 The skewness is, 3.6099921326988996882 The kurtosis is, 27.160822538719054176 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 3, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99999880040623804065 Conditioned that you are succesful the expected number of rounds for that go\ al is, 47.773213031801911010, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 45.268210577400365750 The skewness is, 2.8664093082601612415 The kurtosis is, 17.744179055767240489 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 4, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99999069999322427795 Conditioned that you are succesful the expected number of rounds for that go\ al is, 77.644399039956538638, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 64.075725255250177670 The skewness is, 2.4212383740086583025 The kurtosis is, 13.167575720755458170 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 5, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99994669315553765177 Conditioned that you are succesful the expected number of rounds for that go\ al is, 114.47898067216927850, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 84.654990941637226402 The skewness is, 2.0998090787341932824 The kurtosis is, 10.312320548627905746 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 6, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99975898211641157177 Conditioned that you are succesful the expected number of rounds for that go\ al is, 158.17498380334702026, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 106.50934736554591180 The skewness is, 1.8303812967683515248 The kurtosis is, 8.2195443620600161224 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 7, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99910435751004394729 Conditioned that you are succesful the expected number of rounds for that go\ al is, 208.44591474592019988, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 128.82099258360937107 The skewness is, 1.5788255913329604315 The kurtosis is, 6.5509263331192671457 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 8, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99718523020798579414 Conditioned that you are succesful the expected number of rounds for that go\ al is, 264.64007666635563327, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 150.34936593077562480 The skewness is, 1.3307612842153711328 The kurtosis is, 5.1969242744640540586 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 9, times The prob. of reaching that goal in <=, 1000, rounds is, 0.99235892797551402036 Conditioned that you are succesful the expected number of rounds for that go\ al is, 325.54475448120442165, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 169.50909788092777004 The skewness is, 1.0837031870722797761 The kurtosis is, 4.1272446831176380239 ----------------------- If you roll a fair die with, 6, faces starting at, 2, and look at the runnin\ g total and keep going until that running total has the property, "n -> type(sqrt(n),integer)", for , 10, times The prob. of reaching that goal in <=, 1000, rounds is, 0.98178605739764033486 Conditioned that you are succesful the expected number of rounds for that go\ al is, 389.30088959933961971, (and this is a good estimate for unconditi\ onal expectation) and the standard deviation is, 184.68493319773211757 The skewness is, 0.84107934086250405549 The kurtosis is, 3.3242723228733754550 ----------------------- To sum up the expected time it takes to visit, "n -> type(sqrt(n),integer)", k times for k from 1 to , 10, are [9.019198763, 24.88260990, 47.77321303, 77.64439904, 114.4789807, 158.1749838, 208.4459147, 264.6400767, 325.5447545, 389.3008896] ---------------- This ends this paper that took, 99958.777, seconds to produce.