418:, where alpha is non-negative coefficient, and E(.) is the expected value of the argument. The lower bound can be easily found based on Jensen's inequality, i.e., E(exp{-alpha*x^2}) \geq exp(-alpha*^2), but the upperbound is more difficult to find. Does anyone have idea about what the upperbound is? If the general upperbound is hard to be solved, the upperbound for the case of a large alpha is of my interest.
741:
1137:
546:
946:
350:
Oh, I appear to have been 10 days and ~44 minutes off, while
Gandalf was off by a few days more. If only we could find the most accurate formula to calculate when the next # of articles will arrive... --
223:
Perhaps
Knowledge's growth has looked exponential so far, but I doubt it will remain this way for long. There aren't that many things to write about, and I doubt they grow any faster than quadratic. --
1478:
1330:
1015:
736:{\displaystyle X={\begin{cases}0&{\mbox{with probability }}1-{\frac {E(X)^{2}}{E(X^{2})}}\\{\frac {E(X^{2})}{E(X)}}&{\mbox{with probability }}{\frac {E(X)^{2}}{E(X^{2})}}\end{cases}}}
1810:
1056:
1749:
66:
59:
45:
1656:
799:
55:
51:
1624:
416:
1414:
794:
539:
1248:
1187:
1518:
1498:
1035:
1582:
490:
1547:
455:
1676:
1350:
25:
85:
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
258:, which suggests that in the last few years things have slowed down to something more linear. Personally I'd like to pass the whole thing through
251:
1352:
is chosen to be optimal. I don't think the final solution can be represented algebraically, but it can be approximted one way or another. --
37:
121:
863,700 / ((488 + (9*60+11) = 551 / 1440) = 488.38263888888888888888888888888889) = 1768.4903827969587826030079443063 articles per day.
127:
1,000,000 / 1768.4903827969587826030079443063 = 565.45402210129546010060077444585 days after the creation of the 1,000,000th article.
339:
173:
21:
115:
The duration since the creation of the 1,000,000th article was 488 days, 9 hours, 11 minutes and 0 seconds as of the above date.
1816:
1682:
1588:
1419:
1361:
1046:
422:
354:
300:
287:
266:
233:
218:
201:
188:
1132:{\displaystyle X={\begin{cases}a&{\mbox{with probability }}p_{1}\\b&{\mbox{with probability }}p_{2}\end{cases}}}
109:
296:
Well now, we're just 2 months away from the tentative 2,000,000th article date! Someone ought to start a countdown. --
255:
112:
was created on March 2, 2006 at 5:09 PM Central time. As of 2:20 AM Central on July 4, there was 1,863,700 articles.
1688:
X can be normally distributed or log-normally distributed random variables. The later case is of my interest. Thus
1253:
951:
941:{\displaystyle 1+{\frac {E(X)^{2}}{E(X^{2})}}\left(e^{-\alpha \left({\frac {E(X^{2})}{E(X)}}\right)^{2}}-1\right)}
1754:
1691:
1357:
1042:
229:
86:
17:
1629:
1597:
368:
1370:
750:
495:
351:
297:
185:
1813:
1585:
1192:
419:
1071:
561:
1353:
1038:
225:
1146:
1051:
Actually, that's not true. A higher value (perhaps this one is the maximum?) can be obtained with
184:
Is this an accurate estimate, or did I miss another formula? If I did, can you recalculate it? --
130:
To convert to hours: .45402210129546010060077444585 * 24 = 10.8965304310910424144185867004 hours
1503:
1483:
1020:
1552:
460:
143:
be created 565 days, 10 hours, 53 minutes, and ~48 seconds after the 1,000,000th article.
1679:
1523:
1500:
is large enough. However, I cannot prove it mathematically as well as quantify how large
431:
140:
133:
To minutes: .8965304310910424144185867004 * 60 = 53.791825865462544865115202024 minutes
310:
215:
1661:
1335:
136:
To seconds: .791825865462544865115202024 * 60 = 47.50955192775269190691212144 seconds
280:
284:
263:
195:
259:
147:
Therefore, the estimated date and time of the 2,000,000th article creation is:
210:... and an exponential growth model based on these two data points predicts
74:
194:
The thing is, Knowledge article creation is more exponential than linear.
262:
to see if there's a decent (non-deterministic) model you can fit to it.
313:, the estimated time & date of the 2,000,000th article creation is:
254:, a somewhat old article that suggests an exponential model, and
1812:. Therefore, Var(x), E(x), moments, and distribution are known.
79:
Welcome to the
Knowledge Mathematics Reference Desk Archives
1125:
729:
1367:
Further results: I have run simulations and realized that
1658:
it seems. Is that all, or is perhaps the distribution of
1106:
1080:
674:
570:
118:
At the rate the articles are being created, which is:
1757:
1694:
1664:
1632:
1600:
1555:
1526:
1506:
1486:
1473:{\displaystyle {\frac {1}{\alpha }}e^{-(E{(x)})^{2}}}
1422:
1373:
1338:
1256:
1195:
1149:
1059:
1023:
954:
802:
753:
549:
498:
463:
434:
371:
139:
Which calculates that: The 2,000,000th article will
124:We should have the 2,000,000th article created on:
1804:
1743:
1670:
1650:
1618:
1576:
1541:
1512:
1492:
1472:
1408:
1344:
1324:
1242:
1181:
1131:
1029:
1009:
940:
788:
735:
533:
484:
449:
410:
365:I am having trouble to quantify the upperbound of
1594:What information is given here, really? We have
1325:{\displaystyle p_{1}a^{2}+p_{2}b^{2}=E(X^{2})}
1010:{\displaystyle 1-{\frac {E(X)^{2}}{E(X^{2})}}}
8:
1805:{\displaystyle e^{k\mu +k^{2}\sigma ^{2}/2}}
361:What is the upperbound of E(exp{-alpha*x^2})
214:days to go from 1m articles to 2m articles.
1744:{\displaystyle E(x)=e^{\mu +\sigma ^{2}/2}}
1792:
1786:
1776:
1762:
1756:
1731:
1725:
1714:
1693:
1663:
1631:
1599:
1554:
1525:
1505:
1485:
1462:
1447:
1437:
1423:
1421:
1395:
1384:
1372:
1337:
1313:
1294:
1284:
1271:
1261:
1255:
1216:
1200:
1194:
1167:
1154:
1148:
1116:
1105:
1090:
1079:
1066:
1058:
1022:
995:
977:
961:
953:
919:
889:
876:
864:
843:
825:
809:
801:
775:
764:
752:
714:
696:
680:
673:
645:
632:
616:
598:
582:
569:
556:
548:
520:
509:
497:
462:
433:
398:
387:
370:
49:
36:
1651:{\displaystyle \operatorname {Var} (X)}
65:
252:Knowledge:Modelling Knowledge's growth
43:
1619:{\displaystyle \operatorname {E} (X)}
1584:. Does anybody have idea about this?
411:{\displaystyle E(exp{-\alpha x^{2}})}
104:Estimated Date of 2,000,000th Article
7:
1409:{\displaystyle E(e^{-\alpha x^{2}})}
789:{\displaystyle E(e^{-\alpha X^{2}})}
534:{\displaystyle E(e^{-\alpha X^{2}})}
1601:
1243:{\displaystyle p_{1}a+p_{2}b=E(X)}
283:for estimates of other people. –
32:
492:are given, I have a hunch that
1704:
1698:
1645:
1639:
1613:
1607:
1571:
1565:
1536:
1530:
1459:
1454:
1448:
1441:
1403:
1377:
1319:
1306:
1237:
1231:
1001:
988:
974:
967:
909:
903:
895:
882:
849:
836:
822:
815:
783:
757:
720:
707:
693:
686:
665:
659:
651:
638:
622:
609:
595:
588:
528:
502:
479:
473:
444:
438:
405:
375:
355:06:51, 11 September 2007 (UTC)
1:
1182:{\displaystyle p_{1}+p_{2}=1}
157:Wednesday, September 19, 2007
33:
1751:and the k-th moment of x is
256:Knowledge:Size of Knowledge
1832:
301:05:05, 19 July 2007 (UTC)
250:You may be interested in
1817:14:08, 5 July 2007 (UTC)
1683:12:53, 5 July 2007 (UTC)
1589:07:46, 5 July 2007 (UTC)
1520:should be, depending on
1362:14:26, 4 July 2007 (UTC)
1047:14:05, 4 July 2007 (UTC)
423:13:37, 4 July 2007 (UTC)
337:
332:
328:
323:Tuesday, August 28, 2007
321:
288:14:41, 4 July 2007 (UTC)
267:11:40, 4 July 2007 (UTC)
234:10:56, 4 July 2007 (UTC)
219:10:15, 4 July 2007 (UTC)
202:08:05, 4 July 2007 (UTC)
189:07:58, 4 July 2007 (UTC)
171:
166:
162:
155:
108:First, we all know that
18:Knowledge:Reference desk
1513:{\displaystyle \alpha }
1493:{\displaystyle \alpha }
1030:{\displaystyle \alpha }
541:might be maximal when:
264:Confusing Manifestation
110:the 1,000,000th article
1806:
1745:
1672:
1652:
1620:
1578:
1577:{\displaystyle Var(X)}
1543:
1514:
1494:
1474:
1410:
1346:
1326:
1244:
1183:
1133:
1031:
1011:
942:
790:
737:
535:
486:
485:{\displaystyle Var(X)}
451:
412:
87:current reference desk
1807:
1746:
1673:
1653:
1621:
1579:
1544:
1515:
1495:
1475:
1411:
1347:
1327:
1245:
1184:
1134:
1032:
1012:
943:
791:
738:
536:
487:
452:
413:
1755:
1692:
1662:
1630:
1598:
1553:
1542:{\displaystyle E(X)}
1524:
1504:
1484:
1420:
1371:
1336:
1254:
1193:
1147:
1057:
1021:
952:
800:
751:
547:
496:
461:
450:{\displaystyle E(X)}
432:
369:
1416:is upperbounded by
948:, which reduces to
1802:
1741:
1668:
1648:
1616:
1574:
1539:
1510:
1490:
1470:
1406:
1342:
1322:
1240:
1179:
1129:
1124:
1110:
1084:
1037:is very large. --
1027:
1007:
938:
786:
733:
728:
678:
574:
531:
482:
447:
408:
347:
181:
1671:{\displaystyle X}
1431:
1345:{\displaystyle a}
1109:
1108:with probability
1083:
1082:with probability
1005:
913:
853:
724:
677:
676:with probability
669:
626:
573:
572:with probability
316:
150:
93:
92:
73:
72:
1823:
1811:
1809:
1808:
1803:
1801:
1800:
1796:
1791:
1790:
1781:
1780:
1750:
1748:
1747:
1742:
1740:
1739:
1735:
1730:
1729:
1677:
1675:
1674:
1669:
1657:
1655:
1654:
1649:
1625:
1623:
1622:
1617:
1583:
1581:
1580:
1575:
1548:
1546:
1545:
1540:
1519:
1517:
1516:
1511:
1499:
1497:
1496:
1491:
1479:
1477:
1476:
1471:
1469:
1468:
1467:
1466:
1457:
1432:
1424:
1415:
1413:
1412:
1407:
1402:
1401:
1400:
1399:
1351:
1349:
1348:
1343:
1331:
1329:
1328:
1323:
1318:
1317:
1299:
1298:
1289:
1288:
1276:
1275:
1266:
1265:
1249:
1247:
1246:
1241:
1221:
1220:
1205:
1204:
1188:
1186:
1185:
1180:
1172:
1171:
1159:
1158:
1138:
1136:
1135:
1130:
1128:
1127:
1121:
1120:
1111:
1107:
1095:
1094:
1085:
1081:
1036:
1034:
1033:
1028:
1016:
1014:
1013:
1008:
1006:
1004:
1000:
999:
983:
982:
981:
962:
947:
945:
944:
939:
937:
933:
926:
925:
924:
923:
918:
914:
912:
898:
894:
893:
877:
854:
852:
848:
847:
831:
830:
829:
810:
795:
793:
792:
787:
782:
781:
780:
779:
742:
740:
739:
734:
732:
731:
725:
723:
719:
718:
702:
701:
700:
681:
679:
675:
670:
668:
654:
650:
649:
633:
627:
625:
621:
620:
604:
603:
602:
583:
575:
571:
540:
538:
537:
532:
527:
526:
525:
524:
491:
489:
488:
483:
456:
454:
453:
448:
417:
415:
414:
409:
404:
403:
402:
345:
335:
330:
325:
199:
179:
169:
164:
159:
75:
38:Mathematics desk
34:
1831:
1830:
1826:
1825:
1824:
1822:
1821:
1820:
1782:
1772:
1758:
1753:
1752:
1721:
1710:
1690:
1689:
1660:
1659:
1628:
1627:
1596:
1595:
1551:
1550:
1522:
1521:
1502:
1501:
1482:
1481:
1458:
1433:
1418:
1417:
1391:
1380:
1369:
1368:
1334:
1333:
1309:
1290:
1280:
1267:
1257:
1252:
1251:
1212:
1196:
1191:
1190:
1163:
1150:
1145:
1144:
1123:
1122:
1112:
1103:
1097:
1096:
1086:
1077:
1067:
1055:
1054:
1019:
1018:
991:
984:
973:
963:
950:
949:
899:
885:
878:
872:
871:
860:
859:
855:
839:
832:
821:
811:
798:
797:
796:is, of course,
771:
760:
749:
748:
727:
726:
710:
703:
692:
682:
671:
655:
641:
634:
629:
628:
612:
605:
594:
584:
567:
557:
545:
544:
516:
505:
494:
493:
459:
458:
430:
429:
394:
367:
366:
363:
197:
106:
101:
30:
29:
28:
12:
11:
5:
1829:
1827:
1799:
1795:
1789:
1785:
1779:
1775:
1771:
1768:
1765:
1761:
1738:
1734:
1728:
1724:
1720:
1717:
1713:
1709:
1706:
1703:
1700:
1697:
1686:
1685:
1678:fully known? —
1667:
1647:
1644:
1641:
1638:
1635:
1615:
1612:
1609:
1606:
1603:
1573:
1570:
1567:
1564:
1561:
1558:
1538:
1535:
1532:
1529:
1509:
1489:
1465:
1461:
1456:
1453:
1450:
1446:
1443:
1440:
1436:
1430:
1427:
1405:
1398:
1394:
1390:
1387:
1383:
1379:
1376:
1365:
1364:
1354:Meni Rosenfeld
1341:
1321:
1316:
1312:
1308:
1305:
1302:
1297:
1293:
1287:
1283:
1279:
1274:
1270:
1264:
1260:
1239:
1236:
1233:
1230:
1227:
1224:
1219:
1215:
1211:
1208:
1203:
1199:
1178:
1175:
1170:
1166:
1162:
1157:
1153:
1141:
1140:
1139:
1126:
1119:
1115:
1104:
1102:
1099:
1098:
1093:
1089:
1078:
1076:
1073:
1072:
1070:
1065:
1062:
1049:
1039:Meni Rosenfeld
1026:
1003:
998:
994:
990:
987:
980:
976:
972:
969:
966:
960:
957:
936:
932:
929:
922:
917:
911:
908:
905:
902:
897:
892:
888:
884:
881:
875:
870:
867:
863:
858:
851:
846:
842:
838:
835:
828:
824:
820:
817:
814:
808:
805:
785:
778:
774:
770:
767:
763:
759:
756:
747:In this case,
745:
744:
743:
730:
722:
717:
713:
709:
706:
699:
695:
691:
688:
685:
672:
667:
664:
661:
658:
653:
648:
644:
640:
637:
631:
630:
624:
619:
615:
611:
608:
601:
597:
593:
590:
587:
581:
578:
568:
566:
563:
562:
560:
555:
552:
530:
523:
519:
515:
512:
508:
504:
501:
481:
478:
475:
472:
469:
466:
446:
443:
440:
437:
407:
401:
397:
393:
390:
386:
383:
380:
377:
374:
362:
359:
358:
357:
352:70.179.175.240
329:
318:
306:
305:
304:
303:
298:70.179.170.119
291:
290:
276:
275:
274:
273:
272:
271:
270:
269:
241:
240:
239:
238:
237:
236:
226:Meni Rosenfeld
205:
204:
183:
163:
152:
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
1828:
1819:
1818:
1815:
1797:
1793:
1787:
1783:
1777:
1773:
1769:
1766:
1763:
1759:
1736:
1732:
1726:
1722:
1718:
1715:
1711:
1707:
1701:
1695:
1684:
1681:
1665:
1642:
1636:
1633:
1610:
1604:
1593:
1592:
1591:
1590:
1587:
1568:
1562:
1559:
1556:
1533:
1527:
1507:
1487:
1463:
1451:
1444:
1438:
1434:
1428:
1425:
1396:
1392:
1388:
1385:
1381:
1374:
1363:
1359:
1355:
1339:
1314:
1310:
1303:
1300:
1295:
1291:
1285:
1281:
1277:
1272:
1268:
1262:
1258:
1234:
1228:
1225:
1222:
1217:
1213:
1209:
1206:
1201:
1197:
1176:
1173:
1168:
1164:
1160:
1155:
1151:
1142:
1117:
1113:
1100:
1091:
1087:
1074:
1068:
1063:
1060:
1053:
1052:
1050:
1048:
1044:
1040:
1024:
996:
992:
985:
978:
970:
964:
958:
955:
934:
930:
927:
920:
915:
906:
900:
890:
886:
879:
873:
868:
865:
861:
856:
844:
840:
833:
826:
818:
812:
806:
803:
776:
772:
768:
765:
761:
754:
746:
715:
711:
704:
697:
689:
683:
662:
656:
646:
642:
635:
617:
613:
606:
599:
591:
585:
579:
576:
564:
558:
553:
550:
543:
542:
521:
517:
513:
510:
506:
499:
476:
470:
467:
464:
441:
435:
427:
426:
425:
424:
421:
399:
395:
391:
388:
384:
381:
378:
372:
360:
356:
353:
349:
348:
346:
344:
341:
334:
331:
326:
324:
319:
315:
314:
312:
309:According to
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