127:
ultimately succeeding. Even if the safe-cracker has just failed 499 consecutive times (or 4,999 times), we expect to wait 500 more attempts until we observe the next success. If, instead, this person focused their attempts on a single safe, and "remembered" their previous attempts to open it, they would be guaranteed to open the safe after, at most, 500 attempts (and, in fact, at onset would only expect to need 250 attempts, not 500).
126:
will always be equal to the value of 500, regardless of how many attempts have already been made. Each new attempt has a (1/500) chance of succeeding, so the person is likely to open exactly one safe sometime in the next 500 attempts – but with each new failure they make no "progress" toward
117:
In contrast, let us examine a situation which would exhibit memorylessness. Imagine a long hallway, lined on one wall with thousands of safes. Each safe has a dial with 500 positions, and each has been assigned an opening position at random. Imagine that an eccentric person walks down the hallway,
394:, which describes the number of failed trials before the first success in a sequence of independent and identically distributed bernoulli trials. These random variables do not satisfy the memoryless condition stated above; however they do satisfy a slightly modified memoryless condition:
69:. It describes situations where the time already spent waiting for an event does not affect how much longer the wait will be. To model memoryless situations accurately, we have to disregard the past state of the system – the probabilities remain unaffected by the history of the process.
105:, the lifetime of a car engine, expressed in terms of "number of miles driven until the engine breaks down". It is clear, based on our intuition, that an engine which has already been driven for 300,000 miles will have a much lower
680:. It models random processes like time between consecutive events. The memorylessness property asserts that the amount of time since the previous event has no effect on the future time until the next event occurs.
1045:
835:
471:
607:
240:
356:
1458:
1775:
1408:
1123:
739:
502:
392:
902:
1263:
1177:
956:
1360:
353:
133:, which describes the time until a given radioactive particle decays, is a real-life example of memorylessness. An often used (theoretical) example of memorylessness in
671:
304:
324:
1309:
1283:
1197:
1085:
1065:
976:
647:
627:
533:
280:
260:
166:
1770:
109:
than would a second (equivalent) engine which has only been driven for 1,000 miles. Hence, this random variable would not have the memorylessness property.
362:
occurs. The memorylessness property asserts that the number of previously failed trials has no effect on the number of future trials needed for a success.
475:
Similar to the first definition, only discrete random variables that satisfy this memoryless condition are geometric random variables taking values in
985:
752:
122:
as the lifetime of their search, expressed in terms of "number of attempts the person must make until they successfully open a safe". In this case,
1748:
1713:
1678:
1643:
1583:
1548:
1513:
399:
538:
171:
97:
Most phenomena are not memoryless, which means that observers will obtain information about them over time. For example, suppose that
1365:
688:
The only memoryless continuous probability distribution is the exponential distribution, shown in the following proof:
677:
845:
118:
stopping once at each safe to make a single random attempt to open it. In this case, we might define random variable
1214:
1128:
907:
1314:
513:
330:
1477:
146:
81:
66:
1413:
746:
1500:. Synthesis Lectures on Mathematics & Statistics. Cham: Springer International Publishing. p. 71.
77:
1093:
694:
478:
368:
1529:
Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005).
1665:. Texts in Applied Mathematics. Vol. 77. Cham: Springer International Publishing. p. 84.
1200:
840:
336:
1565:
39:
1744:
1709:
1674:
1639:
1579:
1544:
1509:
742:
130:
31:
1736:
1701:
1666:
1631:
1571:
1536:
1501:
656:
289:
1088:
359:
309:
134:
102:
47:
1288:
1268:
1182:
1070:
1050:
979:
961:
632:
612:
518:
283:
265:
245:
151:
1730:
1660:
1570:. Wiley Series in Probability and Statistics (1st ed.). Wiley. pp. 260–261.
1495:
1764:
1695:
1625:
355:. This random variable describes when the first success in an infinite sequence of
43:
504:. In the continuous case, these two definitions of memorylessness are equivalent.
1530:
1567:
Probability and
Conditional Expectation: Fundamentals for the Empirical Sciences
1208:
650:
54:
1740:
1705:
1670:
1635:
1505:
58:
35:
137:
is the time a storekeeper must wait before the arrival of the next customer.
1204:
1575:
1540:
17:
1535:. Springer Texts in Statistics. London: Springer London. p. 50.
365:
Geometric random variables can also be defined as taking values in
1600:
1040:{\displaystyle S\left({\frac {t}{q}}\right)=S(t)^{\frac {1}{q}}}
830:{\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X>t)}}=\Pr(X>s)}
1497:
676:
The only continuous random variable that is memoryless is the
329:
The only discrete random variable that is memoryless is the
27:
Waiting time property of certain probability distributions
745:. From the memorylessness property and the definition of
1700:. Cham: Springer International Publishing. p. 74.
1104:
466:{\displaystyle \Pr(X>m+n\mid X\geq m)=\Pr(X>n).}
1416:
1368:
1317:
1291:
1271:
1217:
1185:
1131:
1096:
1073:
1053:
988:
964:
910:
848:
755:
697:
659:
635:
615:
541:
521:
481:
402:
371:
339:
312:
292:
268:
248:
174:
154:
602:{\displaystyle \Pr(X>s+t\mid X>t)=\Pr(X>s)}
235:{\displaystyle \Pr(X>m+n\mid X>m)=\Pr(X>n)}
1735:. Cham: Springer Nature Switzerland. pp. 8–9.
1532:
1494:Chattamvelli, Rajan; Shanmugam, Ramalingam (2020).
1452:
1402:
1354:
1303:
1277:
1257:
1191:
1171:
1117:
1079:
1059:
1039:
970:
950:
896:
829:
733:
665:
641:
621:
601:
527:
496:
465:
386:
347:
318:
298:
274:
254:
234:
160:
684:Exponential distribution and memorylessness proof
1630:. New York, NY: Springer New York. p. 279.
809:
785:
759:
713:
581:
542:
442:
403:
214:
175:
1732:Some Fundamentals of Mathematics of Blockchain
1697:Basics of Probability and Stochastic Processes
1776:Characterization of probability distributions
8:
1564:Nagel, Werner; Steyer, Rolf (2017-04-04).
1415:
1388:
1367:
1346:
1316:
1290:
1270:
1249:
1216:
1184:
1163:
1130:
1103:
1095:
1072:
1052:
1026:
996:
987:
963:
942:
909:
847:
756:
754:
696:
658:
634:
614:
540:
520:
488:
484:
483:
480:
401:
378:
374:
373:
370:
341:
340:
338:
311:
291:
267:
247:
173:
153:
1453:{\displaystyle \lambda =-\ln S(1)\geq 0}
1469:
357:independent and identically distributed
326:on the left hand side of the equation.
1662:An Introduction to Applied Probability
1478:"Notes on Memoryless Random Variables"
131:The universal law of radioactive decay
7:
72:Only two kinds of distributions are
1771:Theory of probability distributions
1403:{\displaystyle S(x)=e^{-\lambda x}}
1285:is a nonnegative real number. When
1203:and the set of rational numbers is
741:, also known as the distribution's
653:. The equality is still true when
286:. The equality is still true when
25:
1118:{\displaystyle a={\tfrac {p}{q}}}
168:is memoryless, then it satisfies
734:{\displaystyle S(t)=\Pr(X>t)}
535:is memoryless, then it satisfies
497:{\displaystyle \mathbb {N} _{0}}
387:{\displaystyle \mathbb {N} _{0}}
1067:is a natural number, excluding
897:{\displaystyle S(t+s)=S(t)S(s)}
1441:
1435:
1378:
1372:
1343:
1336:
1327:
1321:
1258:{\displaystyle S(xt)=S(t)^{x}}
1246:
1239:
1230:
1221:
1172:{\displaystyle S(at)=S(t)^{a}}
1160:
1153:
1144:
1135:
1023:
1016:
951:{\displaystyle S(pt)=S(t)^{p}}
939:
932:
923:
914:
891:
885:
879:
873:
864:
852:
824:
812:
800:
788:
780:
762:
728:
716:
707:
701:
596:
584:
575:
545:
457:
445:
436:
406:
229:
217:
208:
178:
1:
1355:{\displaystyle S(x)=S(1)^{x}}
348:{\displaystyle \mathbb {N} }
678:exponential random variable
84:probability distributions.
1792:
514:continuous random variable
29:
1741:10.1007/978-3-031-31323-3
1706:10.1007/978-3-030-32323-3
1671:10.1007/978-3-031-49306-5
1636:10.1007/978-1-4612-4374-8
1506:10.1007/978-3-031-02425-2
508:Continuous memorylessness
331:geometric random variable
67:probability distributions
65:is a property of certain
38:. For use of the term in
1659:Brémaud, Pierre (2024).
147:discrete random variable
1729:Riposo, Julien (2023).
747:conditional probability
141:Discrete memorylessness
30:For use of the term in
1454:
1404:
1356:
1305:
1279:
1259:
1193:
1173:
1119:
1081:
1061:
1041:
972:
952:
898:
831:
735:
667:
643:
623:
603:
529:
498:
467:
388:
349:
320:
300:
276:
256:
236:
162:
1605:mathworld.wolfram.com
1576:10.1002/9781119243496
1541:10.1007/1-84628-168-7
1455:
1405:
1357:
1306:
1280:
1260:
1194:
1174:
1120:
1082:
1062:
1042:
973:
953:
899:
832:
736:
668:
666:{\displaystyle \geq }
644:
624:
604:
530:
499:
468:
389:
350:
321:
301:
299:{\displaystyle \geq }
277:
257:
237:
163:
88:Waiting time examples
1624:Pitman, Jim (1993).
1414:
1366:
1315:
1289:
1269:
1215:
1183:
1129:
1094:
1071:
1051:
986:
962:
908:
846:
753:
695:
657:
633:
613:
539:
519:
479:
400:
369:
337:
319:{\displaystyle >}
310:
290:
266:
246:
172:
152:
40:stochastic processes
1599:Weisstein, Eric W.
1304:{\displaystyle t=1}
841:functional equation
306:is substituted for
1694:Bas, Esra (2019).
1450:
1400:
1352:
1301:
1275:
1255:
1189:
1169:
1115:
1113:
1077:
1057:
1037:
968:
948:
894:
827:
731:
663:
639:
619:
599:
525:
494:
463:
384:
345:
316:
296:
272:
252:
232:
158:
1750:978-3-031-31322-6
1715:978-3-030-32322-6
1680:978-3-031-49305-8
1645:978-0-387-94594-1
1585:978-1-119-24352-6
1550:978-1-85233-896-1
1515:978-3-031-01297-6
1278:{\displaystyle x}
1192:{\displaystyle S}
1112:
1087:. Therefore, all
1080:{\displaystyle 0}
1060:{\displaystyle q}
1034:
1004:
971:{\displaystyle p}
804:
749:, it follows that
743:survival function
642:{\displaystyle t}
622:{\displaystyle s}
528:{\displaystyle X}
333:taking values in
275:{\displaystyle n}
255:{\displaystyle m}
161:{\displaystyle X}
32:materials science
16:(Redirected from
1783:
1755:
1754:
1726:
1720:
1719:
1691:
1685:
1684:
1656:
1650:
1649:
1621:
1615:
1614:
1612:
1611:
1596:
1590:
1589:
1561:
1555:
1554:
1526:
1520:
1519:
1491:
1485:
1484:
1482:
1474:
1459:
1457:
1456:
1451:
1409:
1407:
1406:
1401:
1399:
1398:
1361:
1359:
1358:
1353:
1351:
1350:
1310:
1308:
1307:
1302:
1284:
1282:
1281:
1276:
1264:
1262:
1261:
1256:
1254:
1253:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1168:
1167:
1124:
1122:
1121:
1116:
1114:
1105:
1089:rational numbers
1086:
1084:
1083:
1078:
1066:
1064:
1063:
1058:
1046:
1044:
1043:
1038:
1036:
1035:
1027:
1009:
1005:
997:
977:
975:
974:
969:
957:
955:
954:
949:
947:
946:
903:
901:
900:
895:
836:
834:
833:
828:
805:
803:
783:
757:
740:
738:
737:
732:
673:is substituted.
672:
670:
669:
664:
649:are nonnegative
648:
646:
645:
640:
628:
626:
625:
620:
608:
606:
605:
600:
534:
532:
531:
526:
503:
501:
500:
495:
493:
492:
487:
472:
470:
469:
464:
393:
391:
390:
385:
383:
382:
377:
360:Bernoulli trials
354:
352:
351:
346:
344:
325:
323:
322:
317:
305:
303:
302:
297:
281:
279:
278:
273:
261:
259:
258:
253:
241:
239:
238:
233:
167:
165:
164:
159:
125:
121:
108:
100:
21:
1791:
1790:
1786:
1785:
1784:
1782:
1781:
1780:
1761:
1760:
1759:
1758:
1751:
1728:
1727:
1723:
1716:
1693:
1692:
1688:
1681:
1658:
1657:
1653:
1646:
1623:
1622:
1618:
1609:
1607:
1598:
1597:
1593:
1586:
1563:
1562:
1558:
1551:
1528:
1527:
1523:
1516:
1493:
1492:
1488:
1480:
1476:
1475:
1471:
1466:
1412:
1411:
1384:
1364:
1363:
1342:
1313:
1312:
1287:
1286:
1267:
1266:
1245:
1213:
1212:
1181:
1180:
1159:
1127:
1126:
1092:
1091:
1069:
1068:
1049:
1048:
1022:
992:
984:
983:
960:
959:
938:
906:
905:
844:
843:
839:This gives the
784:
758:
751:
750:
693:
692:
686:
655:
654:
631:
630:
611:
610:
537:
536:
517:
516:
510:
482:
477:
476:
398:
397:
372:
367:
366:
335:
334:
308:
307:
288:
287:
284:natural numbers
264:
263:
244:
243:
170:
169:
150:
149:
143:
135:queueing theory
123:
119:
115:
106:
103:random variable
98:
95:
90:
51:
48:Markov property
28:
23:
22:
15:
12:
11:
5:
1789:
1787:
1779:
1778:
1773:
1763:
1762:
1757:
1756:
1749:
1721:
1714:
1686:
1679:
1651:
1644:
1616:
1591:
1584:
1556:
1549:
1521:
1514:
1486:
1468:
1467:
1465:
1462:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1397:
1394:
1391:
1387:
1383:
1380:
1377:
1374:
1371:
1349:
1345:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1300:
1297:
1294:
1274:
1252:
1248:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1207:in the set of
1188:
1166:
1162:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1111:
1108:
1102:
1099:
1076:
1056:
1033:
1030:
1025:
1021:
1018:
1015:
1012:
1008:
1003:
1000:
995:
991:
980:natural number
967:
945:
941:
937:
934:
931:
928:
925:
922:
919:
916:
913:
904:which implies
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
826:
823:
820:
817:
814:
811:
808:
802:
799:
796:
793:
790:
787:
782:
779:
776:
773:
770:
767:
764:
761:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
691:First, define
685:
682:
662:
638:
618:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
524:
509:
506:
491:
486:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
381:
376:
343:
315:
295:
271:
251:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
157:
142:
139:
114:
113:Without memory
111:
94:
91:
89:
86:
63:memorylessness
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1788:
1777:
1774:
1772:
1769:
1768:
1766:
1752:
1746:
1742:
1738:
1734:
1733:
1725:
1722:
1717:
1711:
1707:
1703:
1699:
1698:
1690:
1687:
1682:
1676:
1672:
1668:
1664:
1663:
1655:
1652:
1647:
1641:
1637:
1633:
1629:
1628:
1620:
1617:
1606:
1602:
1595:
1592:
1587:
1581:
1577:
1573:
1569:
1568:
1560:
1557:
1552:
1546:
1542:
1538:
1534:
1533:
1525:
1522:
1517:
1511:
1507:
1503:
1499:
1498:
1490:
1487:
1479:
1473:
1470:
1463:
1461:
1447:
1444:
1438:
1432:
1429:
1426:
1423:
1420:
1417:
1395:
1392:
1389:
1385:
1381:
1375:
1369:
1362:As a result,
1347:
1339:
1333:
1330:
1324:
1318:
1298:
1295:
1292:
1272:
1250:
1242:
1236:
1233:
1227:
1224:
1218:
1210:
1206:
1202:
1186:
1164:
1156:
1150:
1147:
1141:
1138:
1132:
1109:
1106:
1100:
1097:
1090:
1074:
1054:
1031:
1028:
1019:
1013:
1010:
1006:
1001:
998:
993:
989:
982:. Similarly,
981:
965:
943:
935:
929:
926:
920:
917:
911:
888:
882:
876:
870:
867:
861:
858:
855:
849:
842:
837:
821:
818:
815:
806:
797:
794:
791:
777:
774:
771:
768:
765:
748:
744:
725:
722:
719:
710:
704:
698:
689:
683:
681:
679:
674:
660:
652:
636:
616:
593:
590:
587:
578:
572:
569:
566:
563:
560:
557:
554:
551:
548:
522:
515:
507:
505:
489:
473:
460:
454:
451:
448:
439:
433:
430:
427:
424:
421:
418:
415:
412:
409:
395:
379:
363:
361:
358:
332:
327:
313:
293:
285:
269:
249:
226:
223:
220:
211:
205:
202:
199:
196:
193:
190:
187:
184:
181:
155:
148:
140:
138:
136:
132:
128:
112:
110:
104:
92:
87:
85:
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44:Markov chains
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1608:. Retrieved
1604:
1601:"Memoryless"
1594:
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1627:Probability
93:With memory
82:exponential
55:probability
1765:Categories
1610:2024-07-25
1464:References
1201:continuous
74:memoryless
59:statistics
36:hysteresis
18:Memoryless
1445:≥
1430:
1424:−
1418:λ
1393:λ
1390:−
661:≥
564:∣
431:≥
425:∣
294:≥
197:∣
78:geometric
1125:satisfy
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1179:Since
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1481:(PDF)
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