2132:
1274:
36:
is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a
2407:
2027:
728:
1160:
2296:
1538:
619:
2212:
2176:
998:. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.
1620:
372:
109:
1752:
950:
219:
2494:
1685:
443:
889:
1307:
2041:
1105:
531:
325:
1916:
57:. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "meta-calculus".
1415:
833:
1815:
1356:
1944:
1847:
1435:
1379:
178:
984:
798:
1574:
1493:
1775:
1177:
1055:
771:
273:
253:
135:
1470:
2307:
1955:
637:
1017:
function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a
1115:
2556:
2223:
2527:
2517:
2504:
2464:
2546:
1778:
1359:
994:
of a random variable is the weighted average of the possible values it might take on, with the weights being the respective
1502:
2561:
557:
2181:
2145:
2566:
1591:
336:
73:
1699:
1021:
specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
190:
897:
1642:
388:
2551:
2127:{\displaystyle {\frac {\displaystyle \int _{\Omega }f(x)\,w(x)\,dx}{\displaystyle \int _{\Omega }w(x)\,dx}}}
1283:
839:
2413:
1438:
1064:
488:
282:
2459:
2434:
1870:
1627:
1544:
1319:
449:
226:
184:
situation in which all elements have equal weight. One can then apply this weight to various concepts.
54:
50:
1391:
2417:
1018:
1010:
806:
112:
2454:
379:
2449:
1006:
955:
2523:
2500:
1784:
1325:
1002:
750:
115:
1929:
1832:
1420:
1364:
835:, the best estimate of the signal is obtained by averaging all the measurements with weight
148:
1496:
629:
42:
962:
776:
2474:
1550:
1386:
1269:{\displaystyle {\frac {\sum _{i=1}^{n}w_{i}{\boldsymbol {x}}_{i}}{\sum _{i=1}^{n}w_{i}}},}
1014:
1475:
2429:
1760:
1169:
1040:
991:
756:
258:
238:
120:
1443:
2540:
2444:
2439:
2215:
625:
1623:
995:
893:
and the resulting variance is smaller than each of the independent measurements
468:
222:
1541:
746:
541:
460:
138:
46:
2402:{\displaystyle {\langle f,g\rangle }_{w}:=\int _{\Omega }f(x)g(x)\ w(x)\ dx.}
2469:
1034:
232:
142:
17:
2022:{\displaystyle {\frac {1}{\mathrm {vol} (\Omega )}}\int _{\Omega }f(x)\ dx}
1009:
is assumed to be affected by both current and lagged (past) values of the
958:
method weights the difference between fit and data using the same weights
1634:
801:
1946:
has finite non-zero weighted volume, then we can replace the unweighted
1947:
1577:
549:
1850:
1382:
1108:
723:{\displaystyle {\frac {\sum _{a\in A}f(a)w(a)}{\sum _{a\in A}w(a)}}.}
1155:{\displaystyle {\boldsymbol {x}}_{1},\dotsc ,{\boldsymbol {x}}_{n}}
1165:
1058:
2496:
The First
Systems of Weighted Differential and Integral Calculus
2291:{\displaystyle \langle f,g\rangle :=\int _{\Omega }f(x)g(x)\ dx}
545:
1111:
is now interpreted in the physical sense) and locations
900:
842:
2310:
2226:
2214:
are two functions, one can generalize the unweighted
2184:
2148:
2091:
2047:
2044:
1958:
1932:
1873:
1835:
1787:
1763:
1702:
1645:
1594:
1553:
1505:
1478:
1446:
1423:
1394:
1367:
1328:
1286:
1180:
1118:
1067:
1043:
965:
809:
779:
759:
640:
560:
491:
391:
339:
285:
261:
241:
193:
151:
123:
76:
1279:
which is also the weighted average of the positions
1533:{\displaystyle w\colon \Omega \to \mathbb {R} ^{+}}
2401:
2290:
2206:
2170:
2126:
2021:
1938:
1910:
1841:
1809:
1769:
1746:
1679:
1614:
1568:
1532:
1487:
1464:
1429:
1409:
1373:
1350:
1318:In the continuous setting, a weight is a positive
1301:
1268:
1154:
1099:
1049:
978:
944:
883:
827:
792:
765:
722:
614:{\displaystyle {\frac {1}{|A|}}\sum _{a\in A}f(a)}
613:
525:
448:One common application of weighted sums arises in
437:
366:
319:
267:
247:
213:
172:
129:
103:
2207:{\displaystyle g\colon \Omega \to {\mathbb {R} }}
2171:{\displaystyle f\colon \Omega \to {\mathbb {R} }}
1615:{\displaystyle f\colon \Omega \to \mathbb {R} }
29:Construct related to weighted sums and averages
2493:Jane Grossman, Michael Grossman, Robert Katz.
544:non-empty set, one can replace the unweighted
53:, and are closely related to the concept of a
367:{\displaystyle w\colon A\to \mathbb {R} ^{+}}
104:{\displaystyle w\colon A\to \mathbb {R} ^{+}}
8:
2325:
2313:
2239:
2227:
1747:{\displaystyle \int _{\Omega }f(x)w(x)\,dx}
70:In the discrete setting, a weight function
2342:
2329:
2312:
2309:
2249:
2225:
2199:
2198:
2197:
2183:
2163:
2162:
2161:
2147:
2114:
2096:
2083:
2070:
2052:
2045:
2043:
1992:
1965:
1959:
1957:
1931:
1878:
1872:
1834:
1817:in order for this integral to be finite.
1800:
1786:
1762:
1737:
1707:
1701:
1650:
1644:
1608:
1607:
1593:
1552:
1524:
1520:
1519:
1504:
1477:
1445:
1422:
1401:
1397:
1396:
1393:
1366:
1341:
1327:
1293:
1288:
1285:
1254:
1244:
1233:
1221:
1216:
1209:
1199:
1188:
1181:
1179:
1164:then the lever will be in balance if the
1146:
1141:
1125:
1120:
1117:
1091:
1072:
1066:
1042:
970:
964:
945:{\textstyle \sigma ^{2}=1/\sum _{i}w_{i}}
936:
926:
917:
905:
899:
874:
869:
864:
859:
847:
841:
819:
814:
808:
784:
778:
758:
690:
648:
641:
639:
587:
575:
567:
561:
559:
496:
490:
396:
390:
358:
354:
353:
338:
290:
284:
260:
240:
214:{\displaystyle f\colon A\to \mathbb {R} }
207:
206:
192:
150:
122:
95:
91:
90:
75:
2519:Meta-Calculus: Differential and Integral
1547:. In this context, the weight function
2486:
1680:{\displaystyle \int _{\Omega }f(x)\ dx}
1289:
1217:
1142:
1121:
438:{\displaystyle \sum _{a\in A}f(a)w(a).}
45:. Weight functions occur frequently in
884:{\textstyle w_{i}=1/{\sigma _{i}^{2}}}
1302:{\displaystyle {\boldsymbol {x}}_{i}}
7:
773:measured multiple independent times
745:Weighted means are commonly used in
111:is a positive function defined on a
1100:{\displaystyle w_{1},\ldots ,w_{n}}
526:{\displaystyle \sum _{a\in B}w(a).}
320:{\displaystyle \sum _{a\in A}f(a);}
2343:
2250:
2191:
2155:
2097:
2053:
1993:
1979:
1972:
1969:
1966:
1933:
1911:{\displaystyle \int _{E}w(x)\ dx,}
1836:
1757:Note that one may need to require
1708:
1651:
1601:
1512:
1424:
1368:
749:to compensate for the presence of
25:
467:, one can replace the unweighted
1410:{\displaystyle \mathbb {R} ^{n}}
828:{\displaystyle \sigma _{i}^{2}}
2384:
2378:
2369:
2363:
2357:
2351:
2276:
2270:
2264:
2258:
2194:
2158:
2111:
2105:
2080:
2074:
2067:
2061:
2007:
2001:
1982:
1976:
1893:
1887:
1797:
1791:
1734:
1728:
1722:
1716:
1665:
1659:
1604:
1576:is sometimes referred to as a
1563:
1557:
1515:
1459:
1447:
1338:
1332:
711:
705:
681:
675:
669:
663:
608:
602:
576:
568:
517:
511:
429:
423:
417:
411:
349:
311:
305:
203:
161:
155:
86:
1:
2301:to a weighted bilinear form
1037:: if one has a collection of
1781:with respect to the weight
2583:
2557:Combinatorial optimization
2465:Riemann–Stieltjes integral
1861:can be generalized to the
1690:can be generalized to the
2416:for examples of weighted
1810:{\displaystyle w(x)\,dx}
1351:{\displaystyle w(x)\,dx}
1939:{\displaystyle \Omega }
1842:{\displaystyle \Omega }
1430:{\displaystyle \Omega }
1381:, which is typically a
1374:{\displaystyle \Omega }
1168:of the lever is at the
173:{\displaystyle w(a):=1}
145:. The weight function
2414:orthogonal polynomials
2403:
2292:
2208:
2172:
2128:
2023:
1940:
1912:
1843:
1811:
1771:
1748:
1681:
1616:
1570:
1534:
1489:
1466:
1431:
1411:
1375:
1352:
1303:
1270:
1249:
1204:
1156:
1101:
1051:
980:
946:
885:
829:
794:
767:
737:weights are relevant.
733:In this case only the
724:
615:
527:
439:
368:
321:
269:
249:
215:
174:
131:
105:
2547:Mathematical analysis
2460:Measure (mathematics)
2435:Numerical integration
2404:
2293:
2209:
2173:
2129:
2024:
1941:
1913:
1844:
1812:
1779:absolutely integrable
1772:
1749:
1682:
1617:
1571:
1535:
1490:
1467:
1432:
1412:
1376:
1353:
1304:
1271:
1229:
1184:
1157:
1102:
1052:
981:
979:{\displaystyle w_{i}}
947:
886:
830:
795:
793:{\displaystyle f_{i}}
768:
725:
616:
528:
450:numerical integration
440:
369:
322:
270:
250:
216:
175:
137:, which is typically
132:
106:
2418:orthogonal functions
2308:
2224:
2182:
2146:
2042:
1956:
1930:
1871:
1833:
1785:
1761:
1700:
1643:
1592:
1569:{\displaystyle w(x)}
1551:
1503:
1476:
1444:
1421:
1392:
1365:
1326:
1284:
1178:
1116:
1065:
1041:
1019:moving average model
1011:independent variable
963:
898:
840:
807:
777:
757:
638:
558:
489:
481:weighted cardinality
389:
337:
283:
259:
239:
191:
149:
121:
74:
2562:Functional analysis
2455:Kernel (statistics)
879:
824:
380:conical combination
180:corresponds to the
2567:Types of functions
2450:Linear combination
2399:
2288:
2204:
2168:
2124:
2121:
2090:
2019:
1936:
1908:
1839:
1807:
1767:
1744:
1677:
1612:
1584:General definition
1566:
1540:is a non-negative
1530:
1488:{\displaystyle dx}
1485:
1462:
1427:
1407:
1371:
1348:
1314:Continuous weights
1299:
1266:
1152:
1097:
1047:
1007:dependent variable
976:
956:maximum likelihood
942:
931:
881:
865:
825:
810:
790:
763:
753:. For a quantity
720:
701:
659:
611:
598:
523:
507:
435:
407:
364:
317:
301:
265:
245:
211:
170:
127:
101:
66:General definition
2412:See the entry on
2389:
2374:
2281:
2122:
2012:
1986:
1898:
1770:{\displaystyle f}
1692:weighted integral
1670:
1261:
1050:{\displaystyle n}
922:
766:{\displaystyle f}
715:
686:
644:
583:
581:
492:
392:
286:
268:{\displaystyle A}
248:{\displaystyle f}
130:{\displaystyle A}
16:(Redirected from
2574:
2531:
2514:
2508:
2491:
2408:
2406:
2405:
2400:
2387:
2372:
2347:
2346:
2334:
2333:
2328:
2297:
2295:
2294:
2289:
2279:
2254:
2253:
2213:
2211:
2210:
2205:
2203:
2202:
2177:
2175:
2174:
2169:
2167:
2166:
2133:
2131:
2130:
2125:
2123:
2101:
2100:
2057:
2056:
2046:
2034:weighted average
2028:
2026:
2025:
2020:
2010:
1997:
1996:
1987:
1985:
1975:
1960:
1945:
1943:
1942:
1937:
1922:Weighted average
1917:
1915:
1914:
1909:
1896:
1883:
1882:
1848:
1846:
1845:
1840:
1816:
1814:
1813:
1808:
1776:
1774:
1773:
1768:
1753:
1751:
1750:
1745:
1712:
1711:
1686:
1684:
1683:
1678:
1668:
1655:
1654:
1621:
1619:
1618:
1613:
1611:
1575:
1573:
1572:
1567:
1539:
1537:
1536:
1531:
1529:
1528:
1523:
1497:Lebesgue measure
1494:
1492:
1491:
1486:
1471:
1469:
1468:
1465:{\displaystyle }
1463:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1406:
1405:
1400:
1380:
1378:
1377:
1372:
1357:
1355:
1354:
1349:
1310:
1308:
1306:
1305:
1300:
1298:
1297:
1292:
1275:
1273:
1272:
1267:
1262:
1260:
1259:
1258:
1248:
1243:
1227:
1226:
1225:
1220:
1214:
1213:
1203:
1198:
1182:
1163:
1161:
1159:
1158:
1153:
1151:
1150:
1145:
1130:
1129:
1124:
1106:
1104:
1103:
1098:
1096:
1095:
1077:
1076:
1056:
1054:
1053:
1048:
1029:The terminology
987:
985:
983:
982:
977:
975:
974:
953:
951:
949:
948:
943:
941:
940:
930:
921:
910:
909:
892:
890:
888:
887:
882:
880:
878:
873:
863:
852:
851:
834:
832:
831:
826:
823:
818:
799:
797:
796:
791:
789:
788:
772:
770:
769:
764:
729:
727:
726:
721:
716:
714:
700:
684:
658:
642:
630:weighted average
620:
618:
617:
612:
597:
582:
580:
579:
571:
562:
532:
530:
529:
524:
506:
444:
442:
441:
436:
406:
373:
371:
370:
365:
363:
362:
357:
326:
324:
323:
318:
300:
274:
272:
271:
266:
254:
252:
251:
246:
220:
218:
217:
212:
210:
187:If the function
179:
177:
176:
171:
136:
134:
133:
128:
110:
108:
107:
102:
100:
99:
94:
61:Discrete weights
43:weighted average
21:
2582:
2581:
2577:
2576:
2575:
2573:
2572:
2571:
2537:
2536:
2535:
2534:
2515:
2511:
2492:
2488:
2483:
2475:Window function
2426:
2338:
2311:
2306:
2305:
2245:
2222:
2221:
2180:
2179:
2144:
2143:
2140:
2092:
2048:
2040:
2039:
1988:
1964:
1954:
1953:
1928:
1927:
1924:
1874:
1869:
1868:
1863:weighted volume
1831:
1830:
1829:is a subset of
1823:
1821:Weighted volume
1783:
1782:
1759:
1758:
1703:
1698:
1697:
1646:
1641:
1640:
1590:
1589:
1586:
1549:
1548:
1518:
1501:
1500:
1474:
1473:
1442:
1441:
1419:
1418:
1417:, for instance
1395:
1390:
1389:
1387:Euclidean space
1363:
1362:
1324:
1323:
1316:
1287:
1282:
1281:
1280:
1250:
1228:
1215:
1205:
1183:
1176:
1175:
1140:
1119:
1114:
1113:
1112:
1087:
1068:
1063:
1062:
1061:, with weights
1039:
1038:
1031:weight function
1027:
1015:distributed lag
966:
961:
960:
959:
932:
901:
896:
895:
894:
843:
838:
837:
836:
805:
804:
780:
775:
774:
755:
754:
743:
685:
643:
636:
635:
566:
556:
555:
487:
486:
387:
386:
352:
335:
334:
332:weight function
281:
280:
257:
256:
237:
236:
189:
188:
147:
146:
119:
118:
89:
72:
71:
68:
63:
34:weight function
30:
23:
22:
15:
12:
11:
5:
2580:
2578:
2570:
2569:
2564:
2559:
2554:
2552:Measure theory
2549:
2539:
2538:
2533:
2532:
2516:Jane Grossman.
2509:
2485:
2484:
2482:
2479:
2478:
2477:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2430:Center of mass
2425:
2422:
2410:
2409:
2398:
2395:
2392:
2386:
2383:
2380:
2377:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2345:
2341:
2337:
2332:
2327:
2324:
2321:
2318:
2315:
2299:
2298:
2287:
2284:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2252:
2248:
2244:
2241:
2238:
2235:
2232:
2229:
2201:
2196:
2193:
2190:
2187:
2165:
2160:
2157:
2154:
2151:
2139:
2136:
2135:
2134:
2120:
2117:
2113:
2110:
2107:
2104:
2099:
2095:
2089:
2086:
2082:
2079:
2076:
2073:
2069:
2066:
2063:
2060:
2055:
2051:
2030:
2029:
2018:
2015:
2009:
2006:
2003:
2000:
1995:
1991:
1984:
1981:
1978:
1974:
1971:
1968:
1963:
1935:
1923:
1920:
1919:
1918:
1907:
1904:
1901:
1895:
1892:
1889:
1886:
1881:
1877:
1838:
1822:
1819:
1806:
1803:
1799:
1796:
1793:
1790:
1766:
1755:
1754:
1743:
1740:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1710:
1706:
1688:
1687:
1676:
1673:
1667:
1664:
1661:
1658:
1653:
1649:
1610:
1606:
1603:
1600:
1597:
1585:
1582:
1565:
1562:
1559:
1556:
1527:
1522:
1517:
1514:
1511:
1508:
1484:
1481:
1461:
1458:
1455:
1452:
1449:
1426:
1404:
1399:
1370:
1347:
1344:
1340:
1337:
1334:
1331:
1315:
1312:
1296:
1291:
1277:
1276:
1265:
1257:
1253:
1247:
1242:
1239:
1236:
1232:
1224:
1219:
1212:
1208:
1202:
1197:
1194:
1191:
1187:
1170:center of mass
1149:
1144:
1139:
1136:
1133:
1128:
1123:
1094:
1090:
1086:
1083:
1080:
1075:
1071:
1046:
1026:
1023:
992:expected value
973:
969:
939:
935:
929:
925:
920:
916:
913:
908:
904:
877:
872:
868:
862:
858:
855:
850:
846:
822:
817:
813:
787:
783:
762:
742:
739:
731:
730:
719:
713:
710:
707:
704:
699:
696:
693:
689:
683:
680:
677:
674:
671:
668:
665:
662:
657:
654:
651:
647:
622:
621:
610:
607:
604:
601:
596:
593:
590:
586:
578:
574:
570:
565:
534:
533:
522:
519:
516:
513:
510:
505:
502:
499:
495:
446:
445:
434:
431:
428:
425:
422:
419:
416:
413:
410:
405:
402:
399:
395:
382:is defined as
361:
356:
351:
348:
345:
342:
328:
327:
316:
313:
310:
307:
304:
299:
296:
293:
289:
276:is defined as
264:
244:
209:
205:
202:
199:
196:
169:
166:
163:
160:
157:
154:
126:
98:
93:
88:
85:
82:
79:
67:
64:
62:
59:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2579:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2544:
2542:
2529:
2528:0-9771170-2-2
2525:
2521:
2520:
2513:
2510:
2506:
2505:0-9771170-1-4
2502:
2498:
2497:
2490:
2487:
2480:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2445:Weighted mean
2443:
2441:
2440:Orthogonality
2438:
2436:
2433:
2431:
2428:
2427:
2423:
2421:
2419:
2415:
2396:
2393:
2390:
2381:
2375:
2366:
2360:
2354:
2348:
2339:
2335:
2330:
2322:
2319:
2316:
2304:
2303:
2302:
2285:
2282:
2273:
2267:
2261:
2255:
2246:
2242:
2236:
2233:
2230:
2220:
2219:
2218:
2217:
2216:bilinear form
2188:
2185:
2152:
2149:
2138:Bilinear form
2137:
2118:
2115:
2108:
2102:
2093:
2087:
2084:
2077:
2071:
2064:
2058:
2049:
2038:
2037:
2036:
2035:
2016:
2013:
2004:
1998:
1989:
1961:
1952:
1951:
1950:
1949:
1921:
1905:
1902:
1899:
1890:
1884:
1879:
1875:
1867:
1866:
1865:
1864:
1860:
1856:
1852:
1828:
1820:
1818:
1804:
1801:
1794:
1788:
1780:
1764:
1741:
1738:
1731:
1725:
1719:
1713:
1704:
1696:
1695:
1694:
1693:
1674:
1671:
1662:
1656:
1647:
1639:
1638:
1637:
1636:
1633:
1629:
1625:
1598:
1595:
1583:
1581:
1579:
1560:
1554:
1546:
1543:
1525:
1509:
1506:
1498:
1482:
1479:
1456:
1453:
1450:
1440:
1402:
1388:
1384:
1361:
1345:
1342:
1335:
1329:
1321:
1313:
1311:
1294:
1263:
1255:
1251:
1245:
1240:
1237:
1234:
1230:
1222:
1210:
1206:
1200:
1195:
1192:
1189:
1185:
1174:
1173:
1172:
1171:
1167:
1147:
1137:
1134:
1131:
1126:
1110:
1092:
1088:
1084:
1081:
1078:
1073:
1069:
1060:
1057:objects on a
1044:
1036:
1032:
1024:
1022:
1020:
1016:
1012:
1008:
1005:in which the
1004:
999:
997:
996:probabilities
993:
988:
971:
967:
957:
937:
933:
927:
923:
918:
914:
911:
906:
902:
875:
870:
866:
860:
856:
853:
848:
844:
820:
815:
811:
803:
785:
781:
760:
752:
748:
740:
738:
736:
717:
708:
702:
697:
694:
691:
687:
678:
672:
666:
660:
655:
652:
649:
645:
634:
633:
632:
631:
627:
626:weighted mean
605:
599:
594:
591:
588:
584:
572:
563:
554:
553:
552:
551:
547:
543:
539:
520:
514:
508:
503:
500:
497:
493:
485:
484:
483:
482:
478:
474:
470:
466:
462:
458:
453:
451:
432:
426:
420:
414:
408:
403:
400:
397:
393:
385:
384:
383:
381:
377:
359:
346:
343:
340:
333:
314:
308:
302:
297:
294:
291:
287:
279:
278:
277:
275:
262:
242:
234:
228:
224:
200:
197:
194:
185:
183:
167:
164:
158:
152:
144:
140:
124:
117:
114:
96:
83:
80:
77:
65:
60:
58:
56:
52:
48:
44:
40:
35:
27:
19:
2518:
2512:
2495:
2489:
2411:
2300:
2141:
2033:
2031:
1925:
1862:
1858:
1854:
1826:
1824:
1756:
1691:
1689:
1631:
1587:
1437:could be an
1317:
1278:
1033:arises from
1030:
1028:
1000:
989:
744:
734:
732:
623:
537:
535:
480:
476:
472:
464:
456:
454:
447:
376:weighted sum
375:
331:
330:but given a
329:
230:
186:
181:
69:
39:weighted sum
38:
33:
31:
26:
18:Weighted sum
1849:, then the
1630:, then the
1003:regressions
469:cardinality
231:unweighted
229:, then the
2541:Categories
2481:References
1632:unweighted
1542:measurable
747:statistics
741:Statistics
463:subset of
182:unweighted
47:statistics
2470:Weighting
2344:Ω
2340:∫
2326:⟩
2314:⟨
2251:Ω
2247:∫
2240:⟩
2228:⟨
2195:→
2192:Ω
2189::
2159:→
2156:Ω
2153::
2098:Ω
2094:∫
2054:Ω
2050:∫
1994:Ω
1990:∫
1980:Ω
1934:Ω
1876:∫
1837:Ω
1709:Ω
1705:∫
1652:Ω
1648:∫
1605:→
1602:Ω
1599::
1516:→
1513:Ω
1510::
1425:Ω
1369:Ω
1231:∑
1186:∑
1135:…
1082:…
1035:mechanics
1025:Mechanics
924:∑
903:σ
867:σ
812:σ
695:∈
688:∑
653:∈
646:∑
592:∈
585:∑
501:∈
494:∑
401:∈
394:∑
350:→
344::
295:∈
288:∑
204:→
198::
143:countable
87:→
81::
2424:See also
1635:integral
1628:function
1626:-valued
1545:function
1472:. Here
1439:interval
1358:on some
1322:such as
802:variance
735:relative
227:function
225:-valued
113:discrete
51:analysis
2530:, 1981.
2507:, 1980.
2032:by the
1948:average
1578:density
1320:measure
1166:fulcrum
1107:(where
624:by the
550:average
479:by the
55:measure
2526:
2503:
2388:
2373:
2280:
2011:
1897:
1851:volume
1777:to be
1669:
1383:subset
1360:domain
1109:weight
542:finite
461:finite
374:, the
139:finite
1857:) of
1622:is a
1385:of a
1059:lever
800:with
540:is a
475:| of
459:is a
221:is a
2524:ISBN
2501:ISBN
2178:and
1853:vol(
1624:real
1499:and
1013:, a
990:The
954:The
751:bias
546:mean
223:real
49:and
2142:If
1926:If
1825:If
1588:If
1495:is
1001:In
628:or
548:or
536:If
455:If
378:or
255:on
235:of
233:sum
141:or
116:set
41:or
2543::
2522:,
2499:,
2420:.
2336::=
2243::=
1580:.
452:.
165::=
32:A
2397:.
2394:x
2391:d
2385:)
2382:x
2379:(
2376:w
2370:)
2367:x
2364:(
2361:g
2358:)
2355:x
2352:(
2349:f
2331:w
2323:g
2320:,
2317:f
2286:x
2283:d
2277:)
2274:x
2271:(
2268:g
2265:)
2262:x
2259:(
2256:f
2237:g
2234:,
2231:f
2200:R
2186:g
2164:R
2150:f
2119:x
2116:d
2112:)
2109:x
2106:(
2103:w
2088:x
2085:d
2081:)
2078:x
2075:(
2072:w
2068:)
2065:x
2062:(
2059:f
2017:x
2014:d
2008:)
2005:x
2002:(
1999:f
1983:)
1977:(
1973:l
1970:o
1967:v
1962:1
1906:,
1903:x
1900:d
1894:)
1891:x
1888:(
1885:w
1880:E
1859:E
1855:E
1827:E
1805:x
1802:d
1798:)
1795:x
1792:(
1789:w
1765:f
1742:x
1739:d
1735:)
1732:x
1729:(
1726:w
1723:)
1720:x
1717:(
1714:f
1675:x
1672:d
1666:)
1663:x
1660:(
1657:f
1609:R
1596:f
1564:)
1561:x
1558:(
1555:w
1526:+
1521:R
1507:w
1483:x
1480:d
1460:]
1457:b
1454:,
1451:a
1448:[
1403:n
1398:R
1346:x
1343:d
1339:)
1336:x
1333:(
1330:w
1309:.
1295:i
1290:x
1264:,
1256:i
1252:w
1246:n
1241:1
1238:=
1235:i
1223:i
1218:x
1211:i
1207:w
1201:n
1196:1
1193:=
1190:i
1162:,
1148:n
1143:x
1138:,
1132:,
1127:1
1122:x
1093:n
1089:w
1085:,
1079:,
1074:1
1070:w
1045:n
986:.
972:i
968:w
952:.
938:i
934:w
928:i
919:/
915:1
912:=
907:2
891:,
876:2
871:i
861:/
857:1
854:=
849:i
845:w
821:2
816:i
786:i
782:f
761:f
718:.
712:)
709:a
706:(
703:w
698:A
692:a
682:)
679:a
676:(
673:w
670:)
667:a
664:(
661:f
656:A
650:a
609:)
606:a
603:(
600:f
595:A
589:a
577:|
573:A
569:|
564:1
538:A
521:.
518:)
515:a
512:(
509:w
504:B
498:a
477:B
473:B
471:|
465:A
457:B
433:.
430:)
427:a
424:(
421:w
418:)
415:a
412:(
409:f
404:A
398:a
360:+
355:R
347:A
341:w
315:;
312:)
309:a
306:(
303:f
298:A
292:a
263:A
243:f
208:R
201:A
195:f
168:1
162:)
159:a
156:(
153:w
125:A
97:+
92:R
84:A
78:w
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.