2121:
1263:
25:
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
2396:
2016:
717:
1149:
2285:
1527:
608:
2201:
2165:
987:. 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.
1609:
361:
98:
1741:
939:
208:
2483:
1674:
432:
878:
1296:
2030:
1094:
520:
314:
1905:
46:. 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".
1404:
822:
1804:
1345:
1933:
1836:
1424:
1368:
167:
973:
787:
1563:
1482:
1764:
1166:
1044:
760:
262:
242:
124:
1459:
2296:
1944:
626:
1006:
function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a
1104:
2545:
2212:
2516:
2506:
2493:
2453:
2535:
1767:
1348:
983:
of a random variable is the weighted average of the possible values it might take on, with the weights being the respective
1491:
2550:
546:
2170:
2134:
2555:
1580:
325:
62:
1688:
1010:
specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
179:
886:
1631:
377:
2540:
2116:{\displaystyle {\frac {\displaystyle \int _{\Omega }f(x)\,w(x)\,dx}{\displaystyle \int _{\Omega }w(x)\,dx}}}
1272:
828:
2402:
1427:
1053:
477:
271:
2448:
2423:
1859:
1616:
1533:
1308:
438:
215:
173:
situation in which all elements have equal weight. One can then apply this weight to various concepts.
43:
39:
1380:
2406:
1007:
999:
795:
101:
2443:
368:
2438:
995:
944:
2512:
2489:
1773:
1314:
991:
739:
104:
1918:
1821:
1409:
1353:
824:, the best estimate of the signal is obtained by averaging all the measurements with weight
137:
1485:
618:
31:
951:
765:
2463:
1539:
1375:
1258:{\displaystyle {\frac {\sum _{i=1}^{n}w_{i}{\boldsymbol {x}}_{i}}{\sum _{i=1}^{n}w_{i}}},}
1003:
1464:
2418:
1749:
1158:
1029:
980:
745:
247:
227:
109:
1432:
2529:
2433:
2428:
2204:
614:
1612:
984:
882:
and the resulting variance is smaller than each of the independent measurements
457:
211:
1530:
735:
530:
449:
127:
35:
2391:{\displaystyle {\langle f,g\rangle }_{w}:=\int _{\Omega }f(x)g(x)\ w(x)\ dx.}
2458:
1023:
221:
131:
2011:{\displaystyle {\frac {1}{\mathrm {vol} (\Omega )}}\int _{\Omega }f(x)\ dx}
998:
is assumed to be affected by both current and lagged (past) values of the
947:
method weights the difference between fit and data using the same weights
1623:
790:
1935:
has finite non-zero weighted volume, then we can replace the unweighted
1936:
1566:
538:
1839:
1371:
1097:
712:{\displaystyle {\frac {\sum _{a\in A}f(a)w(a)}{\sum _{a\in A}w(a)}}.}
1144:{\displaystyle {\boldsymbol {x}}_{1},\dotsc ,{\boldsymbol {x}}_{n}}
1154:
1047:
2485:
The First
Systems of Weighted Differential and Integral Calculus
2280:{\displaystyle \langle f,g\rangle :=\int _{\Omega }f(x)g(x)\ dx}
534:
1100:
is now interpreted in the physical sense) and locations
889:
831:
2299:
2215:
2203:
are two functions, one can generalize the unweighted
2173:
2137:
2080:
2036:
2033:
1947:
1921:
1862:
1824:
1776:
1752:
1691:
1634:
1583:
1542:
1494:
1467:
1435:
1412:
1383:
1356:
1317:
1275:
1169:
1107:
1056:
1032:
954:
798:
768:
748:
629:
549:
480:
380:
328:
274:
250:
230:
182:
140:
112:
65:
1268:
which is also the weighted average of the positions
1522:{\displaystyle w\colon \Omega \to \mathbb {R} ^{+}}
2390:
2279:
2195:
2159:
2115:
2010:
1927:
1899:
1830:
1798:
1758:
1735:
1668:
1603:
1557:
1521:
1476:
1453:
1418:
1398:
1362:
1339:
1307:In the continuous setting, a weight is a positive
1290:
1257:
1143:
1088:
1038:
967:
933:
872:
816:
781:
754:
711:
603:{\displaystyle {\frac {1}{|A|}}\sum _{a\in A}f(a)}
602:
514:
437:One common application of weighted sums arises in
426:
355:
308:
256:
236:
202:
161:
118:
92:
2196:{\displaystyle g\colon \Omega \to {\mathbb {R} }}
2160:{\displaystyle f\colon \Omega \to {\mathbb {R} }}
1604:{\displaystyle f\colon \Omega \to \mathbb {R} }
18:Construct related to weighted sums and averages
2482:Jane Grossman, Michael Grossman, Robert Katz.
533:non-empty set, one can replace the unweighted
42:, and are closely related to the concept of a
356:{\displaystyle w\colon A\to \mathbb {R} ^{+}}
93:{\displaystyle w\colon A\to \mathbb {R} ^{+}}
8:
2314:
2302:
2228:
2216:
1736:{\displaystyle \int _{\Omega }f(x)w(x)\,dx}
59:In the discrete setting, a weight function
2331:
2318:
2301:
2298:
2238:
2214:
2188:
2187:
2186:
2172:
2152:
2151:
2150:
2136:
2103:
2085:
2072:
2059:
2041:
2034:
2032:
1981:
1954:
1948:
1946:
1920:
1867:
1861:
1823:
1806:in order for this integral to be finite.
1789:
1775:
1751:
1726:
1696:
1690:
1639:
1633:
1597:
1596:
1582:
1541:
1513:
1509:
1508:
1493:
1466:
1434:
1411:
1390:
1386:
1385:
1382:
1355:
1330:
1316:
1282:
1277:
1274:
1243:
1233:
1222:
1210:
1205:
1198:
1188:
1177:
1170:
1168:
1153:then the lever will be in balance if the
1135:
1130:
1114:
1109:
1106:
1080:
1061:
1055:
1031:
959:
953:
934:{\textstyle \sigma ^{2}=1/\sum _{i}w_{i}}
925:
915:
906:
894:
888:
863:
858:
853:
848:
836:
830:
808:
803:
797:
773:
767:
747:
679:
637:
630:
628:
576:
564:
556:
550:
548:
485:
479:
385:
379:
347:
343:
342:
327:
279:
273:
249:
229:
203:{\displaystyle f\colon A\to \mathbb {R} }
196:
195:
181:
139:
111:
84:
80:
79:
64:
2508:Meta-Calculus: Differential and Integral
1536:. In this context, the weight function
2475:
1669:{\displaystyle \int _{\Omega }f(x)\ dx}
1278:
1206:
1131:
1110:
427:{\displaystyle \sum _{a\in A}f(a)w(a).}
34:. Weight functions occur frequently in
873:{\textstyle w_{i}=1/{\sigma _{i}^{2}}}
1291:{\displaystyle {\boldsymbol {x}}_{i}}
7:
762:measured multiple independent times
734:Weighted means are commonly used in
100:is a positive function defined on a
1089:{\displaystyle w_{1},\ldots ,w_{n}}
515:{\displaystyle \sum _{a\in B}w(a).}
309:{\displaystyle \sum _{a\in A}f(a);}
2332:
2239:
2180:
2144:
2086:
2042:
1982:
1968:
1961:
1958:
1955:
1922:
1900:{\displaystyle \int _{E}w(x)\ dx,}
1825:
1746:Note that one may need to require
1697:
1640:
1590:
1501:
1413:
1357:
738:to compensate for the presence of
14:
456:, one can replace the unweighted
1399:{\displaystyle \mathbb {R} ^{n}}
817:{\displaystyle \sigma _{i}^{2}}
2373:
2367:
2358:
2352:
2346:
2340:
2265:
2259:
2253:
2247:
2183:
2147:
2100:
2094:
2069:
2063:
2056:
2050:
1996:
1990:
1971:
1965:
1882:
1876:
1786:
1780:
1723:
1717:
1711:
1705:
1654:
1648:
1593:
1565:is sometimes referred to as a
1552:
1546:
1504:
1448:
1436:
1327:
1321:
700:
694:
670:
664:
658:
652:
597:
591:
565:
557:
506:
500:
418:
412:
406:
400:
338:
300:
294:
192:
150:
144:
75:
1:
2290:to a weighted bilinear form
1026:: if one has a collection of
1770:with respect to the weight
2572:
2546:Combinatorial optimization
2454:Riemann–Stieltjes integral
1850:can be generalized to the
1679:can be generalized to the
2405:for examples of weighted
1799:{\displaystyle w(x)\,dx}
1340:{\displaystyle w(x)\,dx}
1928:{\displaystyle \Omega }
1831:{\displaystyle \Omega }
1419:{\displaystyle \Omega }
1370:, which is typically a
1363:{\displaystyle \Omega }
1157:of the lever is at the
162:{\displaystyle w(a):=1}
134:. The weight function
2403:orthogonal polynomials
2392:
2281:
2197:
2161:
2117:
2012:
1929:
1901:
1832:
1800:
1760:
1737:
1670:
1605:
1559:
1523:
1478:
1455:
1420:
1400:
1364:
1341:
1292:
1259:
1238:
1193:
1145:
1090:
1040:
969:
935:
874:
818:
783:
756:
726:weights are relevant.
722:In this case only the
713:
604:
516:
428:
357:
310:
258:
238:
204:
163:
120:
94:
2536:Mathematical analysis
2449:Measure (mathematics)
2424:Numerical integration
2393:
2282:
2198:
2162:
2118:
2013:
1930:
1902:
1833:
1801:
1768:absolutely integrable
1761:
1738:
1671:
1606:
1560:
1524:
1479:
1456:
1421:
1401:
1365:
1342:
1293:
1260:
1218:
1173:
1146:
1091:
1041:
970:
968:{\displaystyle w_{i}}
936:
875:
819:
784:
782:{\displaystyle f_{i}}
757:
714:
605:
517:
439:numerical integration
429:
358:
311:
259:
239:
205:
164:
126:, which is typically
121:
95:
2407:orthogonal functions
2297:
2213:
2171:
2135:
2031:
1945:
1919:
1860:
1822:
1774:
1750:
1689:
1632:
1581:
1558:{\displaystyle w(x)}
1540:
1492:
1465:
1433:
1410:
1381:
1354:
1315:
1273:
1167:
1105:
1054:
1030:
1008:moving average model
1000:independent variable
952:
887:
829:
796:
766:
746:
627:
547:
478:
470:weighted cardinality
378:
326:
272:
248:
228:
180:
138:
110:
63:
2551:Functional analysis
2444:Kernel (statistics)
868:
813:
369:conical combination
169:corresponds to the
2556:Types of functions
2439:Linear combination
2388:
2277:
2193:
2157:
2113:
2110:
2079:
2008:
1925:
1897:
1828:
1796:
1756:
1733:
1666:
1601:
1573:General definition
1555:
1529:is a non-negative
1519:
1477:{\displaystyle dx}
1474:
1451:
1416:
1396:
1360:
1337:
1303:Continuous weights
1288:
1255:
1141:
1086:
1036:
996:dependent variable
965:
945:maximum likelihood
931:
920:
870:
854:
814:
799:
779:
752:
742:. For a quantity
709:
690:
648:
600:
587:
512:
496:
424:
396:
353:
306:
290:
254:
234:
200:
159:
116:
90:
55:General definition
2401:See the entry on
2378:
2363:
2270:
2111:
2001:
1975:
1887:
1759:{\displaystyle f}
1681:weighted integral
1659:
1250:
1039:{\displaystyle n}
911:
755:{\displaystyle f}
704:
675:
633:
572:
570:
481:
381:
275:
257:{\displaystyle A}
237:{\displaystyle f}
119:{\displaystyle A}
2563:
2520:
2503:
2497:
2480:
2397:
2395:
2394:
2389:
2376:
2361:
2336:
2335:
2323:
2322:
2317:
2286:
2284:
2283:
2278:
2268:
2243:
2242:
2202:
2200:
2199:
2194:
2192:
2191:
2166:
2164:
2163:
2158:
2156:
2155:
2122:
2120:
2119:
2114:
2112:
2090:
2089:
2046:
2045:
2035:
2023:weighted average
2017:
2015:
2014:
2009:
1999:
1986:
1985:
1976:
1974:
1964:
1949:
1934:
1932:
1931:
1926:
1911:Weighted average
1906:
1904:
1903:
1898:
1885:
1872:
1871:
1837:
1835:
1834:
1829:
1805:
1803:
1802:
1797:
1765:
1763:
1762:
1757:
1742:
1740:
1739:
1734:
1701:
1700:
1675:
1673:
1672:
1667:
1657:
1644:
1643:
1610:
1608:
1607:
1602:
1600:
1564:
1562:
1561:
1556:
1528:
1526:
1525:
1520:
1518:
1517:
1512:
1486:Lebesgue measure
1483:
1481:
1480:
1475:
1460:
1458:
1457:
1454:{\displaystyle }
1452:
1425:
1423:
1422:
1417:
1405:
1403:
1402:
1397:
1395:
1394:
1389:
1369:
1367:
1366:
1361:
1346:
1344:
1343:
1338:
1299:
1297:
1295:
1294:
1289:
1287:
1286:
1281:
1264:
1262:
1261:
1256:
1251:
1249:
1248:
1247:
1237:
1232:
1216:
1215:
1214:
1209:
1203:
1202:
1192:
1187:
1171:
1152:
1150:
1148:
1147:
1142:
1140:
1139:
1134:
1119:
1118:
1113:
1095:
1093:
1092:
1087:
1085:
1084:
1066:
1065:
1045:
1043:
1042:
1037:
1018:The terminology
976:
974:
972:
971:
966:
964:
963:
942:
940:
938:
937:
932:
930:
929:
919:
910:
899:
898:
881:
879:
877:
876:
871:
869:
867:
862:
852:
841:
840:
823:
821:
820:
815:
812:
807:
788:
786:
785:
780:
778:
777:
761:
759:
758:
753:
718:
716:
715:
710:
705:
703:
689:
673:
647:
631:
619:weighted average
609:
607:
606:
601:
586:
571:
569:
568:
560:
551:
521:
519:
518:
513:
495:
433:
431:
430:
425:
395:
362:
360:
359:
354:
352:
351:
346:
315:
313:
312:
307:
289:
263:
261:
260:
255:
243:
241:
240:
235:
209:
207:
206:
201:
199:
176:If the function
168:
166:
165:
160:
125:
123:
122:
117:
99:
97:
96:
91:
89:
88:
83:
50:Discrete weights
32:weighted average
2571:
2570:
2566:
2565:
2564:
2562:
2561:
2560:
2526:
2525:
2524:
2523:
2504:
2500:
2481:
2477:
2472:
2464:Window function
2415:
2327:
2300:
2295:
2294:
2234:
2211:
2210:
2169:
2168:
2133:
2132:
2129:
2081:
2037:
2029:
2028:
1977:
1953:
1943:
1942:
1917:
1916:
1913:
1863:
1858:
1857:
1852:weighted volume
1820:
1819:
1818:is a subset of
1812:
1810:Weighted volume
1772:
1771:
1748:
1747:
1692:
1687:
1686:
1635:
1630:
1629:
1579:
1578:
1575:
1538:
1537:
1507:
1490:
1489:
1463:
1462:
1431:
1430:
1408:
1407:
1406:, for instance
1384:
1379:
1378:
1376:Euclidean space
1352:
1351:
1313:
1312:
1305:
1276:
1271:
1270:
1269:
1239:
1217:
1204:
1194:
1172:
1165:
1164:
1129:
1108:
1103:
1102:
1101:
1076:
1057:
1052:
1051:
1050:, with weights
1028:
1027:
1020:weight function
1016:
1004:distributed lag
955:
950:
949:
948:
921:
890:
885:
884:
883:
832:
827:
826:
825:
794:
793:
769:
764:
763:
744:
743:
732:
674:
632:
625:
624:
555:
545:
544:
476:
475:
376:
375:
341:
324:
323:
321:weight function
270:
269:
246:
245:
226:
225:
178:
177:
136:
135:
108:
107:
78:
61:
60:
57:
52:
23:weight function
19:
12:
11:
5:
2569:
2567:
2559:
2558:
2553:
2548:
2543:
2541:Measure theory
2538:
2528:
2527:
2522:
2521:
2505:Jane Grossman.
2498:
2474:
2473:
2471:
2468:
2467:
2466:
2461:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2419:Center of mass
2414:
2411:
2399:
2398:
2387:
2384:
2381:
2375:
2372:
2369:
2366:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2334:
2330:
2326:
2321:
2316:
2313:
2310:
2307:
2304:
2288:
2287:
2276:
2273:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2241:
2237:
2233:
2230:
2227:
2224:
2221:
2218:
2190:
2185:
2182:
2179:
2176:
2154:
2149:
2146:
2143:
2140:
2128:
2125:
2124:
2123:
2109:
2106:
2102:
2099:
2096:
2093:
2088:
2084:
2078:
2075:
2071:
2068:
2065:
2062:
2058:
2055:
2052:
2049:
2044:
2040:
2019:
2018:
2007:
2004:
1998:
1995:
1992:
1989:
1984:
1980:
1973:
1970:
1967:
1963:
1960:
1957:
1952:
1924:
1912:
1909:
1908:
1907:
1896:
1893:
1890:
1884:
1881:
1878:
1875:
1870:
1866:
1827:
1811:
1808:
1795:
1792:
1788:
1785:
1782:
1779:
1755:
1744:
1743:
1732:
1729:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1699:
1695:
1677:
1676:
1665:
1662:
1656:
1653:
1650:
1647:
1642:
1638:
1599:
1595:
1592:
1589:
1586:
1574:
1571:
1554:
1551:
1548:
1545:
1516:
1511:
1506:
1503:
1500:
1497:
1473:
1470:
1450:
1447:
1444:
1441:
1438:
1415:
1393:
1388:
1359:
1336:
1333:
1329:
1326:
1323:
1320:
1304:
1301:
1285:
1280:
1266:
1265:
1254:
1246:
1242:
1236:
1231:
1228:
1225:
1221:
1213:
1208:
1201:
1197:
1191:
1186:
1183:
1180:
1176:
1159:center of mass
1138:
1133:
1128:
1125:
1122:
1117:
1112:
1083:
1079:
1075:
1072:
1069:
1064:
1060:
1035:
1015:
1012:
981:expected value
962:
958:
928:
924:
918:
914:
909:
905:
902:
897:
893:
866:
861:
857:
851:
847:
844:
839:
835:
811:
806:
802:
776:
772:
751:
731:
728:
720:
719:
708:
702:
699:
696:
693:
688:
685:
682:
678:
672:
669:
666:
663:
660:
657:
654:
651:
646:
643:
640:
636:
611:
610:
599:
596:
593:
590:
585:
582:
579:
575:
567:
563:
559:
554:
523:
522:
511:
508:
505:
502:
499:
494:
491:
488:
484:
435:
434:
423:
420:
417:
414:
411:
408:
405:
402:
399:
394:
391:
388:
384:
371:is defined as
350:
345:
340:
337:
334:
331:
317:
316:
305:
302:
299:
296:
293:
288:
285:
282:
278:
265:is defined as
253:
233:
198:
194:
191:
188:
185:
158:
155:
152:
149:
146:
143:
115:
87:
82:
77:
74:
71:
68:
56:
53:
51:
48:
17:
13:
10:
9:
6:
4:
3:
2:
2568:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2533:
2531:
2518:
2517:0-9771170-2-2
2514:
2510:
2509:
2502:
2499:
2495:
2494:0-9771170-1-4
2491:
2487:
2486:
2479:
2476:
2469:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2434:Weighted mean
2432:
2430:
2429:Orthogonality
2427:
2425:
2422:
2420:
2417:
2416:
2412:
2410:
2408:
2404:
2385:
2382:
2379:
2370:
2364:
2355:
2349:
2343:
2337:
2328:
2324:
2319:
2311:
2308:
2305:
2293:
2292:
2291:
2274:
2271:
2262:
2256:
2250:
2244:
2235:
2231:
2225:
2222:
2219:
2209:
2208:
2207:
2206:
2205:bilinear form
2177:
2174:
2141:
2138:
2127:Bilinear form
2126:
2107:
2104:
2097:
2091:
2082:
2076:
2073:
2066:
2060:
2053:
2047:
2038:
2027:
2026:
2025:
2024:
2005:
2002:
1993:
1987:
1978:
1950:
1941:
1940:
1939:
1938:
1910:
1894:
1891:
1888:
1879:
1873:
1868:
1864:
1856:
1855:
1854:
1853:
1849:
1845:
1841:
1817:
1809:
1807:
1793:
1790:
1783:
1777:
1769:
1753:
1730:
1727:
1720:
1714:
1708:
1702:
1693:
1685:
1684:
1683:
1682:
1663:
1660:
1651:
1645:
1636:
1628:
1627:
1626:
1625:
1622:
1618:
1614:
1587:
1584:
1572:
1570:
1568:
1549:
1543:
1535:
1532:
1514:
1498:
1495:
1487:
1471:
1468:
1445:
1442:
1439:
1429:
1391:
1377:
1373:
1350:
1334:
1331:
1324:
1318:
1310:
1302:
1300:
1283:
1252:
1244:
1240:
1234:
1229:
1226:
1223:
1219:
1211:
1199:
1195:
1189:
1184:
1181:
1178:
1174:
1163:
1162:
1161:
1160:
1156:
1136:
1126:
1123:
1120:
1115:
1099:
1081:
1077:
1073:
1070:
1067:
1062:
1058:
1049:
1046:objects on a
1033:
1025:
1021:
1013:
1011:
1009:
1005:
1001:
997:
994:in which the
993:
988:
986:
985:probabilities
982:
977:
960:
956:
946:
926:
922:
916:
912:
907:
903:
900:
895:
891:
864:
859:
855:
849:
845:
842:
837:
833:
809:
804:
800:
792:
774:
770:
749:
741:
737:
729:
727:
725:
706:
697:
691:
686:
683:
680:
676:
667:
661:
655:
649:
644:
641:
638:
634:
623:
622:
621:
620:
616:
615:weighted mean
594:
588:
583:
580:
577:
573:
561:
552:
543:
542:
541:
540:
536:
532:
528:
509:
503:
497:
492:
489:
486:
482:
474:
473:
472:
471:
467:
463:
459:
455:
451:
447:
442:
440:
421:
415:
409:
403:
397:
392:
389:
386:
382:
374:
373:
372:
370:
366:
348:
335:
332:
329:
322:
303:
297:
291:
286:
283:
280:
276:
268:
267:
266:
264:
251:
231:
223:
217:
213:
189:
186:
183:
174:
172:
156:
153:
147:
141:
133:
129:
113:
106:
103:
85:
72:
69:
66:
54:
49:
47:
45:
41:
37:
33:
29:
24:
16:
2507:
2501:
2484:
2478:
2400:
2289:
2130:
2022:
2020:
1914:
1851:
1847:
1843:
1815:
1813:
1745:
1680:
1678:
1620:
1576:
1426:could be an
1306:
1267:
1022:arises from
1019:
1017:
989:
978:
733:
723:
721:
612:
526:
524:
469:
465:
461:
453:
445:
443:
436:
365:weighted sum
364:
320:
319:but given a
318:
219:
175:
170:
58:
28:weighted sum
27:
22:
20:
15:
1838:, then the
1619:, then the
992:regressions
458:cardinality
220:unweighted
218:, then the
2530:Categories
2470:References
1621:unweighted
1531:measurable
736:statistics
730:Statistics
452:subset of
171:unweighted
36:statistics
2459:Weighting
2333:Ω
2329:∫
2315:⟩
2303:⟨
2240:Ω
2236:∫
2229:⟩
2217:⟨
2184:→
2181:Ω
2178::
2148:→
2145:Ω
2142::
2087:Ω
2083:∫
2043:Ω
2039:∫
1983:Ω
1979:∫
1969:Ω
1923:Ω
1865:∫
1826:Ω
1698:Ω
1694:∫
1641:Ω
1637:∫
1594:→
1591:Ω
1588::
1505:→
1502:Ω
1499::
1414:Ω
1358:Ω
1220:∑
1175:∑
1124:…
1071:…
1024:mechanics
1014:Mechanics
913:∑
892:σ
856:σ
801:σ
684:∈
677:∑
642:∈
635:∑
581:∈
574:∑
490:∈
483:∑
390:∈
383:∑
339:→
333::
284:∈
277:∑
193:→
187::
132:countable
76:→
70::
2413:See also
1624:integral
1617:function
1615:-valued
1534:function
1461:. Here
1428:interval
1347:on some
1311:such as
791:variance
724:relative
216:function
214:-valued
102:discrete
40:analysis
2519:, 1981.
2496:, 1980.
2021:by the
1937:average
1567:density
1309:measure
1155:fulcrum
1096:(where
613:by the
539:average
468:by the
44:measure
2515:
2492:
2377:
2362:
2269:
2000:
1886:
1840:volume
1766:to be
1658:
1372:subset
1349:domain
1098:weight
531:finite
450:finite
363:, the
128:finite
1846:) of
1611:is a
1374:of a
1048:lever
789:with
529:is a
464:| of
448:is a
210:is a
2513:ISBN
2490:ISBN
2167:and
1842:vol(
1613:real
1488:and
1002:, a
979:The
943:The
740:bias
535:mean
212:real
38:and
2131:If
1915:If
1814:If
1577:If
1484:is
990:In
617:or
537:or
525:If
444:If
367:or
244:on
224:of
222:sum
130:or
105:set
30:or
2532::
2511:,
2488:,
2409:.
2325::=
2232::=
1569:.
441:.
154::=
21:A
2386:.
2383:x
2380:d
2374:)
2371:x
2368:(
2365:w
2359:)
2356:x
2353:(
2350:g
2347:)
2344:x
2341:(
2338:f
2320:w
2312:g
2309:,
2306:f
2275:x
2272:d
2266:)
2263:x
2260:(
2257:g
2254:)
2251:x
2248:(
2245:f
2226:g
2223:,
2220:f
2189:R
2175:g
2153:R
2139:f
2108:x
2105:d
2101:)
2098:x
2095:(
2092:w
2077:x
2074:d
2070:)
2067:x
2064:(
2061:w
2057:)
2054:x
2051:(
2048:f
2006:x
2003:d
1997:)
1994:x
1991:(
1988:f
1972:)
1966:(
1962:l
1959:o
1956:v
1951:1
1895:,
1892:x
1889:d
1883:)
1880:x
1877:(
1874:w
1869:E
1848:E
1844:E
1816:E
1794:x
1791:d
1787:)
1784:x
1781:(
1778:w
1754:f
1731:x
1728:d
1724:)
1721:x
1718:(
1715:w
1712:)
1709:x
1706:(
1703:f
1664:x
1661:d
1655:)
1652:x
1649:(
1646:f
1598:R
1585:f
1553:)
1550:x
1547:(
1544:w
1515:+
1510:R
1496:w
1472:x
1469:d
1449:]
1446:b
1443:,
1440:a
1437:[
1392:n
1387:R
1335:x
1332:d
1328:)
1325:x
1322:(
1319:w
1298:.
1284:i
1279:x
1253:,
1245:i
1241:w
1235:n
1230:1
1227:=
1224:i
1212:i
1207:x
1200:i
1196:w
1190:n
1185:1
1182:=
1179:i
1151:,
1137:n
1132:x
1127:,
1121:,
1116:1
1111:x
1082:n
1078:w
1074:,
1068:,
1063:1
1059:w
1034:n
975:.
961:i
957:w
941:.
927:i
923:w
917:i
908:/
904:1
901:=
896:2
880:,
865:2
860:i
850:/
846:1
843:=
838:i
834:w
810:2
805:i
775:i
771:f
750:f
707:.
701:)
698:a
695:(
692:w
687:A
681:a
671:)
668:a
665:(
662:w
659:)
656:a
653:(
650:f
645:A
639:a
598:)
595:a
592:(
589:f
584:A
578:a
566:|
562:A
558:|
553:1
527:A
510:.
507:)
504:a
501:(
498:w
493:B
487:a
466:B
462:B
460:|
454:A
446:B
422:.
419:)
416:a
413:(
410:w
407:)
404:a
401:(
398:f
393:A
387:a
349:+
344:R
336:A
330:w
304:;
301:)
298:a
295:(
292:f
287:A
281:a
252:A
232:f
197:R
190:A
184:f
157:1
151:)
148:a
145:(
142:w
114:A
86:+
81:R
73:A
67:w
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