318:: I'm trying to verify the formula for the Expectation of \sigma^2 E which is given now as \beta/(\alpha-1/2), but was just recently changed. Just a few days ago it was \beta/(2 (\alpha-1)). But I tried to carry out the integral myself and come to the old result. The issue is whether we assume this is a density function of (x, \sigma^2) or of (x, \sigma). If I assume it's a function of (x,\sigma) I get the old result E = \beta/(2(\alpha-1)). However if I assume that it is a function of (x,\sigma^2) I need to do a different variable substitution in the integral and get the result E = \beta^{1/2} \Gamma(\alpha-3/2)/\Gamma(\alpha). Very confusing. So I wonder if this distribution should be indeed seen as function of (x, \sigma), that is, a joint density over the domain (x, \sigma)
71:
53:
1555:
22:
1214:
1850:
1223:
938:
929:
1588:
1550:{\displaystyle ={\frac {{\sqrt {\lambda }}\beta ^{\alpha }}{{\sqrt {2\pi }}\Gamma (\alpha )}}\int _{0}^{\infty }{\frac {1}{\sqrt {}}}^{-(\alpha +1)}\exp \left(-{\frac {\beta }{}}\right)\int _{-\infty }^{\infty }\exp \left(-{\frac {(x-\mu )^{2}}{2\lambda ^{-1}}}\right)\ {\text{d}}x\,{\text{d}}}
2010:
2172:
1209:{\displaystyle =\int _{0}^{\infty }\int _{-\infty }^{\infty }{\frac {\sqrt {\lambda }}{\sqrt {2\pi }}}{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}^{-(\alpha +1)}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2}}\right)\ {\text{d}}x\,{\text{d}}}
738:
1845:{\displaystyle ={\frac {{\sqrt {\lambda }}\beta ^{\alpha }}{{\sqrt {2\pi }}\Gamma (\alpha )}}\int _{0}^{\infty }{\frac {1}{\sqrt {}}}^{-(\alpha +1)}\exp \left(-{\frac {\beta }{}}\right){\sqrt {2\pi \lambda ^{-1}}}\ {\text{d}}}
2411:
1859:
117:
2416:
2414:
2215:
We can further confirm this result by generating samples from the normal-inverse-gamma distribution (the sampling procedure is described in the article) and estimating the expected value of the
2211:
2275:
488:
299:
448:
2089:
244:
2240:
672:
612:
582:
515:
369:
2343:
209:
924:{\displaystyle E]=\int _{0}^{\infty }\int _{-\infty }^{\infty }\ {\text{Normal-inverse-gamma}}\left(x,\ |\ \mu ,\lambda ,\alpha ,\beta \right)\ {\text{d}}x\,{\text{d}}}
692:
535:
189:
2446:
111:
2317:
1577:
731:
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264:
2082:
2045:
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2417:
https://bookdown.org/aramir21/IntroductionBayesianEconometricsGuidedTour/sec42.html#sec42:~:text=%2C%20respectively.-,The%20marginal%20posterior%20of,is,-%CF%80
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2441:
2355:
2289:
87:
2005:{\displaystyle ={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\int _{0}^{\infty }^{-\alpha }\exp \left(-{\frac {\beta }{}}\right)\ {\text{d}}}
78:
58:
333:
135:
The CDF does not look right, if sigma^2 to Infty, the CDF goes to zero if I am not mistaken, which does not make sense, no?
33:
2048:
2178:
2245:
140:
21:
2285:
2167:{\displaystyle ={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}{\frac {\Gamma (\alpha -1)}{\beta ^{\alpha -1}}}}
136:
2423:
158:
2319:
is wrong (which can be readily observed by comparison with the result of the proof reported for the case
453:
269:
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329:
39:
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52:
2419:
321:
154:
1580:
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86:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
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2345:
below) and is taken from the posterior predictive formula that one can find in the table of
677:
520:
2346:
150:
174:
2302:
1562:
700:
540:
377:
249:
2435:
2054:
2017:
617:
1579:, which appears as a squared exponential function (proportional to the pdf of the
490:(incorrect). To get this right, you have to remember to integrate with respect to
2406:{\displaystyle t_{2\alpha }\left(\mu ,{\frac {\beta }{\alpha \lambda }}\right)}
83:
2413:. A derivation with a different notation for the parameters can be found in
246:, but in the section about generating values from the distribution, mean is
301:. I don't have a reference handy so I'm not sure which one is correct.
149:
We probably ought to make the parameters match with those listed in the
933:
From the definition of the normal-inverse-gamma distribution:
15:
537:
since the support of the distribution is defined in terms of
171:
Notation is inconsistent with regard to the distribution of
343:
2295:
On the marginal distribution of x in the univariate case
2358:
2325:
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2248:
2221:
2181:
2092:
2057:
2020:
1862:
1591:
1565:
1226:
941:
741:
703:
680:
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217:
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82:, a collaborative effort to improve the coverage of
2405:
2337:
2311:
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2205:
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2076:
2047:, which appears in the same form as the pdf of an
2039:
2004:
1844:
1571:
1549:
1208:
923:
725:
686:
666:
639:
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549:
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442:
402:
374:There seems to be some disagreement about whether
363:
293:
258:
238:
203:
183:
116:This article has not yet received a rating on the
2299:The formula for the marginal distribution of
8:
2206:{\displaystyle ={\frac {\beta }{\alpha -1}}}
2270:{\displaystyle {\frac {\beta }{\alpha -1}}}
19:
587:To make the above clear, all instances of
319:
47:
2383:
2363:
2357:
2324:
2304:
2249:
2247:
2226:
2220:
2185:
2180:
2150:
2124:
2102:
2096:
2091:
2065:
2056:
2028:
2019:
1993:
1981:
1962:
1949:
1926:
1916:
1903:
1898:
1872:
1866:
1861:
1833:
1821:
1808:
1792:
1780:
1763:
1750:
1715:
1705:
1686:
1673:
1664:
1651:
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1617:
1609:
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1538:
1526:
1516:
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994:
985:
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844:
821:
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797:
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628:
619:
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1524:
1183:
898:
49:
2447:Unknown-importance Statistics articles
2242:argument empirically. It converges to
7:
483:{\displaystyle \beta /(\alpha -0.5)}
294:{\displaystyle \sigma ^{2}/\lambda }
76:This article is within the scope of
38:It is of interest to the following
2127:
2109:
1904:
1879:
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1287:
1262:
1040:
973:
968:
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780:
443:{\displaystyle \beta /(\alpha -1)}
14:
344:Derivation of expected value of
96:Knowledge:WikiProject Statistics
69:
51:
20:
2452:WikiProject Statistics articles
2442:Start-Class Statistics articles
2352:I think the correct formula is
99:Template:WikiProject Statistics
2428:19:05, 21 September 2023 (UTC)
2290:14:05, 27 September 2018 (UTC)
2142:
2130:
2118:
2112:
2071:
2058:
2034:
2021:
1999:
1986:
1968:
1955:
1923:
1909:
1888:
1882:
1839:
1826:
1814:
1801:
1769:
1756:
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1149:
1135:
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1088:
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1021:
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991:
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918:
905:
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764:
761:
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707:
634:
621:
477:
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425:
397:
384:
191:. In the definition, mean is
1:
239:{\displaystyle \sigma ^{2}/v}
90:and see a list of open tasks.
145:19:57, 8 November 2017 (UTC)
2235:{\displaystyle \sigma ^{2}}
667:{\displaystyle \sigma ^{2}}
607:{\displaystyle \sigma ^{2}}
577:{\displaystyle \sigma ^{2}}
510:{\displaystyle \sigma ^{2}}
364:{\displaystyle \sigma ^{2}}
2468:
2338:{\displaystyle \lambda =1}
2049:inverse-gamma distribution
311:23:40, 12 June 2009 (UTC)
153:article or vice-versa. --
115:
64:
46:
674:is our variable and not
338:00:40, 8 June 2018 (UTC)
204:{\displaystyle \lambda }
163:16:56, 29 May 2008 (UTC)
697:From the definition of
687:{\displaystyle \sigma }
530:{\displaystyle \sigma }
2407:
2339:
2313:
2271:
2236:
2207:
2168:
2078:
2041:
2006:
1846:
1573:
1551:
1210:
925:
727:
688:
668:
641:
608:
578:
551:
531:
511:
484:
444:
404:
365:
295:
260:
240:
205:
185:
79:WikiProject Statistics
28:This article is rated
2408:
2340:
2314:
2272:
2237:
2208:
2169:
2079:
2042:
2007:
1847:
1574:
1552:
1211:
926:
728:
689:
669:
642:
614:below are written as
609:
579:
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532:
512:
485:
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405:
366:
296:
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186:
2356:
2323:
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2219:
2179:
2090:
2055:
2018:
1860:
1589:
1563:
1224:
939:
823:Normal-inverse-gamma
739:
701:
678:
651:
618:
591:
561:
541:
521:
494:
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378:
348:
270:
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215:
195:
184:{\displaystyle \mu }
175:
1908:
1656:
1581:normal distribution
1432:
1291:
977:
959:
802:
784:
102:Statistics articles
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921:
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547:
527:
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480:
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361:
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256:
236:
201:
181:
34:content assessment
2396:
2312:{\displaystyle x}
2265:
2201:
2162:
2122:
1984:
1980:
1972:
1892:
1824:
1820:
1817:
1773:
1696:
1695:
1640:
1625:
1603:
1572:{\displaystyle x}
1529:
1519:
1515:
1507:
1408:
1331:
1330:
1275:
1260:
1238:
1188:
1178:
1174:
1166:
1053:
1025:
1024:
1000:
903:
893:
889:
861:
854:
824:
820:
726:{\displaystyle E}
647:to indicate that
550:{\displaystyle x}
403:{\displaystyle E}
340:
324:comment added by
259:{\displaystyle v}
132:
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2459:
2412:
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2250:
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2239:
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2170:
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2161:
2160:
2145:
2125:
2123:
2121:
2107:
2106:
2097:
2083:
2081:
2080:
2077:{\displaystyle }
2075:
2070:
2069:
2046:
2044:
2043:
2040:{\displaystyle }
2038:
2033:
2032:
2011:
2009:
2008:
2003:
1998:
1997:
1985:
1982:
1979:
1978:
1974:
1973:
1971:
1967:
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1950:
1934:
1933:
1921:
1920:
1907:
1902:
1893:
1891:
1877:
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1867:
1851:
1849:
1848:
1843:
1838:
1837:
1825:
1822:
1819:
1818:
1813:
1812:
1800:
1799:
1781:
1779:
1775:
1774:
1772:
1768:
1767:
1751:
1735:
1734:
1710:
1709:
1697:
1691:
1690:
1678:
1674:
1669:
1668:
1655:
1650:
1641:
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1626:
1618:
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1614:
1613:
1604:
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1578:
1576:
1575:
1570:
1556:
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1553:
1548:
1543:
1542:
1530:
1527:
1520:
1517:
1514:
1513:
1509:
1508:
1506:
1502:
1501:
1489:
1488:
1472:
1471:
1470:
1448:
1431:
1426:
1414:
1410:
1409:
1407:
1403:
1402:
1386:
1370:
1369:
1345:
1344:
1332:
1326:
1325:
1313:
1309:
1304:
1303:
1290:
1285:
1276:
1274:
1261:
1253:
1250:
1249:
1248:
1239:
1234:
1231:
1215:
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1207:
1202:
1201:
1189:
1186:
1179:
1176:
1173:
1172:
1168:
1167:
1165:
1161:
1160:
1144:
1143:
1142:
1108:
1092:
1091:
1067:
1066:
1054:
1052:
1038:
1037:
1028:
1026:
1020:
1019:
1001:
996:
995:
990:
989:
976:
971:
958:
953:
930:
928:
927:
922:
917:
916:
904:
901:
894:
891:
888:
887:
883:
860:
859:
853:
849:
848:
825:
822:
819:
815:
814:
801:
796:
783:
778:
760:
759:
732:
730:
729:
724:
719:
718:
693:
691:
690:
685:
673:
671:
670:
665:
663:
662:
646:
644:
643:
640:{\displaystyle }
638:
633:
632:
613:
611:
610:
605:
603:
602:
583:
581:
580:
575:
573:
572:
556:
554:
553:
548:
536:
534:
533:
528:
516:
514:
513:
508:
506:
505:
489:
487:
486:
481:
464:
449:
447:
446:
441:
424:
409:
407:
406:
401:
396:
395:
370:
368:
367:
362:
360:
359:
300:
298:
297:
292:
287:
282:
281:
266:and variance is
265:
263:
262:
257:
245:
243:
242:
237:
232:
227:
226:
211:and variance is
210:
208:
207:
202:
190:
188:
187:
182:
137:DoubleMatchPoint
118:importance scale
104:
103:
100:
97:
94:
73:
66:
65:
55:
48:
31:
25:
24:
16:
2467:
2466:
2462:
2461:
2460:
2458:
2457:
2456:
2432:
2431:
2388:
2376:
2372:
2359:
2354:
2353:
2347:Conjugate prior
2321:
2320:
2301:
2300:
2297:
2254:
2244:
2243:
2222:
2217:
2216:
2190:
2177:
2176:
2146:
2126:
2108:
2098:
2088:
2087:
2061:
2053:
2052:
2024:
2016:
2015:
1989:
1958:
1954:
1945:
1941:
1922:
1912:
1878:
1868:
1858:
1857:
1829:
1804:
1788:
1759:
1755:
1746:
1742:
1711:
1701:
1682:
1660:
1616:
1605:
1597:
1587:
1586:
1561:
1560:
1534:
1493:
1477:
1473:
1462:
1449:
1443:
1439:
1394:
1390:
1381:
1377:
1346:
1336:
1317:
1295:
1251:
1240:
1232:
1222:
1221:
1193:
1152:
1145:
1134:
1109:
1103:
1099:
1068:
1058:
1039:
1029:
1011:
981:
937:
936:
908:
840:
830:
826:
806:
751:
737:
736:
710:
699:
698:
676:
675:
654:
649:
648:
624:
616:
615:
594:
589:
588:
564:
559:
558:
539:
538:
519:
518:
497:
492:
491:
452:
451:
412:
411:
387:
376:
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351:
346:
345:
273:
268:
267:
248:
247:
218:
213:
212:
193:
192:
173:
172:
151:Conjugate prior
101:
98:
95:
92:
91:
32:on Knowledge's
29:
12:
11:
5:
2465:
2463:
2455:
2454:
2449:
2444:
2434:
2433:
2401:
2394:
2391:
2387:
2382:
2379:
2375:
2369:
2366:
2362:
2334:
2331:
2328:
2308:
2296:
2293:
2263:
2260:
2257:
2253:
2229:
2225:
2199:
2196:
2193:
2189:
2184:
2159:
2156:
2153:
2149:
2144:
2141:
2138:
2135:
2132:
2129:
2120:
2117:
2114:
2111:
2105:
2101:
2095:
2073:
2068:
2064:
2060:
2051:with argument
2036:
2031:
2027:
2023:
2014:Integrate out
2001:
1996:
1992:
1988:
1977:
1970:
1965:
1961:
1957:
1953:
1948:
1944:
1940:
1937:
1932:
1929:
1925:
1919:
1915:
1911:
1906:
1901:
1897:
1890:
1887:
1884:
1881:
1875:
1871:
1865:
1841:
1836:
1832:
1828:
1816:
1811:
1807:
1803:
1798:
1795:
1791:
1787:
1784:
1778:
1771:
1766:
1762:
1758:
1754:
1749:
1745:
1741:
1738:
1733:
1730:
1727:
1724:
1721:
1718:
1714:
1708:
1704:
1700:
1694:
1689:
1685:
1681:
1677:
1672:
1667:
1663:
1659:
1654:
1649:
1645:
1638:
1635:
1632:
1629:
1624:
1621:
1612:
1608:
1602:
1594:
1568:
1559:Integrate out
1546:
1541:
1537:
1533:
1523:
1512:
1505:
1500:
1496:
1492:
1487:
1484:
1480:
1476:
1469:
1465:
1461:
1458:
1455:
1452:
1446:
1442:
1438:
1435:
1430:
1425:
1422:
1418:
1413:
1406:
1401:
1397:
1393:
1389:
1384:
1380:
1376:
1373:
1368:
1365:
1362:
1359:
1356:
1353:
1349:
1343:
1339:
1335:
1329:
1324:
1320:
1316:
1312:
1307:
1302:
1298:
1294:
1289:
1284:
1280:
1273:
1270:
1267:
1264:
1259:
1256:
1247:
1243:
1237:
1229:
1205:
1200:
1196:
1192:
1182:
1171:
1164:
1159:
1155:
1151:
1148:
1141:
1137:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1106:
1102:
1098:
1095:
1090:
1087:
1084:
1081:
1078:
1075:
1071:
1065:
1061:
1057:
1051:
1048:
1045:
1042:
1036:
1032:
1023:
1018:
1014:
1010:
1007:
1004:
999:
993:
988:
984:
980:
975:
970:
967:
963:
957:
952:
948:
944:
920:
915:
911:
907:
897:
886:
882:
879:
876:
873:
870:
867:
864:
858:
852:
847:
843:
839:
836:
833:
829:
818:
813:
809:
805:
800:
795:
792:
788:
782:
777:
773:
769:
766:
763:
758:
754:
750:
747:
744:
722:
717:
713:
709:
706:
683:
661:
657:
636:
631:
627:
623:
601:
597:
571:
567:
546:
526:
504:
500:
479:
476:
473:
470:
467:
463:
459:
439:
436:
433:
430:
427:
423:
419:
399:
394:
390:
386:
383:
371:
358:
354:
342:
290:
286:
280:
276:
255:
235:
231:
225:
221:
200:
180:
169:
167:
134:
130:
129:
126:
125:
122:
121:
114:
108:
107:
105:
88:the discussion
74:
62:
61:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2464:
2453:
2450:
2448:
2445:
2443:
2440:
2439:
2437:
2430:
2429:
2425:
2421:
2418:
2415:
2399:
2392:
2389:
2385:
2380:
2377:
2373:
2367:
2364:
2360:
2350:
2348:
2332:
2329:
2326:
2306:
2294:
2292:
2291:
2287:
2283:
2278:
2261:
2258:
2255:
2251:
2227:
2223:
2213:
2197:
2194:
2191:
2187:
2182:
2174:
2157:
2154:
2151:
2147:
2139:
2136:
2133:
2115:
2103:
2099:
2093:
2085:
2066:
2062:
2050:
2029:
2025:
2012:
1994:
1990:
1975:
1963:
1959:
1951:
1946:
1942:
1938:
1935:
1930:
1927:
1917:
1913:
1899:
1895:
1885:
1873:
1869:
1863:
1855:
1852:
1834:
1830:
1809:
1805:
1796:
1793:
1789:
1785:
1782:
1776:
1764:
1760:
1752:
1747:
1743:
1739:
1736:
1728:
1725:
1722:
1716:
1706:
1702:
1687:
1683:
1675:
1665:
1661:
1647:
1643:
1633:
1622:
1619:
1610:
1606:
1600:
1592:
1584:
1582:
1566:
1557:
1539:
1535:
1521:
1510:
1498:
1494:
1485:
1482:
1478:
1474:
1467:
1459:
1456:
1453:
1444:
1440:
1436:
1433:
1420:
1416:
1411:
1399:
1395:
1387:
1382:
1378:
1374:
1371:
1363:
1360:
1357:
1351:
1341:
1337:
1322:
1318:
1310:
1300:
1296:
1282:
1278:
1268:
1257:
1254:
1245:
1241:
1235:
1227:
1219:
1216:
1198:
1194:
1180:
1169:
1157:
1153:
1146:
1139:
1131:
1128:
1125:
1119:
1116:
1113:
1110:
1104:
1100:
1096:
1093:
1085:
1082:
1079:
1073:
1063:
1059:
1046:
1034:
1030:
1016:
1012:
1005:
1002:
997:
986:
982:
965:
961:
950:
946:
942:
934:
931:
913:
909:
895:
884:
880:
877:
874:
871:
868:
865:
862:
845:
841:
834:
831:
827:
811:
807:
790:
786:
775:
771:
767:
756:
752:
742:
734:
715:
711:
704:
695:
681:
659:
655:
629:
625:
599:
595:
585:
569:
565:
544:
524:
502:
498:
474:
471:
468:
461:
457:
450:(correct) or
434:
431:
428:
421:
417:
392:
388:
381:
356:
352:
341:
339:
335:
331:
327:
323:
317:
316:User:rewtnode
313:
312:
308:
304:
288:
284:
278:
274:
253:
233:
229:
223:
219:
198:
178:
168:
165:
164:
160:
156:
152:
147:
146:
142:
138:
119:
113:
110:
109:
106:
89:
85:
81:
80:
75:
72:
68:
67:
63:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2351:
2298:
2279:
2214:
2175:
2086:
2013:
1856:
1853:
1585:
1558:
1220:
1217:
935:
932:
735:
696:
586:
373:
320:— Preceding
314:
170:
166:
148:
133:
77:
40:WikiProjects
1218:Rearrange:
30:Start-class
2436:Categories
2420:Br1 Ursino
1854:Simplify:
155:Rhaertel80
93:Statistics
84:statistics
59:Statistics
517:and not
334:contribs
326:Rewtnode
322:unsigned
303:Ksimek
36:scale.
2282:CarlS
2424:talk
2286:talk
557:and
330:talk
307:talk
159:talk
141:talk
1936:exp
1737:exp
1583:):
1434:exp
1372:exp
1094:exp
475:0.5
410:is
112:???
2438::
2426:)
2393:λ
2390:α
2386:β
2378:μ
2368:α
2349:.
2327:λ
2288:)
2280:--
2277:.
2259:−
2256:α
2252:β
2224:σ
2195:−
2192:α
2188:β
2155:−
2152:α
2148:β
2137:−
2134:α
2128:Γ
2116:α
2110:Γ
2104:α
2100:β
2084::
2063:σ
2026:σ
1991:σ
1960:σ
1952:β
1947:−
1939:
1931:α
1928:−
1914:σ
1905:∞
1896:∫
1886:α
1880:Γ
1874:α
1870:β
1831:σ
1806:σ
1794:−
1790:λ
1786:π
1761:σ
1753:β
1748:−
1740:
1723:α
1717:−
1703:σ
1684:σ
1662:σ
1653:∞
1644:∫
1634:α
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1623:π
1611:α
1607:β
1601:λ
1536:σ
1495:σ
1483:−
1479:λ
1460:μ
1457:−
1445:−
1437:
1429:∞
1424:∞
1421:−
1417:∫
1396:σ
1388:β
1383:−
1375:
1358:α
1352:−
1338:σ
1319:σ
1297:σ
1288:∞
1279:∫
1269:α
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1242:β
1236:λ
1195:σ
1154:σ
1132:μ
1129:−
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1114:β
1105:−
1097:
1080:α
1074:−
1060:σ
1047:α
1041:Γ
1035:α
1031:β
1013:σ
1006:π
998:λ
983:σ
974:∞
969:∞
966:−
962:∫
956:∞
947:∫
910:σ
881:β
875:α
869:λ
863:μ
842:σ
808:σ
799:∞
794:∞
791:−
787:∫
781:∞
772:∫
753:σ
733::
712:σ
694:.
682:σ
656:σ
626:σ
596:σ
584:.
566:σ
525:σ
499:σ
472:−
469:α
458:β
432:−
429:α
418:β
389:σ
353:σ
336:)
332:•
309:)
289:λ
275:σ
220:σ
199:λ
179:μ
161:)
143:)
2422:(
2400:)
2381:,
2374:(
2365:2
2361:t
2333:1
2330:=
2307:x
2284:(
2262:1
2228:2
2198:1
2183:=
2158:1
2143:)
2140:1
2131:(
2119:)
2113:(
2094:=
2072:]
2067:2
2059:[
2035:]
2030:2
2022:[
2000:]
1995:2
1987:[
1983:d
1976:)
1969:]
1964:2
1956:[
1943:(
1924:]
1918:2
1910:[
1900:0
1889:)
1883:(
1864:=
1840:]
1835:2
1827:[
1823:d
1815:]
1810:2
1802:[
1797:1
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1777:)
1770:]
1765:2
1757:[
1744:(
1732:)
1729:1
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1720:(
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1707:2
1699:[
1693:]
1688:2
1680:[
1676:1
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1666:2
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1540:2
1532:[
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1499:2
1491:[
1486:1
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1451:(
1441:(
1412:)
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1400:2
1392:[
1379:(
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1364:1
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1355:(
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1342:2
1334:[
1328:]
1323:2
1315:[
1311:1
1306:]
1301:2
1293:[
1283:0
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1266:(
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1199:2
1191:[
1187:d
1181:x
1177:d
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1163:]
1158:2
1150:[
1147:2
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1126:x
1123:(
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1070:]
1064:2
1056:[
1050:)
1044:(
1022:]
1017:2
1009:[
1003:2
992:]
987:2
979:[
951:0
943:=
919:]
914:2
906:[
902:d
896:x
892:d
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878:,
872:,
866:,
857:|
851:]
846:2
838:[
835:,
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828:(
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757:2
749:[
746:[
743:E
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716:2
708:[
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660:2
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630:2
622:[
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279:2
254:v
234:v
230:/
224:2
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120:.
42::
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