40:
48:
3988:
1508:
negligible, the transition is very likely not observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a
678:
3998:
1495:
484:
434:. Boltzmann's statistical work is borne out in his paper âOn the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" The distribution was later investigated extensively, in its modern generic form, by
2001:
gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow
MaxwellâBoltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling
1935:
of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a
1087:
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states
1507:
of atoms or molecules undergoing transitions from one state to another. In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is
1350:
1342:
is the total number of particles in the system. We may use the
Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state
1189:
417:
1257:
The
Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over bound states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state
1903:(equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble. Some special cases (derivable from the canonical ensemble) show the Boltzmann distribution in different aspects:
673:{\displaystyle p_{i}={\frac {1}{Q}}\exp \left(-{\frac {\varepsilon _{i}}{kT}}\right)={\frac {\exp \left(-{\tfrac {\varepsilon _{i}}{kT}}\right)}{\displaystyle \sum _{j=1}^{M}\exp \left(-{\tfrac {\varepsilon _{j}}{kT}}\right)}}}
843:
982:
222:
1650:
1826:
310:. Therefore, the Boltzmann distribution can be used to solve a wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied.
1099:
1064:
1322:
94:
2214:(1868). "Studien ĂŒber das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten" [Studies on the balance of living force between moving material points].
1080:
The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the
2117:. The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries.
1490:{\displaystyle {\frac {N_{i}}{N}}={\frac {\exp \left(-{\frac {\varepsilon _{i}}{kT}}\right)}{\displaystyle \sum _{j=1}^{M}\exp \left(-{\tfrac {\varepsilon _{j}}{kT}}\right)}}}
2646:
327:
2100:
in deep learning, as the number of nodes are increased the difficulty of implementing in real time applications becomes critical, so a different type of architecture named
1838:
The
Boltzmann distribution is a special case of the generalized Boltzmann distribution. The generalized Boltzmann distribution is used in statistical mechanics to describe
761:
1020:
2233:
1066:
equals a particular mean energy value, except for two special cases. (These special cases occur when the mean value is either the minimum or maximum of the energies
859:
1949:
In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space.
289:
2282:
1081:
1961:
Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
2775:
3258:
2264:
3166:
3953:
156:
3819:
3031:
2790:
2639:
1984:
interact with each other, then the expected frequencies of subsystem states no longer follow a
Boltzmann distribution, and even may not have an
3714:
3478:
2574:
Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the
Boltzmann distribution. Physica A 391: 4883â4890
1262:
is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state
3152:
2392:
2367:
3473:
3417:
3077:
2715:
1862:
306:
here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a
3223:
1525:
3759:
3493:
3346:
3021:
2765:
4001:
3218:
442:
31:
3991:
3663:
3639:
2632:
1931:
When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the
1873:
1866:
147:
3860:
3488:
2415:"The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy"
3737:
3698:
3670:
3644:
3562:
2911:
2659:
2613:
2330:
2194:
1950:
1943:
1890:
1847:
446:
3848:
3814:
3680:
3675:
3520:
3328:
3026:
2780:
1666:
3598:
3511:
3483:
3392:
3341:
3315:
3213:
2996:
2961:
2121:
4022:
3612:
3529:
3366:
3113:
2991:
2966:
2830:
2825:
2820:
1251:
3290:
2800:
2795:
2240:
3928:
3794:
3502:
3351:
3283:
3268:
3161:
3135:
3067:
2906:
2737:
2722:
292:
3445:
2141:
2007:
3824:
3764:
3754:
3371:
3072:
2931:
2101:
2077:
1851:
756:
3173:
2916:
2845:
3809:
3804:
3749:
3685:
3450:
3228:
3125:
2710:
3629:
3437:
2583:
2146:
2003:
3943:
3719:
3538:
3320:
3273:
3142:
3118:
3098:
2941:
2815:
2695:
2288:
453:
as a function of that state's energy, while the
Maxwell-Boltzmann distributions give the probabilities of particle
150:
as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:
3948:
2891:
4027:
3732:
3693:
3567:
3404:
3248:
3193:
3091:
3055:
2926:
2069:
265:
1973:
is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a
1025:
3634:
3422:
3188:
3147:
3062:
3016:
2956:
2921:
2810:
2705:
2655:
1970:
1843:
474:
139:
2747:
1280:
56:
3933:
3875:
3546:
3333:
3243:
3198:
3183:
3103:
3001:
2951:
2946:
2727:
1974:
1270:
divided by the total number of particles in the system, that is the fraction of particles that occupy state
477:
that gives the probability of a certain state as a function of that state's energy and temperature of the
3799:
3787:
3776:
3658:
3554:
3361:
2805:
2785:
2690:
2085:
1957:
gas of non-interacting particles at equilibrium. This expected number distribution has the
Boltzmann form.
3923:
3880:
3724:
3399:
3253:
3233:
3130:
2700:
2093:
2065:
1992:
states of the entire system considered as a whole, provided the entire system is in thermal equilibrium.
1932:
1896:
1184:{\displaystyle {\frac {p_{i}}{p_{j}}}=\exp \left({\frac {\varepsilon _{j}-\varepsilon _{i}}{kT}}\right)}
427:
412:{\displaystyle {\frac {p_{i}}{p_{j}}}=\exp \left({\frac {\varepsilon _{j}-\varepsilon _{i}}{kT}}\right)}
123:
2482:
3973:
3968:
3963:
3958:
3895:
3865:
3744:
3387:
3278:
3178:
2881:
2840:
2835:
2732:
2504:
2436:
2278:
2151:
1509:
850:
736:
690:
435:
243:
3907:
3432:
3412:
3382:
3356:
3310:
3238:
3050:
2986:
1985:
1954:
1900:
1073:. In these cases, the entropy maximizing distribution is a limit of Boltzmann distributions where
431:
307:
143:
39:
3938:
3427:
3208:
3203:
3108:
3045:
3040:
2896:
2886:
2770:
2562:
2550:
2520:
2494:
2460:
2426:
2081:
1921:
of the various possible states of a closed system of fixed volume, in thermal equilibrium with a
1914:
1907:
1886:
1839:
726:
258:
845:
It results from the constraint that the probabilities of all accessible states must add up to 1.
47:
3836:
3263:
3006:
2936:
2901:
2850:
2609:
2601:
2538:
2452:
2388:
2363:
2326:
2258:
2190:
2114:
2097:
2089:
2073:
2036:
1997:
1937:
989:
2549:
behaviour that can be understood with a single-particle canonical ensemble (resulting in the
3011:
2685:
2512:
2444:
2211:
2156:
2054:
2050:
2032:
1516:
423:
2345:
30:
This article is about system energy states. For particle energy levels and velocities, see
2182:
2129:
2125:
274:
2508:
2440:
3084:
2558:
2542:
2042:
In more general mathematical settings, the
Boltzmann distribution is also known as the
853:, one can prove that the Boltzmann distribution is the distribution that maximizes the
2414:
1925:. The canonical ensemble has a state probability distribution with the Boltzmann form.
449:. The Boltzmann distribution gives the probability that a system will be in a certain
4016:
3707:
3455:
2742:
2524:
2464:
2061:
2043:
2028:
1918:
1504:
2624:
2546:
2189:. Course of Theoretical Physics. Vol. 5 (3 ed.). Oxford: Pergamon Press.
1965:
When a system is in thermodynamic equilibrium with respect to both energy exchange
1953:
give the expected number of particles found in a given single-particle state, in a
1500:
838:{\displaystyle Q=\sum _{j=1}^{M}\exp \left(-{\tfrac {\varepsilon _{j}}{kT}}\right)}
17:
977:{\displaystyle S(p_{1},p_{2},\cdots ,p_{M})=-\sum _{i=1}^{M}p_{i}\log _{2}p_{i}}
127:
2516:
1928:
Statistical frequencies of subsystems' states (in a non-interacting collection)
2178:
1940:
statistical frequency distribution of subsystem states has the
Boltzmann form.
217:{\displaystyle p_{i}\propto \exp \left(-{\frac {\varepsilon _{i}}{kT}}\right)}
1250:
The corresponding ratio of populations of energy levels must also take their
1922:
1519:
commonly used in machine learning is related to the Boltzmann distribution:
2456:
2590:
blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).
2011:
1872:
It is the only distribution that is mathematically consistent with the
854:
2448:
321:
and characteristically only depends on the states' energy difference:
1857:
The generalized Boltzmann distribution has the following properties:
1850:. The generalized Boltzmann distribution is usually derived from the
478:
2113:
The Boltzmann distribution can be introduced to allocate permits in
2499:
2431:
1645:{\displaystyle (p_{1},\ldots ,p_{M})=\operatorname {softmax} \left}
2015:
46:
38:
1899:
when considering closed systems of fixed composition that are in
745:
is the number of all states accessible to the system of interest,
2308:
Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York
1861:
It is the only distribution for which the entropy as defined by
1266:. This probability is equal to the number of particles in state
2628:
146:
that gives the probability that a system will be in a certain
1988:. The canonical ensemble can however still be applied to the
441:
The Boltzmann distribution should not be confused with the
2201:
Translated by J.B. Sykes and M.J. Kearsley. See section 28
1821:{\displaystyle \Pr \left(\omega \right)\propto \exp \left}
426:
who first formulated it in 1868 during his studies of the
2413:
Gao, Xiang; Gallicchio, Emilio; Roitberg, Adrian (2019).
1946:
of classical gases (systems of non-interacting particles)
1969:, the requirement of fixed composition is relaxed and a
1876:
where state functions are described by ensemble average.
43:
Boltzmann's distribution is an exponential distribution.
1456:
1028:
992:
807:
639:
573:
277:
61:
1669:
1528:
1419:
1353:
1283:
1102:
862:
764:
602:
487:
481:
to which the distribution is applied. It is given as
465:
gas however, does follow the Boltzmann distribution.
330:
159:
59:
27:
Probability distribution of energy states of a system
2316:
2314:
2304:
2302:
2300:
2298:
2132:
made the connection to random utility maximization.
2120:
The Boltzmann distribution has the same form as the
1077:
approaches zero from above or below, respectively.)
3916:
3874:
3775:
3611:
3589:
3580:
3464:
3299:
2975:
2872:
2863:
2756:
2676:
2667:
2557:spins can show much more complex behaviour such as
2383:Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006).
2128:model, this is very well known in economics since
1820:
1644:
1489:
1316:
1183:
1058:
1014:
976:
837:
672:
461:in ideal gases. The distribution of energies in a
411:
283:
216:
88:
2362:(9th ed.). Oxford: Oxford University Press.
755:) is the normalization denominator, which is the
1670:
234:is the probability of the system being in state
1347:as a function of the energy of that state is
317:of probabilities of two states is known as the
2284:Elementary Principles in Statistical Mechanics
2640:
2608:. Oxford: Basil Blackwell. pp. 295â299.
2002:statistics of quantum gases are described by
986:subject to the normalization constraint that
98:(vertical axis) as a function of temperature
8:
2064:, the Boltzmann distribution is used in the
253:is the energy of that state, and a constant
2325:. Sausalito, CA: University Science Books.
1059:{\textstyle \sum {p_{i}{\varepsilon }_{i}}}
3586:
2869:
2673:
2647:
2633:
2625:
2476:
2474:
2408:
2406:
2404:
422:The Boltzmann distribution is named after
257:of the distribution is the product of the
2498:
2430:
2010:, depending on whether the particles are
1801:
1782:
1776:
1761:
1741:
1736:
1726:
1719:
1713:
1702:
1668:
1621:
1615:
1587:
1581:
1555:
1536:
1527:
1462:
1455:
1435:
1424:
1397:
1391:
1374:
1360:
1354:
1352:
1303:
1297:
1288:
1282:
1160:
1147:
1140:
1119:
1109:
1103:
1101:
1049:
1044:
1037:
1032:
1027:
1000:
991:
968:
955:
945:
935:
924:
905:
886:
873:
861:
813:
806:
786:
775:
763:
645:
638:
618:
607:
579:
572:
555:
531:
525:
501:
492:
486:
388:
375:
368:
347:
337:
331:
329:
276:
193:
187:
164:
158:
77:
67:
60:
58:
2483:"The Mathematics of the Ensemble Theory"
2346:NIST Atomic Spectra Database Levels Form
2173:
2171:
1499:This equation is of great importance to
1317:{\displaystyle p_{i}={\frac {N_{i}}{N}}}
2167:
1865:matches with the entropy as defined in
89:{\displaystyle {\tfrac {p_{i}}{p_{j}}}}
2263:: CS1 maint: archived copy as title (
2256:
1980:If the subsystems within a collection
1895:The Boltzmann distribution appears in
2584:The Thorny Problem Of Fair Allocation
1936:canonical ensemble. As a result, the
296:
7:
3997:
2358:Atkins, P. W.; de Paula, J. (2009).
1334:is the number of particles in state
2385:Principles of Instrumental Analysis
1854:, but there are other derivations.
299:for the proportionality constant).
1874:fundamental thermodynamic relation
1833:generalized Boltzmann distribution
1656:Generalized Boltzmann distribution
25:
3996:
3987:
3986:
473:The Boltzmann distribution is a
2419:The Journal of Chemical Physics
2092:is considered to be one of the
1503:. In spectroscopy we observe a
102:for several energy differences
1561:
1529:
911:
866:
443:MaxwellâBoltzmann distribution
32:MaxwellâBoltzmann distribution
1:
2541:. Systems of non-interacting
2537:A classic example of this is
2183:Lifshitz, Evgeny Mikhailovich
2102:Restricted Boltzmann machine
2078:restricted Boltzmann machine
1951:MaxwellâBoltzmann statistics
1944:MaxwellâBoltzmann statistics
1891:MaxwellâBoltzmann statistics
1852:principle of maximum entropy
1848:isothermalâisobaric ensemble
1203:is the probability of state
757:canonical partition function
751:(denoted by some authors by
702:is the probability of state
447:Maxwell-Boltzmann statistics
2387:. Boston, MA: Brooks/Cole.
4044:
3820:Wrapped asymmetric Laplace
2791:Extended negative binomial
2517:10.1016/j.rinp.2022.105230
2070:stochastic neural networks
2026:
1884:
29:
3982:
3479:Generalized extreme value
3259:Relativistic BreitâWigner
2656:Probability distributions
2602:"Multinomial Logit Model"
2600:Amemiya, Takeshi (1985).
2481:Gao, Xiang (March 2022).
2096:models. In the design of
1660:Distribution of the form
1216:the probability of state
1084:Atomic Spectra Database.
1015:{\textstyle \sum p_{i}=1}
266:thermodynamic temperature
2142:BoseâEinstein statistics
2088:. In deep learning, the
2008:BoseâEinstein statistics
1971:grand canonical ensemble
1881:In statistical mechanics
1867:classical thermodynamics
1844:grand canonical ensemble
1022:and the constraint that
475:probability distribution
308:natural gas storage tank
140:probability distribution
3474:Generalized chi-squared
3418:Normal-inverse Gaussian
2289:Charles Scribner's Sons
1975:microcanonical ensemble
1242:is the energy of state
1229:is the energy of state
715:is the energy of state
297:§ The distribution
3786:Univariate (circular)
3347:Generalized hyperbolic
2776:ConwayâMaxwellâPoisson
2766:Beta negative binomial
2321:McQuarrie, A. (2000).
2179:Landau, Lev Davidovich
2147:FermiâDirac statistics
2086:deep Boltzmann machine
2004:FermiâDirac statistics
1822:
1718:
1646:
1491:
1440:
1318:
1185:
1060:
1016:
978:
940:
839:
791:
674:
623:
413:
285:
218:
132:Boltzmann distribution
119:
90:
44:
4023:Statistical mechanics
3831:Bivariate (spherical)
3329:Kaniadakis Îș-Gaussian
2606:Advanced Econometrics
2323:Statistical Mechanics
2279:Gibbs, Josiah Willard
2094:unsupervised learning
2066:sampling distribution
1967:and particle exchange
1933:statistical frequency
1897:statistical mechanics
1863:Gibbs entropy formula
1823:
1698:
1647:
1492:
1420:
1319:
1186:
1061:
1017:
979:
920:
840:
771:
675:
603:
428:statistical mechanics
414:
286:
284:{\textstyle \propto }
219:
124:statistical mechanics
91:
50:
42:
3896:Dirac delta function
3843:Bivariate (toroidal)
3800:Univariate von Mises
3671:Multivariate Laplace
3563:Shifted log-logistic
2912:Continuous Bernoulli
2152:Negative temperature
1667:
1526:
1510:forbidden transition
1351:
1281:
1100:
1026:
990:
860:
851:Lagrange multipliers
762:
737:absolute temperature
691:exponential function
485:
436:Josiah Willard Gibbs
328:
275:
244:exponential function
157:
57:
3944:Natural exponential
3849:Bivariate von Mises
3815:Wrapped exponential
3681:Multivariate stable
3676:Multivariate normal
2997:Benktander 2nd kind
2992:Benktander 1st kind
2781:Discrete phase-type
2509:2022ResPh..3405230G
2441:2019JChPh.151c4113G
2187:Statistical Physics
2082:energy-based models
1986:analytical solution
1901:thermal equilibrium
1754:
432:thermal equilibrium
144:probability measure
3599:Rectified Gaussian
3484:Generalized Pareto
3342:Generalized normal
3214:Matrix-exponential
2563:antiferromagnetism
2551:Brillouin function
2487:Results in Physics
2360:Physical Chemistry
2049:In statistics and
1915:canonical ensemble
1908:Canonical ensemble
1887:Canonical ensemble
1840:canonical ensemble
1818:
1732:
1642:
1487:
1484:
1477:
1314:
1181:
1056:
1012:
974:
835:
828:
727:Boltzmann constant
670:
667:
660:
594:
409:
281:
259:Boltzmann constant
214:
136:Gibbs distribution
120:
86:
84:
45:
18:Gibbs distribution
4010:
4009:
3607:
3606:
3576:
3575:
3467:whose type varies
3413:Normal (Gaussian)
3367:Hyperbolic secant
3316:Exponential power
3219:MaxwellâBoltzmann
2967:Wigner semicircle
2859:
2858:
2831:Parabolic fractal
2821:Negative binomial
2588:Technology Review
2539:magnetic ordering
2449:10.1063/1.5111333
2394:978-0-495-12570-9
2369:978-0-19-954337-3
2212:Boltzmann, Ludwig
2122:multinomial logit
2115:emissions trading
2098:Boltzmann machine
2090:Boltzmann machine
2074:Boltzmann machine
2053:, it is called a
2037:Boltzmann machine
1835:by some authors.
1811:
1771:
1635:
1601:
1485:
1476:
1411:
1369:
1312:
1175:
1125:
827:
668:
659:
593:
545:
509:
403:
353:
207:
83:
51:Boltzmann factor
16:(Redirected from
4035:
4028:Ludwig Boltzmann
4000:
3999:
3990:
3989:
3929:Compound Poisson
3904:
3892:
3861:von MisesâFisher
3857:
3845:
3833:
3795:Circular uniform
3791:
3711:
3655:
3626:
3587:
3489:MarchenkoâPastur
3352:Geometric stable
3269:Truncated normal
3162:Inverse Gaussian
3068:Hyperexponential
2907:Beta rectangular
2875:bounded interval
2870:
2738:Discrete uniform
2723:Poisson binomial
2674:
2649:
2642:
2635:
2626:
2620:
2619:
2597:
2591:
2581:
2575:
2572:
2566:
2535:
2529:
2528:
2502:
2478:
2469:
2468:
2434:
2410:
2399:
2398:
2380:
2374:
2373:
2355:
2349:
2343:
2337:
2336:
2318:
2309:
2306:
2293:
2292:
2275:
2269:
2268:
2262:
2254:
2252:
2251:
2245:
2239:. Archived from
2238:
2230:
2224:
2223:
2208:
2202:
2200:
2175:
2157:Softmax function
2055:log-linear model
2051:machine learning
2033:Log-linear model
1977:applies instead.
1827:
1825:
1824:
1819:
1817:
1813:
1812:
1810:
1806:
1805:
1795:
1794:
1793:
1777:
1772:
1770:
1766:
1765:
1755:
1753:
1752:
1740:
1731:
1730:
1720:
1717:
1712:
1683:
1651:
1649:
1648:
1643:
1641:
1637:
1636:
1634:
1626:
1625:
1616:
1602:
1600:
1592:
1591:
1582:
1560:
1559:
1541:
1540:
1517:softmax function
1496:
1494:
1493:
1488:
1486:
1483:
1479:
1478:
1475:
1467:
1466:
1457:
1439:
1434:
1418:
1417:
1413:
1412:
1410:
1402:
1401:
1392:
1375:
1370:
1365:
1364:
1355:
1346:
1341:
1337:
1333:
1323:
1321:
1320:
1315:
1313:
1308:
1307:
1298:
1293:
1292:
1273:
1269:
1265:
1261:
1245:
1241:
1232:
1228:
1219:
1215:
1206:
1202:
1190:
1188:
1187:
1182:
1180:
1176:
1174:
1166:
1165:
1164:
1152:
1151:
1141:
1126:
1124:
1123:
1114:
1113:
1104:
1095:
1091:
1076:
1072:
1065:
1063:
1062:
1057:
1055:
1054:
1053:
1048:
1042:
1041:
1021:
1019:
1018:
1013:
1005:
1004:
983:
981:
980:
975:
973:
972:
960:
959:
950:
949:
939:
934:
910:
909:
891:
890:
878:
877:
844:
842:
841:
836:
834:
830:
829:
826:
818:
817:
808:
790:
785:
754:
750:
744:
734:
724:
718:
714:
705:
701:
688:
679:
677:
676:
671:
669:
666:
662:
661:
658:
650:
649:
640:
622:
617:
601:
600:
596:
595:
592:
584:
583:
574:
556:
551:
547:
546:
544:
536:
535:
526:
510:
502:
497:
496:
469:The distribution
424:Ludwig Boltzmann
418:
416:
415:
410:
408:
404:
402:
394:
393:
392:
380:
379:
369:
354:
352:
351:
342:
341:
332:
319:Boltzmann factor
290:
288:
287:
282:
270:
263:
256:
252:
241:
237:
233:
223:
221:
220:
215:
213:
209:
208:
206:
198:
197:
188:
169:
168:
117:
101:
97:
95:
93:
92:
87:
85:
82:
81:
72:
71:
62:
21:
4043:
4042:
4038:
4037:
4036:
4034:
4033:
4032:
4013:
4012:
4011:
4006:
3978:
3954:Maximum entropy
3912:
3900:
3888:
3878:
3870:
3853:
3841:
3829:
3784:
3771:
3708:Matrix-valued:
3705:
3651:
3622:
3614:
3603:
3591:
3582:
3572:
3466:
3460:
3377:
3303:
3301:
3295:
3224:MaxwellâJĂŒttner
3073:Hypoexponential
2979:
2977:
2976:supported on a
2971:
2932:Noncentral beta
2892:BaldingâNichols
2874:
2873:supported on a
2865:
2855:
2758:
2752:
2748:ZipfâMandelbrot
2678:
2669:
2663:
2653:
2623:
2616:
2599:
2598:
2594:
2582:
2578:
2573:
2569:
2536:
2532:
2480:
2479:
2472:
2412:
2411:
2402:
2395:
2382:
2381:
2377:
2370:
2357:
2356:
2352:
2344:
2340:
2333:
2320:
2319:
2312:
2307:
2296:
2277:
2276:
2272:
2255:
2249:
2247:
2243:
2236:
2234:"Archived copy"
2232:
2231:
2227:
2216:Wiener Berichte
2210:
2209:
2205:
2197:
2177:
2176:
2169:
2165:
2138:
2130:Daniel McFadden
2126:discrete choice
2111:
2039:
2027:Main articles:
2025:
2018:, respectively.
1893:
1885:Main articles:
1883:
1797:
1796:
1783:
1778:
1757:
1756:
1742:
1722:
1721:
1697:
1693:
1673:
1665:
1664:
1658:
1627:
1617:
1593:
1583:
1577:
1573:
1551:
1532:
1524:
1523:
1468:
1458:
1451:
1447:
1403:
1393:
1387:
1383:
1376:
1356:
1349:
1348:
1344:
1339:
1335:
1332:
1328:
1299:
1284:
1279:
1278:
1271:
1267:
1263:
1259:
1243:
1240:
1236:
1230:
1227:
1223:
1217:
1214:
1210:
1204:
1201:
1197:
1167:
1156:
1143:
1142:
1136:
1115:
1105:
1098:
1097:
1093:
1089:
1074:
1071:
1067:
1043:
1033:
1024:
1023:
996:
988:
987:
964:
951:
941:
901:
882:
869:
858:
857:
819:
809:
802:
798:
760:
759:
752:
748:
742:
732:
722:
716:
713:
709:
703:
700:
696:
686:
651:
641:
634:
630:
585:
575:
568:
564:
557:
537:
527:
521:
517:
488:
483:
482:
471:
463:one-dimensional
395:
384:
371:
370:
364:
343:
333:
326:
325:
293:proportionality
273:
272:
268:
261:
254:
251:
247:
239:
235:
232:
228:
199:
189:
183:
179:
160:
155:
154:
115:
108:
103:
99:
73:
63:
55:
54:
52:
35:
28:
23:
22:
15:
12:
11:
5:
4041:
4039:
4031:
4030:
4025:
4015:
4014:
4008:
4007:
4005:
4004:
3994:
3983:
3980:
3979:
3977:
3976:
3971:
3966:
3961:
3956:
3951:
3949:Locationâscale
3946:
3941:
3936:
3931:
3926:
3920:
3918:
3914:
3913:
3911:
3910:
3905:
3898:
3893:
3885:
3883:
3872:
3871:
3869:
3868:
3863:
3858:
3851:
3846:
3839:
3834:
3827:
3822:
3817:
3812:
3810:Wrapped Cauchy
3807:
3805:Wrapped normal
3802:
3797:
3792:
3781:
3779:
3773:
3772:
3770:
3769:
3768:
3767:
3762:
3760:Normal-inverse
3757:
3752:
3742:
3741:
3740:
3730:
3722:
3717:
3712:
3703:
3702:
3701:
3691:
3683:
3678:
3673:
3668:
3667:
3666:
3656:
3649:
3648:
3647:
3642:
3632:
3627:
3619:
3617:
3609:
3608:
3605:
3604:
3602:
3601:
3595:
3593:
3584:
3578:
3577:
3574:
3573:
3571:
3570:
3565:
3560:
3552:
3544:
3536:
3527:
3518:
3509:
3500:
3491:
3486:
3481:
3476:
3470:
3468:
3462:
3461:
3459:
3458:
3453:
3451:Variance-gamma
3448:
3443:
3435:
3430:
3425:
3420:
3415:
3410:
3402:
3397:
3396:
3395:
3385:
3380:
3375:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3331:
3326:
3318:
3313:
3307:
3305:
3297:
3296:
3294:
3293:
3291:Wilks's lambda
3288:
3287:
3286:
3276:
3271:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3229:Mittag-Leffler
3226:
3221:
3216:
3211:
3206:
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3170:
3169:
3159:
3150:
3145:
3140:
3139:
3138:
3128:
3126:gamma/Gompertz
3123:
3122:
3121:
3116:
3106:
3101:
3096:
3095:
3094:
3082:
3081:
3080:
3075:
3070:
3060:
3059:
3058:
3048:
3043:
3038:
3037:
3036:
3035:
3034:
3024:
3014:
3009:
3004:
2999:
2994:
2989:
2983:
2981:
2978:semi-infinite
2973:
2972:
2970:
2969:
2964:
2959:
2954:
2949:
2944:
2939:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2884:
2878:
2876:
2867:
2861:
2860:
2857:
2856:
2854:
2853:
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2768:
2762:
2760:
2757:with infinite
2754:
2753:
2751:
2750:
2745:
2740:
2735:
2730:
2725:
2720:
2719:
2718:
2711:Hypergeometric
2708:
2703:
2698:
2693:
2688:
2682:
2680:
2671:
2665:
2664:
2654:
2652:
2651:
2644:
2637:
2629:
2622:
2621:
2614:
2592:
2576:
2567:
2559:ferromagnetism
2553:). Systems of
2530:
2470:
2400:
2393:
2375:
2368:
2350:
2338:
2331:
2310:
2294:
2270:
2225:
2203:
2195:
2166:
2164:
2161:
2160:
2159:
2154:
2149:
2144:
2137:
2134:
2110:
2107:
2106:
2105:
2104:is introduced.
2058:
2047:
2024:
2023:In mathematics
2021:
2020:
2019:
1993:
1978:
1959:
1958:
1947:
1941:
1929:
1926:
1911:
1910:(general case)
1882:
1879:
1878:
1877:
1870:
1829:
1828:
1816:
1809:
1804:
1800:
1792:
1789:
1786:
1781:
1775:
1769:
1764:
1760:
1751:
1748:
1745:
1739:
1735:
1729:
1725:
1716:
1711:
1708:
1705:
1701:
1696:
1692:
1689:
1686:
1682:
1679:
1676:
1672:
1657:
1654:
1653:
1652:
1640:
1633:
1630:
1624:
1620:
1614:
1611:
1608:
1605:
1599:
1596:
1590:
1586:
1580:
1576:
1572:
1569:
1566:
1563:
1558:
1554:
1550:
1547:
1544:
1539:
1535:
1531:
1482:
1474:
1471:
1465:
1461:
1454:
1450:
1446:
1443:
1438:
1433:
1430:
1427:
1423:
1416:
1409:
1406:
1400:
1396:
1390:
1386:
1382:
1379:
1373:
1368:
1363:
1359:
1330:
1325:
1324:
1311:
1306:
1302:
1296:
1291:
1287:
1254:into account.
1248:
1247:
1238:
1234:
1225:
1221:
1212:
1208:
1199:
1179:
1173:
1170:
1163:
1159:
1155:
1150:
1146:
1139:
1135:
1132:
1129:
1122:
1118:
1112:
1108:
1069:
1052:
1047:
1040:
1036:
1031:
1011:
1008:
1003:
999:
995:
971:
967:
963:
958:
954:
948:
944:
938:
933:
930:
927:
923:
919:
916:
913:
908:
904:
900:
897:
894:
889:
885:
881:
876:
872:
868:
865:
847:
846:
833:
825:
822:
816:
812:
805:
801:
797:
794:
789:
784:
781:
778:
774:
770:
767:
746:
740:
739:of the system,
730:
720:
711:
707:
698:
694:
665:
657:
654:
648:
644:
637:
633:
629:
626:
621:
616:
613:
610:
606:
599:
591:
588:
582:
578:
571:
567:
563:
560:
554:
550:
543:
540:
534:
530:
524:
520:
516:
513:
508:
505:
500:
495:
491:
470:
467:
464:
420:
419:
407:
401:
398:
391:
387:
383:
378:
374:
367:
363:
360:
357:
350:
346:
340:
336:
280:
249:
230:
225:
224:
212:
205:
202:
196:
192:
186:
182:
178:
175:
172:
167:
163:
113:
106:
80:
76:
70:
66:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4040:
4029:
4026:
4024:
4021:
4020:
4018:
4003:
3995:
3993:
3985:
3984:
3981:
3975:
3972:
3970:
3967:
3965:
3962:
3960:
3957:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3921:
3919:
3915:
3909:
3906:
3903:
3899:
3897:
3894:
3891:
3887:
3886:
3884:
3882:
3877:
3873:
3867:
3864:
3862:
3859:
3856:
3852:
3850:
3847:
3844:
3840:
3838:
3835:
3832:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3790:
3789:
3783:
3782:
3780:
3778:
3774:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3748:
3747:
3746:
3743:
3739:
3736:
3735:
3734:
3731:
3729:
3728:
3723:
3721:
3720:Matrix normal
3718:
3716:
3713:
3710:
3709:
3704:
3700:
3697:
3696:
3695:
3692:
3690:
3689:
3686:Multivariate
3684:
3682:
3679:
3677:
3674:
3672:
3669:
3665:
3662:
3661:
3660:
3657:
3654:
3650:
3646:
3643:
3641:
3638:
3637:
3636:
3633:
3631:
3628:
3625:
3621:
3620:
3618:
3616:
3613:Multivariate
3610:
3600:
3597:
3596:
3594:
3588:
3585:
3579:
3569:
3566:
3564:
3561:
3559:
3557:
3553:
3551:
3549:
3545:
3543:
3541:
3537:
3535:
3533:
3528:
3526:
3524:
3519:
3517:
3515:
3510:
3508:
3506:
3501:
3499:
3497:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3471:
3469:
3465:with support
3463:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3441:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3408:
3403:
3401:
3398:
3394:
3391:
3390:
3389:
3386:
3384:
3381:
3379:
3378:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3337:
3332:
3330:
3327:
3325:
3324:
3319:
3317:
3314:
3312:
3309:
3308:
3306:
3302:on the whole
3298:
3292:
3289:
3285:
3282:
3281:
3280:
3277:
3275:
3274:type-2 Gumbel
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3168:
3165:
3164:
3163:
3160:
3158:
3156:
3151:
3149:
3146:
3144:
3143:Half-logistic
3141:
3137:
3134:
3133:
3132:
3129:
3127:
3124:
3120:
3117:
3115:
3112:
3111:
3110:
3107:
3105:
3102:
3100:
3099:Folded normal
3097:
3093:
3090:
3089:
3088:
3087:
3083:
3079:
3076:
3074:
3071:
3069:
3066:
3065:
3064:
3061:
3057:
3054:
3053:
3052:
3049:
3047:
3044:
3042:
3039:
3033:
3030:
3029:
3028:
3025:
3023:
3020:
3019:
3018:
3015:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2984:
2982:
2974:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2942:Raised cosine
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2883:
2880:
2879:
2877:
2871:
2868:
2862:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2816:Mixed Poisson
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2763:
2761:
2755:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2717:
2714:
2713:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2696:Beta-binomial
2694:
2692:
2689:
2687:
2684:
2683:
2681:
2675:
2672:
2666:
2661:
2657:
2650:
2645:
2643:
2638:
2636:
2631:
2630:
2627:
2617:
2615:0-631-13345-3
2611:
2607:
2603:
2596:
2593:
2589:
2585:
2580:
2577:
2571:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2534:
2531:
2526:
2522:
2518:
2514:
2510:
2506:
2501:
2496:
2492:
2488:
2484:
2477:
2475:
2471:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2433:
2428:
2425:(3): 034113.
2424:
2420:
2416:
2409:
2407:
2405:
2401:
2396:
2390:
2386:
2379:
2376:
2371:
2365:
2361:
2354:
2351:
2347:
2342:
2339:
2334:
2332:1-891389-15-7
2328:
2324:
2317:
2315:
2311:
2305:
2303:
2301:
2299:
2295:
2290:
2286:
2285:
2280:
2274:
2271:
2266:
2260:
2246:on 2021-03-05
2242:
2235:
2229:
2226:
2221:
2217:
2213:
2207:
2204:
2198:
2196:0-7506-3372-7
2192:
2188:
2184:
2180:
2174:
2172:
2168:
2162:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2139:
2135:
2133:
2131:
2127:
2123:
2118:
2116:
2108:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2062:deep learning
2059:
2056:
2052:
2048:
2045:
2044:Gibbs measure
2041:
2040:
2038:
2034:
2030:
2029:Gibbs measure
2022:
2017:
2013:
2009:
2005:
2000:
1999:
1994:
1991:
1987:
1983:
1979:
1976:
1972:
1968:
1964:
1963:
1962:
1956:
1952:
1948:
1945:
1942:
1939:
1934:
1930:
1927:
1924:
1920:
1919:probabilities
1916:
1912:
1909:
1906:
1905:
1904:
1902:
1898:
1892:
1888:
1880:
1875:
1871:
1868:
1864:
1860:
1859:
1858:
1855:
1853:
1849:
1845:
1841:
1836:
1834:
1814:
1807:
1802:
1798:
1790:
1787:
1784:
1779:
1773:
1767:
1762:
1758:
1749:
1746:
1743:
1737:
1733:
1727:
1723:
1714:
1709:
1706:
1703:
1699:
1694:
1690:
1687:
1684:
1680:
1677:
1674:
1663:
1662:
1661:
1655:
1638:
1631:
1628:
1622:
1618:
1612:
1609:
1606:
1603:
1597:
1594:
1588:
1584:
1578:
1574:
1570:
1567:
1564:
1556:
1552:
1548:
1545:
1542:
1537:
1533:
1522:
1521:
1520:
1518:
1513:
1511:
1506:
1505:spectral line
1502:
1497:
1480:
1472:
1469:
1463:
1459:
1452:
1448:
1444:
1441:
1436:
1431:
1428:
1425:
1421:
1414:
1407:
1404:
1398:
1394:
1388:
1384:
1380:
1377:
1371:
1366:
1361:
1357:
1309:
1304:
1300:
1294:
1289:
1285:
1277:
1276:
1275:
1255:
1253:
1235:
1222:
1209:
1196:
1195:
1194:
1191:
1177:
1171:
1168:
1161:
1157:
1153:
1148:
1144:
1137:
1133:
1130:
1127:
1120:
1116:
1110:
1106:
1085:
1083:
1078:
1050:
1045:
1038:
1034:
1029:
1009:
1006:
1001:
997:
993:
984:
969:
965:
961:
956:
952:
946:
942:
936:
931:
928:
925:
921:
917:
914:
906:
902:
898:
895:
892:
887:
883:
879:
874:
870:
863:
856:
852:
831:
823:
820:
814:
810:
803:
799:
795:
792:
787:
782:
779:
776:
772:
768:
765:
758:
747:
741:
738:
731:
728:
721:
708:
695:
692:
685:
684:
683:
680:
663:
655:
652:
646:
642:
635:
631:
627:
624:
619:
614:
611:
608:
604:
597:
589:
586:
580:
576:
569:
565:
561:
558:
552:
548:
541:
538:
532:
528:
522:
518:
514:
511:
506:
503:
498:
493:
489:
480:
476:
468:
466:
462:
460:
456:
452:
448:
444:
439:
437:
433:
429:
425:
405:
399:
396:
389:
385:
381:
376:
372:
365:
361:
358:
355:
348:
344:
338:
334:
324:
323:
322:
320:
316:
311:
309:
305:
300:
298:
294:
278:
271:. The symbol
267:
260:
245:
210:
203:
200:
194:
190:
184:
180:
176:
173:
170:
165:
161:
153:
152:
151:
149:
145:
141:
137:
134:(also called
133:
129:
125:
116:
109:
78:
74:
68:
64:
49:
41:
37:
33:
19:
3901:
3889:
3855:Multivariate
3854:
3842:
3830:
3825:Wrapped LĂ©vy
3785:
3733:Matrix gamma
3726:
3706:
3694:Normal-gamma
3687:
3653:Continuous:
3652:
3623:
3568:Tukey lambda
3555:
3547:
3542:-exponential
3539:
3531:
3522:
3513:
3504:
3498:-exponential
3495:
3439:
3406:
3373:
3335:
3322:
3249:Poly-Weibull
3194:Log-logistic
3154:
3153:Hotelling's
3085:
2927:Logit-normal
2801:GaussâKuzmin
2796:FloryâSchulz
2677:with finite
2605:
2595:
2587:
2579:
2570:
2554:
2547:paramagnetic
2533:
2490:
2486:
2422:
2418:
2384:
2378:
2359:
2353:
2341:
2322:
2287:. New York:
2283:
2273:
2248:. Retrieved
2241:the original
2228:
2219:
2215:
2206:
2186:
2124:model. As a
2119:
2112:
2109:In economics
2072:such as the
1996:
1989:
1981:
1966:
1960:
1894:
1856:
1837:
1832:
1830:
1659:
1514:
1501:spectroscopy
1498:
1326:
1256:
1252:degeneracies
1249:
1192:
1096:is given as
1086:
1079:
985:
848:
681:
472:
458:
454:
450:
440:
430:of gases in
421:
318:
314:
312:
303:
301:
226:
135:
131:
121:
111:
104:
36:
3939:Exponential
3788:directional
3777:Directional
3664:Generalized
3635:Multinomial
3590:continuous-
3530:Kaniadakis
3521:Kaniadakis
3512:Kaniadakis
3503:Kaniadakis
3494:Kaniadakis
3446:TracyâWidom
3423:Skew normal
3405:Noncentral
3189:Log-Laplace
3167:Generalized
3148:Half-normal
3114:Generalized
3078:Logarithmic
3063:Exponential
3017:Chi-squared
2957:U-quadratic
2922:Kumaraswamy
2864:Continuous
2811:Logarithmic
2706:Categorical
2555:interacting
2348:at nist.gov
128:mathematics
4017:Categories
3934:Elliptical
3890:Degenerate
3876:Degenerate
3624:Discrete:
3583:univariate
3438:Student's
3393:Asymmetric
3372:Johnson's
3300:supported
3244:Phase-type
3199:Log-normal
3184:Log-Cauchy
3174:Kolmogorov
3092:Noncentral
3022:Noncentral
3002:Beta prime
2952:Triangular
2947:Reciprocal
2917:IrwinâHall
2866:univariate
2846:YuleâSimon
2728:Rademacher
2670:univariate
2500:2006.00485
2493:: 105230.
2432:1903.02121
2250:2017-05-11
2222:: 517â560.
2163:References
1990:collective
1917:gives the
1831:is called
3659:Dirichlet
3640:Dirichlet
3550:-Gaussian
3525:-Logistic
3362:Holtsmark
3334:Gaussian
3321:Fisher's
3304:real line
2806:Geometric
2786:Delaporte
2691:Bernoulli
2668:Discrete
2525:221978379
2465:118981017
2185:(1980) .
1955:classical
1923:heat bath
1788:ω
1774:−
1747:ω
1738:η
1728:η
1704:η
1700:∑
1691:
1685:∝
1678:ω
1619:ε
1613:−
1607:…
1585:ε
1579:−
1571:
1546:…
1460:ε
1453:−
1445:
1422:∑
1395:ε
1389:−
1381:
1158:ε
1154:−
1145:ε
1134:
1046:ε
1030:∑
994:∑
962:
922:∑
918:−
896:⋯
811:ε
804:−
796:
773:∑
643:ε
636:−
628:
605:∑
577:ε
570:−
562:
529:ε
523:−
515:
438:in 1902.
386:ε
382:−
373:ε
362:
302:The term
279:∝
191:ε
185:−
177:
171:∝
3992:Category
3924:Circular
3917:Families
3902:Singular
3881:singular
3645:Negative
3592:discrete
3558:-Weibull
3516:-Weibull
3400:Logistic
3284:Discrete
3254:Rayleigh
3234:Nakagami
3157:-squared
3131:Gompertz
2980:interval
2716:Negative
2701:Binomial
2457:31325924
2281:(1902).
2259:cite web
2136:See also
2012:fermions
1938:expected
459:energies
291:denotes
4002:Commons
3974:Wrapped
3969:Tweedie
3964:Pearson
3959:Mixture
3866:Bingham
3765:Complex
3755:Inverse
3745:Wishart
3738:Inverse
3725:Matrix
3699:Inverse
3615:(joint)
3534:-Erlang
3388:Laplace
3279:Weibull
3136:Shifted
3119:Inverse
3104:Fréchet
3027:Inverse
2962:Uniform
2882:Arcsine
2841:Skellam
2836:Poisson
2759:support
2733:Soliton
2686:Benford
2679:support
2505:Bibcode
2437:Bibcode
1998:quantum
1568:softmax
1193:where:
855:entropy
735:is the
725:is the
689:is the
682:where:
242:is the
138:) is a
96:
53:
3908:Cantor
3750:Normal
3581:Mixed
3507:-Gamma
3433:Stable
3383:Landau
3357:Gumbel
3311:Cauchy
3239:Pareto
3051:Erlang
3032:Scaled
2987:Benini
2826:Panjer
2612:
2523:
2463:
2455:
2391:
2366:
2329:
2193:
2181:&
2035:, and
2016:bosons
1327:where
849:Using
479:system
455:speeds
304:system
227:where
3630:Ewens
3456:Voigt
3428:Slash
3209:Lomax
3204:Log-t
3109:Gamma
3056:Hyper
3046:Davis
3041:Dagum
2897:Bates
2887:ARGUS
2771:Borel
2545:show
2543:spins
2521:S2CID
2495:arXiv
2461:S2CID
2427:arXiv
2244:(PDF)
2237:(PDF)
1995:With
687:exp()
451:state
315:ratio
295:(see
148:state
3879:and
3837:Kent
3264:Rice
3179:LĂ©vy
3007:Burr
2937:PERT
2902:Beta
2851:Zeta
2743:Zipf
2660:list
2610:ISBN
2453:PMID
2389:ISBN
2364:ISBN
2327:ISBN
2265:link
2191:ISBN
2084:and
1913:The
1889:and
1846:and
1515:The
1338:and
1092:and
1082:NIST
313:The
264:and
130:, a
126:and
3715:LKJ
3012:Chi
2561:or
2513:doi
2445:doi
2423:151
2068:of
2060:In
2014:or
2006:or
1688:exp
1442:exp
1378:exp
1131:exp
953:log
793:exp
625:exp
559:exp
512:exp
457:or
445:or
359:exp
240:exp
174:exp
142:or
122:In
4019::
2604:.
2586:.
2519:.
2511:.
2503:.
2491:34
2489:.
2485:.
2473:^
2459:.
2451:.
2443:.
2435:.
2421:.
2417:.
2403:^
2313:^
2297:^
2261:}}
2257:{{
2220:58
2218:.
2170:^
2080:,
2076:,
2031:,
1982:do
1842:,
1671:Pr
1512:.
1274:.
255:kT
246:,
238:,
110:â
3727:t
3688:t
3556:q
3548:q
3540:q
3532:Îș
3523:Îș
3514:Îș
3505:Îș
3496:Îș
3440:t
3407:t
3376:U
3374:S
3336:q
3323:z
3155:T
3086:F
2662:)
2658:(
2648:e
2641:t
2634:v
2618:.
2565:.
2527:.
2515::
2507::
2497::
2467:.
2447::
2439::
2429::
2397:.
2372:.
2335:.
2291:.
2267:)
2253:.
2199:.
2057:.
2046:.
1869:.
1815:]
1808:T
1803:B
1799:k
1791:)
1785:(
1780:E
1768:T
1763:B
1759:k
1750:)
1744:(
1734:x
1724:X
1715:n
1710:1
1707:=
1695:[
1681:)
1675:(
1639:]
1632:T
1629:k
1623:M
1610:,
1604:,
1598:T
1595:k
1589:1
1575:[
1565:=
1562:)
1557:M
1553:p
1549:,
1543:,
1538:1
1534:p
1530:(
1481:)
1473:T
1470:k
1464:j
1449:(
1437:M
1432:1
1429:=
1426:j
1415:)
1408:T
1405:k
1399:i
1385:(
1372:=
1367:N
1362:i
1358:N
1345:i
1340:N
1336:i
1331:i
1329:N
1310:N
1305:i
1301:N
1295:=
1290:i
1286:p
1272:i
1268:i
1264:i
1260:i
1246:.
1244:j
1239:j
1237:Δ
1233:,
1231:i
1226:i
1224:Δ
1220:,
1218:j
1213:j
1211:p
1207:,
1205:i
1200:i
1198:p
1178:)
1172:T
1169:k
1162:i
1149:j
1138:(
1128:=
1121:j
1117:p
1111:i
1107:p
1094:j
1090:i
1075:T
1070:i
1068:Δ
1051:i
1039:i
1035:p
1010:1
1007:=
1002:i
998:p
970:i
966:p
957:2
947:i
943:p
937:M
932:1
929:=
926:i
915:=
912:)
907:M
903:p
899:,
893:,
888:2
884:p
880:,
875:1
871:p
867:(
864:S
832:)
824:T
821:k
815:j
800:(
788:M
783:1
780:=
777:j
769:=
766:Q
753:Z
749:Q
743:M
733:T
729:,
723:k
719:,
717:i
712:i
710:Δ
706:,
704:i
699:i
697:p
693:,
664:)
656:T
653:k
647:j
632:(
620:M
615:1
612:=
609:j
598:)
590:T
587:k
581:i
566:(
553:=
549:)
542:T
539:k
533:i
519:(
507:Q
504:1
499:=
494:i
490:p
406:)
400:T
397:k
390:i
377:j
366:(
356:=
349:j
345:p
339:i
335:p
269:T
262:k
250:i
248:Δ
236:i
231:i
229:p
211:)
204:T
201:k
195:i
181:(
166:i
162:p
118:.
114:j
112:Δ
107:i
105:Δ
100:T
79:j
75:p
69:i
65:p
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
