1823:
36:
436:
More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the
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1600:
140:, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two
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plane. The intersection of that plane with the joint normal density, once rescaled to give unit area under the intersection, is the relevant conditional density of
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2002:
1973:
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2183:{\displaystyle Y\mid X=70\ \sim \ {\mathcal {N}}\left(\mu _{Y}+{\frac {\sigma _{Y}}{\sigma _{X}}}\rho (70-\mu _{X}),\,(1-\rho ^{2})\sigma _{Y}^{2}\right).}
1651:
783:
4402:
100:
2854:
Additionally, a marginal of a joint distribution can be expressed as the expectation of the corresponding conditional distribution. For instance,
1297:
1827:
72:
274:
is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value
3326:
79:
5600:
4382:{\displaystyle \mathbf {1} _{A}(\omega )={\begin{cases}1\;&{\text{if }}\omega \in A,\\0\;&{\text{if }}\omega \notin A,\end{cases}}}
86:
3139:
4156:
3597:
17:
68:
4671:
An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.
4522:
5555:
5504:
4488:
3003:
4682:
3450:
4726:
4942:
5479:
3317:, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called
1162:
is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that
119:
4808:
5605:
4661:{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=\operatorname {E} (\mathbf {1} _{A}\mid {\mathcal {G}})\;}
1810:
The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem:
93:
5437:
4072:
3318:
2961:
5545:
141:
57:
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4392:
which is a random variable. Note that the expectation of this random variable is equal to the probability of
3865:{\displaystyle \mu _{X\,|\,{\mathcal {G}}}(B\,|\,{\mathcal {G}})=\mathrm {P} (X^{-1}(B)\,|\,{\mathcal {G}}).}
1834:
1814:
shows that conditional probability density functions need not be invariant under coordinate transformations.
359:
362:
is typically used to represent the conditional probability. The conditional distribution contrasts with the
2519:
1227:
414:
3980:{\displaystyle \mu _{X\,|{\mathcal {G}}}(\cdot \,|{\mathcal {G}})(\omega ):{\mathcal {E}}\to \mathbb {R} }
4998:
4249:
3728:
3567:
3290:
3081:
651:{\displaystyle p_{Y|X}(y\mid x)\triangleq P(Y=y\mid X=x)={\frac {P(\{X=x\}\cap \{Y=y\})}{P(X=x)}}\qquad }
5323:
5273:
4103:
4038:
3875:
3692:
3417:
5427:
2324:
1223:
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of a random variable, which is its distribution without reference to the value of the other variable.
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5201:
5137:
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4871:
4784:
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3555:{\displaystyle \operatorname {P} \!\left(A\mid \{\emptyset ,\Omega \}\right)=\operatorname {P} (A).}
3115:
3057:
2857:
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2238:
1563:
450:
231:
4134:
1411:
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46:
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be the sigma-field of all countable sets and sets whose complement is countable. So each set in
2721:
2513:
5075:
1503:
4468:
4083:
3109:
3037:
1214:
is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).
700:
363:
665:
430:
422:
355:
1926:
1900:
8:
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1981:
1952:
1191:
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985:
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476:
5497:
Fundamentals of
Probability and Stochastic Processes with Applications to Communications
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2007:
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392:
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169:
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133:
2787:(or integral if it is a conditional probability density) is 1. Seen as a function of
5551:
5500:
5475:
5442:
5394:(it is independent of it), and in another sense it contains all the information in
1975:
1800:{\displaystyle f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X,Y}(x,y)=f_{X|Y}(x\mid y)f_{Y}(y).}
1537:
426:
5580:
5467:
937:{\displaystyle P(Y=y\mid X=x)P(X=x)=P(\{X=x\}\cap \{Y=y\})=P(X=x\mid Y=y)P(Y=y).}
144:
697:
in the denominator, this is defined only for non-zero (hence strictly positive)
3280:{\displaystyle \int _{G}P(A\mid {\mathcal {G}})(\omega )dP(\omega )=P(A\cap G)}
5594:
4455:{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\operatorname {P} (A).\;}
1978:, and then visualize the plane containing that line and perpendicular to the
1397:{\displaystyle f_{Y\mid X}(y\mid x)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}\qquad }
4781:
can be loosely interpreted as containing a subset of the information in
1822:
4153:), every conditional probability distribution is regular. In this case,
137:
3367:{\displaystyle \operatorname {P} (\cdot \mid {\mathcal {G}})(\omega )}
35:
4674:
5025:. It is incorrect to conclude in general that the information in
3182:{\displaystyle P(A\mid {\mathcal {G}}):\Omega \to \mathbb {R} }
4231:{\displaystyle E=\int _{-\infty }^{\infty }x\,\mu (dx,\cdot )}
3642:{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})=1_{A}}
5474:(Fifth ed.). San Diego: Academic Press. pp. 88–91.
4080:
For a real-valued random variable (with respect to the Borel
421:. The properties of a conditional distribution, such as the
4375:
953:
425:, are often referred to by corresponding names such as the
5049:
does not tell us anything about the probability of event
4554:{\displaystyle \operatorname {P} (A\mid {\mathcal {G}})}
3650:
3489:, the conditional probability is the constant function
2767:
is a probability mass function and so the sum over all
1217:
4512:{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}
4241:
3027:{\displaystyle {\mathcal {G}}\subseteq {\mathcal {F}}}
5400:
5376:
5352:
5326:
5302:
5276:
5252:
5228:
5204:
5184:
5164:
5140:
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occurring. This can be shown with a counter-example:
5055:
5031:
5001:
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4901:
4874:
4854:
4811:
4787:
4763:
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4716:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
4685:
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4525:
4491:
4471:
4405:
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4252:
4159:
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4017:
3997:
3904:
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3761:
3731:
3695:
3663:
3600:
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3495:
3482:{\displaystyle {\mathcal {G}}=\{\emptyset ,\Omega \}}
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172:
152:
4750:{\displaystyle {\mathcal {A}}\subset {\mathcal {F}}}
5072:Consider a probability space on the unit interval,
60:. Unsourced material may be challenged and removed.
5585:(3rd ed.). New York, NY: John Wiley and Sons.
5410:
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2182:
2016:
1996:
1967:
1941:
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1605:The relation with the probability distribution of
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737:The relation with the probability distribution of
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286:
266:
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222:
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4988:{\displaystyle \mathbb {P} (B|A)=\mathbb {P} (B)}
3499:
5592:
5320:occurred is equivalent to knowing exactly which
2953:
4675:Interpretation of conditioning on a Sigma Field
2241:if and only if the conditional distribution of
499:can be written according to its definition as:
453:, the conditional probability mass function of
4841:{\displaystyle \mathbb {P} (B|{\mathcal {A}})}
1034:if the number is prime (i.e., 2, 3, or 5) and
2301:, equal to the unconditional distribution of
982:if the number is even (i.e., 2, 4, or 6) and
5494:
3523:
3511:
3476:
3464:
874:
862:
856:
844:
618:
606:
600:
588:
5543:
2321:. For discrete random variables this means
2192:
5246:also contains all the singleton events in
4657:
4451:
4352:
4325:
2993:{\displaystyle (\Omega ,{\mathcal {F}},P)}
4972:
4947:
4813:
4706:
4206:
4139:
3973:
3934:
3913:
3848:
3842:
3798:
3792:
3776:
3770:
3175:
2831:, so that the sum (or integral) over all
2134:
1560:. Also in this case it is necessary that
120:Learn how and when to remove this message
27:Probability theory and statistics concept
5488:
5270:(those sets which contain only a single
3407:{\displaystyle (\Omega ,{\mathcal {F}})}
1821:
1136:Then the unconditional probability that
5198:and so is independent of each event in
14:
5593:
69:"Conditional probability distribution"
5296:). So knowing which of the events in
2581:{\displaystyle f_{Y}(y|X=x)=f_{Y}(y)}
2281:is, for all possible realizations of
5466:
4915:is independent of a sub-sigma field
4035:. If it is a probability measure on
3989:conditional probability distribution
1218:Conditional continuous distributions
188:conditional probability distribution
58:adding citations to reliable sources
29:
5601:Theory of probability distributions
5544:Billingsley, Patrick (2012-02-28).
5018:{\displaystyle A\in {\mathcal {A}}}
4269:{\displaystyle A\in {\mathcal {F}}}
4242:Relation to conditional expectation
3748:{\displaystyle B\in {\mathcal {E}}}
3587:{\displaystyle A\in {\mathcal {G}}}
3310:{\displaystyle G\in {\mathcal {G}}}
3101:{\displaystyle A\in {\mathcal {F}}}
1923:, one can first visualize the line
369:If the conditional distribution of
24:
5575:
5531:
5519:
5472:Introduction to Probability Models
5403:
5379:
5355:
5339:{\displaystyle \omega \in \Omega }
5333:
5305:
5289:{\displaystyle \omega \in \Omega }
5283:
5255:
5231:
5207:
5143:
5119:
5079:
5034:
5010:
4924:
4877:
4830:
4790:
4766:
4742:
4732:
4697:
4689:
4649:
4623:
4612:
4595:
4543:
4526:
4504:
4494:
4433:
4406:
4261:
4198:
4193:
4174:
4124:{\displaystyle {\mathcal {R}}^{1}}
4110:
4064:{\displaystyle (E,{\mathcal {E}})}
4053:
4020:
3964:
3942:
3921:
3891:{\displaystyle \omega \in \Omega }
3885:
3851:
3813:
3801:
3779:
3740:
3725:-valued random variable. For each
3718:{\displaystyle (E,{\mathcal {E}})}
3707:
3670:
3649:, the indicator function (defined
3618:
3601:
3579:
3534:
3520:
3514:
3496:
3473:
3467:
3456:
3433:{\displaystyle \omega \in \Omega }
3427:
3396:
3388:
3347:
3330:
3302:
3224:
3168:
3157:
3121:
3093:
3063:
3019:
3009:
2976:
2968:
2472:. For continuous random variables
2059:
1250:given the occurrence of the value
445:Conditional discrete distributions
25:
5617:
4805:. For example, we might think of
2384:{\displaystyle P(Y=y|X=x)=P(Y=y)}
4848:as the probability of the event
4634:
4417:
4292:
34:
5438:Regular conditional probability
4679:Consider the probability space
1393:
647:
45:needs additional citations for
5550:. Hoboken, New Jersey: Wiley.
5537:
5525:
5513:
5460:
5411:{\displaystyle {\mathcal {F}}}
5387:{\displaystyle {\mathcal {F}}}
5370:contains no information about
5363:{\displaystyle {\mathcal {G}}}
5313:{\displaystyle {\mathcal {G}}}
5263:{\displaystyle {\mathcal {F}}}
5239:{\displaystyle {\mathcal {G}}}
5215:{\displaystyle {\mathcal {F}}}
5151:{\displaystyle {\mathcal {G}}}
5127:{\displaystyle {\mathcal {G}}}
5097:
5085:
5042:{\displaystyle {\mathcal {A}}}
4982:
4976:
4965:
4958:
4951:
4932:{\displaystyle {\mathcal {A}}}
4885:{\displaystyle {\mathcal {A}}}
4835:
4824:
4817:
4798:{\displaystyle {\mathcal {F}}}
4774:{\displaystyle {\mathcal {A}}}
4710:
4686:
4654:
4629:
4617:
4601:
4565:of the indicator function for
4548:
4532:
4519:, the conditional probability
4445:
4439:
4427:
4412:
4308:
4302:
4225:
4210:
4179:
4163:
4058:
4042:
4028:{\displaystyle {\mathcal {G}}}
3969:
3956:
3950:
3947:
3936:
3928:
3915:
3856:
3844:
3839:
3833:
3817:
3806:
3794:
3786:
3772:
3712:
3696:
3682:{\displaystyle X:\Omega \to E}
3673:
3623:
3607:
3546:
3540:
3447:For the trivial sigma algebra
3401:
3385:
3361:
3355:
3352:
3336:
3274:
3262:
3253:
3247:
3238:
3232:
3229:
3213:
3171:
3162:
3146:
3129:{\displaystyle {\mathcal {G}}}
3071:{\displaystyle {\mathcal {F}}}
2987:
2965:
2943:{\displaystyle p_{X}(x)=E_{Y}}
2937:
2934:
2924:
2914:
2905:
2893:
2877:
2871:
2754:
2741:
2728:
2651:
2645:
2575:
2569:
2553:
2540:
2533:
2453:
2441:
2378:
2366:
2357:
2344:
2331:
2154:
2135:
2128:
2109:
1835:bivariate normal joint density
1791:
1785:
1772:
1760:
1751:
1736:
1724:
1702:
1696:
1683:
1671:
1583:
1577:
1523:
1517:
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1431:
1387:
1381:
1366:
1354:
1329:
1317:
928:
916:
910:
886:
877:
841:
832:
820:
814:
790:
719:
707:
684:
672:
641:
629:
621:
585:
573:
549:
540:
528:
519:
13:
1:
5448:
2954:Measure-theoretic formulation
2673:
2663:{\displaystyle f_{X}(x)>0}
1877:. To see the distribution of
1595:{\displaystyle f_{X}(x)>0}
360:conditional probability table
5453:
4146:{\displaystyle \mathbb {R} }
3136:-measurable random variable
1817:
1449:{\displaystyle f_{X,Y}(x,y)}
1228:probability density function
1008:otherwise. Furthermore, let
952:Consider the roll of a fair
419:conditional density function
415:probability density function
7:
5421:
5346:occurred! So in one sense,
2465:{\displaystyle P(X=x)>0}
1224:continuous random variables
441:of the included variables.
10:
5622:
5568:
5428:Conditioning (probability)
4895:Also recall that an event
2760:{\displaystyle P(Y=y|X=x)}
947:
314:as a parameter. When both
4868:given the information in
3112:implies that there is a
662:Due to the occurrence of
451:discrete random variables
5103:{\displaystyle \Omega =}
3000:be a probability space,
2193:Relation to independence
1529:{\displaystyle f_{X}(x)}
232:probability distribution
18:Conditional distribution
5606:Conditional probability
5582:Probability and Measure
5547:Probability and Measure
5433:Conditional probability
5222:. However, notice that
4563:conditional expectation
4478:{\displaystyle \sigma }
4093:{\displaystyle \sigma }
3191:conditional probability
3047:{\displaystyle \sigma }
728:{\displaystyle P(X=x).}
411:continuous distribution
5412:
5388:
5364:
5340:
5314:
5290:
5264:
5240:
5216:
5192:
5172:
5152:
5128:
5104:
5063:
5043:
5019:
4989:
4933:
4909:
4886:
4862:
4842:
4799:
4775:
4757:. The sub-sigma field
4751:
4723:and a sub-sigma field
4717:
4662:
4579:
4555:
4513:
4479:
4456:
4383:
4270:
4232:
4147:
4125:
4094:
4065:
4029:
4005:
3981:
3892:
3866:
3749:
3719:
3683:
3643:
3588:
3556:
3483:
3434:
3408:
3368:
3311:
3281:
3183:
3130:
3102:
3072:
3048:
3028:
2994:
2944:
2845:
2821:
2801:
2781:
2761:
2712:
2692:
2678:Seen as a function of
2664:
2622:
2602:
2582:
2514:joint density function
2506:
2486:
2466:
2425:
2405:
2385:
2315:
2295:
2275:
2255:
2231:
2211:
2184:
2018:
1998:
1969:
1943:
1917:
1891:
1871:
1851:
1830:
1801:
1639:
1619:
1596:
1554:
1530:
1494:
1474:
1450:
1398:
1284:
1264:
1244:
1208:
1182:
1156:
1054:
1028:
1002:
976:
938:
771:
751:
729:
691:
690:{\displaystyle P(X=x)}
652:
493:
467:
403:
383:
348:
328:
308:
288:
268:
248:
224:
204:
180:
160:
5495:Park, Kun Il (2018).
5413:
5389:
5365:
5341:
5315:
5291:
5265:
5241:
5217:
5193:
5173:
5153:
5129:
5105:
5064:
5044:
5020:
4990:
4934:
4910:
4887:
4863:
4843:
4800:
4776:
4752:
4718:
4663:
4580:
4556:
4514:
4480:
4457:
4384:
4271:
4233:
4148:
4126:
4095:
4066:
4030:
4006:
3982:
3893:
3867:
3750:
3720:
3684:
3644:
3589:
3557:
3484:
3435:
3409:
3369:
3312:
3282:
3184:
3131:
3110:Radon-Nikodym theorem
3103:
3073:
3049:
3029:
2995:
2945:
2846:
2822:
2802:
2782:
2762:
2713:
2693:
2665:
2623:
2603:
2583:
2507:
2487:
2467:
2426:
2406:
2386:
2316:
2296:
2276:
2256:
2232:
2212:
2185:
2019:
1999:
1970:
1944:
1918:
1892:
1872:
1852:
1837:for random variables
1825:
1802:
1640:
1620:
1597:
1555:
1531:
1495:
1475:
1451:
1399:
1285:
1265:
1245:
1209:
1183:
1157:
1055:
1029:
1003:
977:
939:
772:
752:
730:
692:
653:
494:
468:
404:
384:
364:marginal distribution
356:categorical variables
349:
329:
309:
289:
269:
249:
225:
205:
181:
161:
5577:Billingsley, Patrick
5398:
5374:
5350:
5324:
5300:
5274:
5250:
5226:
5202:
5182:
5162:
5138:
5114:
5076:
5053:
5029:
4999:
4943:
4919:
4899:
4872:
4852:
4809:
4785:
4761:
4727:
4683:
4592:
4569:
4561:is a version of the
4523:
4489:
4469:
4403:
4287:
4250:
4157:
4135:
4104:
4084:
4071:, then it is called
4039:
4015:
3995:
3902:
3876:
3759:
3729:
3693:
3661:
3598:
3568:
3493:
3451:
3418:
3382:
3327:
3291:
3197:
3140:
3116:
3082:
3058:
3038:
3004:
2962:
2858:
2835:
2811:
2791:
2771:
2722:
2702:
2682:
2632:
2612:
2592:
2520:
2496:
2476:
2435:
2415:
2395:
2325:
2305:
2285:
2265:
2245:
2221:
2201:
2030:
2008:
1982:
1953:
1942:{\displaystyle X=70}
1927:
1916:{\displaystyle X=70}
1901:
1881:
1861:
1841:
1652:
1629:
1609:
1564:
1544:
1504:
1484:
1464:
1412:
1298:
1274:
1254:
1234:
1192:
1166:
1140:
1038:
1012:
986:
960:
784:
761:
741:
701:
666:
507:
477:
457:
431:conditional variance
393:
373:
338:
318:
298:
278:
258:
238:
214:
194:
170:
150:
54:improve this article
4202:
3376:probability measure
2829:likelihood function
2171:
1997:{\displaystyle X,Y}
1968:{\displaystyle X,Y}
1207:{\displaystyle Y=1}
1181:{\displaystyle X=1}
1155:{\displaystyle X=1}
1053:{\displaystyle Y=0}
1027:{\displaystyle Y=1}
1001:{\displaystyle X=0}
975:{\displaystyle X=1}
492:{\displaystyle X=x}
142:jointly distributed
5532:Billingsley (1995)
5520:Billingsley (1995)
5408:
5384:
5360:
5336:
5310:
5286:
5260:
5236:
5212:
5188:
5168:
5148:
5124:
5100:
5059:
5039:
5015:
4985:
4929:
4905:
4882:
4858:
4838:
4795:
4771:
4747:
4713:
4658:
4575:
4551:
4509:
4475:
4452:
4379:
4374:
4278:indicator function
4266:
4228:
4185:
4143:
4121:
4090:
4061:
4025:
4001:
3977:
3888:
3862:
3745:
3715:
3679:
3639:
3584:
3552:
3479:
3430:
3404:
3364:
3307:
3277:
3179:
3126:
3098:
3068:
3044:
3024:
2990:
2940:
2841:
2817:
2797:
2777:
2757:
2708:
2688:
2660:
2618:
2598:
2578:
2502:
2482:
2462:
2421:
2401:
2381:
2311:
2291:
2271:
2251:
2227:
2207:
2180:
2157:
2014:
1994:
1965:
1939:
1913:
1887:
1867:
1847:
1833:The graph shows a
1831:
1797:
1635:
1615:
1592:
1550:
1526:
1490:
1470:
1446:
1394:
1290:can be written as
1280:
1260:
1240:
1226:, the conditional
1204:
1178:
1152:
1050:
1024:
998:
972:
934:
767:
747:
725:
687:
648:
489:
463:
439:joint distribution
399:
379:
344:
324:
304:
284:
264:
244:
220:
200:
176:
156:
134:probability theory
5557:978-1-118-12237-2
5506:978-3-319-68074-3
5191:{\displaystyle 1}
5171:{\displaystyle 0}
5062:{\displaystyle B}
4908:{\displaystyle B}
4861:{\displaystyle B}
4578:{\displaystyle A}
4358:
4331:
4004:{\displaystyle X}
2930:
2922:
2844:{\displaystyle x}
2820:{\displaystyle y}
2800:{\displaystyle x}
2780:{\displaystyle y}
2711:{\displaystyle x}
2691:{\displaystyle y}
2621:{\displaystyle x}
2601:{\displaystyle y}
2588:for all possible
2505:{\displaystyle Y}
2485:{\displaystyle X}
2424:{\displaystyle x}
2404:{\displaystyle y}
2391:for all possible
2314:{\displaystyle Y}
2294:{\displaystyle X}
2274:{\displaystyle X}
2254:{\displaystyle Y}
2230:{\displaystyle Y}
2210:{\displaystyle X}
2197:Random variables
2104:
2056:
2050:
2017:{\displaystyle Y}
1890:{\displaystyle Y}
1870:{\displaystyle Y}
1850:{\displaystyle X}
1826:Bivariate normal
1638:{\displaystyle Y}
1618:{\displaystyle X}
1553:{\displaystyle X}
1493:{\displaystyle Y}
1473:{\displaystyle X}
1391:
1283:{\displaystyle X}
1263:{\displaystyle x}
1243:{\displaystyle Y}
1134:
1133:
770:{\displaystyle Y}
750:{\displaystyle X}
645:
466:{\displaystyle Y}
402:{\displaystyle X}
382:{\displaystyle Y}
347:{\displaystyle Y}
327:{\displaystyle X}
307:{\displaystyle X}
287:{\displaystyle x}
267:{\displaystyle X}
247:{\displaystyle Y}
223:{\displaystyle X}
203:{\displaystyle Y}
179:{\displaystyle Y}
159:{\displaystyle X}
130:
129:
122:
104:
16:(Redirected from
5613:
5586:
5562:
5561:
5541:
5535:
5529:
5523:
5517:
5511:
5510:
5492:
5486:
5485:
5468:Ross, Sheldon M.
5464:
5417:
5415:
5414:
5409:
5407:
5406:
5393:
5391:
5390:
5385:
5383:
5382:
5369:
5367:
5366:
5361:
5359:
5358:
5345:
5343:
5342:
5337:
5319:
5317:
5316:
5311:
5309:
5308:
5295:
5293:
5292:
5287:
5269:
5267:
5266:
5261:
5259:
5258:
5245:
5243:
5242:
5237:
5235:
5234:
5221:
5219:
5218:
5213:
5211:
5210:
5197:
5195:
5194:
5189:
5177:
5175:
5174:
5169:
5157:
5155:
5154:
5149:
5147:
5146:
5133:
5131:
5130:
5125:
5123:
5122:
5109:
5107:
5106:
5101:
5068:
5066:
5065:
5060:
5048:
5046:
5045:
5040:
5038:
5037:
5024:
5022:
5021:
5016:
5014:
5013:
4994:
4992:
4991:
4986:
4975:
4961:
4950:
4938:
4936:
4935:
4930:
4928:
4927:
4914:
4912:
4911:
4906:
4891:
4889:
4888:
4883:
4881:
4880:
4867:
4865:
4864:
4859:
4847:
4845:
4844:
4839:
4834:
4833:
4827:
4816:
4804:
4802:
4801:
4796:
4794:
4793:
4780:
4778:
4777:
4772:
4770:
4769:
4756:
4754:
4753:
4748:
4746:
4745:
4736:
4735:
4722:
4720:
4719:
4714:
4709:
4701:
4700:
4667:
4665:
4664:
4659:
4653:
4652:
4643:
4642:
4637:
4616:
4615:
4584:
4582:
4581:
4576:
4560:
4558:
4557:
4552:
4547:
4546:
4518:
4516:
4515:
4510:
4508:
4507:
4498:
4497:
4484:
4482:
4481:
4476:
4461:
4459:
4458:
4453:
4426:
4425:
4420:
4388:
4386:
4385:
4380:
4378:
4377:
4359:
4356:
4332:
4329:
4301:
4300:
4295:
4275:
4273:
4272:
4267:
4265:
4264:
4237:
4235:
4234:
4229:
4201:
4196:
4178:
4177:
4152:
4150:
4149:
4144:
4142:
4130:
4128:
4127:
4122:
4120:
4119:
4114:
4113:
4099:
4097:
4096:
4091:
4070:
4068:
4067:
4062:
4057:
4056:
4034:
4032:
4031:
4026:
4024:
4023:
4010:
4008:
4007:
4002:
3986:
3984:
3983:
3978:
3976:
3968:
3967:
3946:
3945:
3939:
3927:
3926:
3925:
3924:
3918:
3897:
3895:
3894:
3889:
3871:
3869:
3868:
3863:
3855:
3854:
3847:
3832:
3831:
3816:
3805:
3804:
3797:
3785:
3784:
3783:
3782:
3775:
3754:
3752:
3751:
3746:
3744:
3743:
3724:
3722:
3721:
3716:
3711:
3710:
3688:
3686:
3685:
3680:
3648:
3646:
3645:
3640:
3638:
3637:
3622:
3621:
3593:
3591:
3590:
3585:
3583:
3582:
3561:
3559:
3558:
3553:
3530:
3526:
3488:
3486:
3485:
3480:
3460:
3459:
3439:
3437:
3436:
3431:
3413:
3411:
3410:
3405:
3400:
3399:
3373:
3371:
3370:
3365:
3351:
3350:
3316:
3314:
3313:
3308:
3306:
3305:
3286:
3284:
3283:
3278:
3228:
3227:
3209:
3208:
3188:
3186:
3185:
3180:
3178:
3161:
3160:
3135:
3133:
3132:
3127:
3125:
3124:
3107:
3105:
3104:
3099:
3097:
3096:
3077:
3075:
3074:
3069:
3067:
3066:
3053:
3051:
3050:
3045:
3033:
3031:
3030:
3025:
3023:
3022:
3013:
3012:
2999:
2997:
2996:
2991:
2980:
2979:
2949:
2947:
2946:
2941:
2928:
2927:
2920:
2913:
2912:
2908:
2892:
2891:
2870:
2869:
2850:
2848:
2847:
2842:
2826:
2824:
2823:
2818:
2806:
2804:
2803:
2798:
2786:
2784:
2783:
2778:
2766:
2764:
2763:
2758:
2744:
2717:
2715:
2714:
2709:
2697:
2695:
2694:
2689:
2669:
2667:
2666:
2661:
2644:
2643:
2627:
2625:
2624:
2619:
2607:
2605:
2604:
2599:
2587:
2585:
2584:
2579:
2568:
2567:
2543:
2532:
2531:
2511:
2509:
2508:
2503:
2491:
2489:
2488:
2483:
2471:
2469:
2468:
2463:
2430:
2428:
2427:
2422:
2410:
2408:
2407:
2402:
2390:
2388:
2387:
2382:
2347:
2320:
2318:
2317:
2312:
2300:
2298:
2297:
2292:
2280:
2278:
2277:
2272:
2260:
2258:
2257:
2252:
2236:
2234:
2233:
2228:
2216:
2214:
2213:
2208:
2189:
2187:
2186:
2181:
2176:
2172:
2170:
2165:
2153:
2152:
2127:
2126:
2105:
2103:
2102:
2093:
2092:
2083:
2078:
2077:
2063:
2062:
2054:
2048:
2023:
2021:
2020:
2015:
2003:
2001:
2000:
1995:
1974:
1972:
1971:
1966:
1948:
1946:
1945:
1940:
1922:
1920:
1919:
1914:
1896:
1894:
1893:
1888:
1876:
1874:
1873:
1868:
1856:
1854:
1853:
1848:
1806:
1804:
1803:
1798:
1784:
1783:
1759:
1758:
1754:
1723:
1722:
1695:
1694:
1670:
1669:
1644:
1642:
1641:
1636:
1624:
1622:
1621:
1616:
1601:
1599:
1598:
1593:
1576:
1575:
1559:
1557:
1556:
1551:
1538:marginal density
1535:
1533:
1532:
1527:
1516:
1515:
1499:
1497:
1496:
1491:
1479:
1477:
1476:
1471:
1455:
1453:
1452:
1447:
1430:
1429:
1403:
1401:
1400:
1395:
1392:
1390:
1380:
1379:
1369:
1353:
1352:
1336:
1316:
1315:
1289:
1287:
1286:
1281:
1269:
1267:
1266:
1261:
1249:
1247:
1246:
1241:
1213:
1211:
1210:
1205:
1187:
1185:
1184:
1179:
1161:
1159:
1158:
1153:
1063:
1062:
1059:
1057:
1056:
1051:
1033:
1031:
1030:
1025:
1007:
1005:
1004:
999:
981:
979:
978:
973:
943:
941:
940:
935:
776:
774:
773:
768:
756:
754:
753:
748:
734:
732:
731:
726:
696:
694:
693:
688:
657:
655:
654:
649:
646:
644:
624:
580:
527:
526:
522:
498:
496:
495:
490:
472:
470:
469:
464:
427:conditional mean
417:is known as the
408:
406:
405:
400:
388:
386:
385:
380:
353:
351:
350:
345:
333:
331:
330:
325:
313:
311:
310:
305:
293:
291:
290:
285:
273:
271:
270:
265:
253:
251:
250:
245:
229:
227:
226:
221:
209:
207:
206:
201:
185:
183:
182:
177:
165:
163:
162:
157:
145:random variables
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
5621:
5620:
5616:
5615:
5614:
5612:
5611:
5610:
5591:
5590:
5589:
5571:
5566:
5565:
5558:
5542:
5538:
5530:
5526:
5518:
5514:
5507:
5493:
5489:
5482:
5465:
5461:
5456:
5451:
5424:
5402:
5401:
5399:
5396:
5395:
5378:
5377:
5375:
5372:
5371:
5354:
5353:
5351:
5348:
5347:
5325:
5322:
5321:
5304:
5303:
5301:
5298:
5297:
5275:
5272:
5271:
5254:
5253:
5251:
5248:
5247:
5230:
5229:
5227:
5224:
5223:
5206:
5205:
5203:
5200:
5199:
5183:
5180:
5179:
5163:
5160:
5159:
5142:
5141:
5139:
5136:
5135:
5118:
5117:
5115:
5112:
5111:
5077:
5074:
5073:
5054:
5051:
5050:
5033:
5032:
5030:
5027:
5026:
5009:
5008:
5000:
4997:
4996:
4971:
4957:
4946:
4944:
4941:
4940:
4923:
4922:
4920:
4917:
4916:
4900:
4897:
4896:
4876:
4875:
4873:
4870:
4869:
4853:
4850:
4849:
4829:
4828:
4823:
4812:
4810:
4807:
4806:
4789:
4788:
4786:
4783:
4782:
4765:
4764:
4762:
4759:
4758:
4741:
4740:
4731:
4730:
4728:
4725:
4724:
4705:
4696:
4695:
4684:
4681:
4680:
4677:
4648:
4647:
4638:
4633:
4632:
4611:
4610:
4593:
4590:
4589:
4570:
4567:
4566:
4542:
4541:
4524:
4521:
4520:
4503:
4502:
4493:
4492:
4490:
4487:
4486:
4470:
4467:
4466:
4421:
4416:
4415:
4404:
4401:
4400:
4373:
4372:
4355:
4353:
4346:
4345:
4328:
4326:
4315:
4314:
4296:
4291:
4290:
4288:
4285:
4284:
4260:
4259:
4251:
4248:
4247:
4244:
4238:almost surely.
4197:
4189:
4173:
4172:
4158:
4155:
4154:
4138:
4136:
4133:
4132:
4115:
4109:
4108:
4107:
4105:
4102:
4101:
4085:
4082:
4081:
4052:
4051:
4040:
4037:
4036:
4019:
4018:
4016:
4013:
4012:
3996:
3993:
3992:
3972:
3963:
3962:
3941:
3940:
3935:
3920:
3919:
3914:
3909:
3905:
3903:
3900:
3899:
3898:, the function
3877:
3874:
3873:
3850:
3849:
3843:
3824:
3820:
3812:
3800:
3799:
3793:
3778:
3777:
3771:
3766:
3762:
3760:
3757:
3756:
3739:
3738:
3730:
3727:
3726:
3706:
3705:
3694:
3691:
3690:
3662:
3659:
3658:
3633:
3629:
3617:
3616:
3599:
3596:
3595:
3578:
3577:
3569:
3566:
3565:
3504:
3500:
3494:
3491:
3490:
3455:
3454:
3452:
3449:
3448:
3443:Special cases:
3419:
3416:
3415:
3395:
3394:
3383:
3380:
3379:
3346:
3345:
3328:
3325:
3324:
3301:
3300:
3292:
3289:
3288:
3223:
3222:
3204:
3200:
3198:
3195:
3194:
3174:
3156:
3155:
3141:
3138:
3137:
3120:
3119:
3117:
3114:
3113:
3092:
3091:
3083:
3080:
3079:
3062:
3061:
3059:
3056:
3055:
3039:
3036:
3035:
3018:
3017:
3008:
3007:
3005:
3002:
3001:
2975:
2974:
2963:
2960:
2959:
2956:
2923:
2904:
2900:
2896:
2887:
2883:
2865:
2861:
2859:
2856:
2855:
2851:need not be 1.
2836:
2833:
2832:
2812:
2809:
2808:
2792:
2789:
2788:
2772:
2769:
2768:
2740:
2723:
2720:
2719:
2703:
2700:
2699:
2683:
2680:
2679:
2676:
2639:
2635:
2633:
2630:
2629:
2613:
2610:
2609:
2593:
2590:
2589:
2563:
2559:
2539:
2527:
2523:
2521:
2518:
2517:
2497:
2494:
2493:
2477:
2474:
2473:
2436:
2433:
2432:
2416:
2413:
2412:
2396:
2393:
2392:
2343:
2326:
2323:
2322:
2306:
2303:
2302:
2286:
2283:
2282:
2266:
2263:
2262:
2246:
2243:
2242:
2222:
2219:
2218:
2202:
2199:
2198:
2195:
2166:
2161:
2148:
2144:
2122:
2118:
2098:
2094:
2088:
2084:
2082:
2073:
2069:
2068:
2064:
2058:
2057:
2031:
2028:
2027:
2009:
2006:
2005:
1983:
1980:
1979:
1954:
1951:
1950:
1928:
1925:
1924:
1902:
1899:
1898:
1897:conditional on
1882:
1879:
1878:
1862:
1859:
1858:
1842:
1839:
1838:
1820:
1812:Borel's paradox
1779:
1775:
1750:
1746:
1742:
1712:
1708:
1690:
1686:
1659:
1655:
1653:
1650:
1649:
1630:
1627:
1626:
1610:
1607:
1606:
1571:
1567:
1565:
1562:
1561:
1545:
1542:
1541:
1511:
1507:
1505:
1502:
1501:
1485:
1482:
1481:
1465:
1462:
1461:
1419:
1415:
1413:
1410:
1409:
1406:
1375:
1371:
1370:
1342:
1338:
1337:
1335:
1305:
1301:
1299:
1296:
1295:
1275:
1272:
1271:
1255:
1252:
1251:
1235:
1232:
1231:
1220:
1193:
1190:
1189:
1188:conditional on
1167:
1164:
1163:
1141:
1138:
1137:
1039:
1036:
1035:
1013:
1010:
1009:
987:
984:
983:
961:
958:
957:
950:
785:
782:
781:
762:
759:
758:
742:
739:
738:
702:
699:
698:
667:
664:
663:
660:
625:
581:
579:
518:
514:
510:
508:
505:
504:
478:
475:
474:
458:
455:
454:
447:
394:
391:
390:
374:
371:
370:
339:
336:
335:
319:
316:
315:
299:
296:
295:
279:
276:
275:
259:
256:
255:
239:
236:
235:
215:
212:
211:
195:
192:
191:
171:
168:
167:
151:
148:
147:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
5619:
5609:
5608:
5603:
5588:
5587:
5572:
5570:
5567:
5564:
5563:
5556:
5536:
5524:
5512:
5505:
5487:
5480:
5458:
5457:
5455:
5452:
5450:
5447:
5446:
5445:
5443:Bayes' theorem
5440:
5435:
5430:
5423:
5420:
5405:
5381:
5357:
5335:
5332:
5329:
5307:
5285:
5282:
5279:
5257:
5233:
5209:
5187:
5167:
5145:
5121:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5058:
5036:
5012:
5007:
5004:
4984:
4981:
4978:
4974:
4970:
4967:
4964:
4960:
4956:
4953:
4949:
4926:
4904:
4879:
4857:
4837:
4832:
4826:
4822:
4819:
4815:
4792:
4768:
4744:
4739:
4734:
4712:
4708:
4704:
4699:
4694:
4691:
4688:
4676:
4673:
4669:
4668:
4656:
4651:
4646:
4641:
4636:
4631:
4628:
4625:
4622:
4619:
4614:
4609:
4606:
4603:
4600:
4597:
4574:
4550:
4545:
4540:
4537:
4534:
4531:
4528:
4506:
4501:
4496:
4474:
4463:
4462:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4424:
4419:
4414:
4411:
4408:
4390:
4389:
4376:
4371:
4368:
4365:
4362:
4354:
4351:
4348:
4347:
4344:
4341:
4338:
4335:
4327:
4324:
4321:
4320:
4318:
4313:
4310:
4307:
4304:
4299:
4294:
4263:
4258:
4255:
4246:For any event
4243:
4240:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4205:
4200:
4195:
4192:
4188:
4184:
4181:
4176:
4171:
4168:
4165:
4162:
4141:
4118:
4112:
4089:
4060:
4055:
4050:
4047:
4044:
4022:
4000:
3987:is called the
3975:
3971:
3966:
3961:
3958:
3955:
3952:
3949:
3944:
3938:
3933:
3930:
3923:
3917:
3912:
3908:
3887:
3884:
3881:
3861:
3858:
3853:
3846:
3841:
3838:
3835:
3830:
3827:
3823:
3819:
3815:
3811:
3808:
3803:
3796:
3791:
3788:
3781:
3774:
3769:
3765:
3742:
3737:
3734:
3714:
3709:
3704:
3701:
3698:
3678:
3675:
3672:
3669:
3666:
3655:
3654:
3636:
3632:
3628:
3625:
3620:
3615:
3612:
3609:
3606:
3603:
3581:
3576:
3573:
3562:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3529:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3503:
3498:
3478:
3475:
3472:
3469:
3466:
3463:
3458:
3429:
3426:
3423:
3403:
3398:
3393:
3390:
3387:
3363:
3360:
3357:
3354:
3349:
3344:
3341:
3338:
3335:
3332:
3304:
3299:
3296:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3226:
3221:
3218:
3215:
3212:
3207:
3203:
3177:
3173:
3170:
3167:
3164:
3159:
3154:
3151:
3148:
3145:
3123:
3095:
3090:
3087:
3065:
3043:
3021:
3016:
3011:
2989:
2986:
2983:
2978:
2973:
2970:
2967:
2955:
2952:
2939:
2936:
2933:
2926:
2919:
2916:
2911:
2907:
2903:
2899:
2895:
2890:
2886:
2882:
2879:
2876:
2873:
2868:
2864:
2840:
2816:
2796:
2776:
2756:
2753:
2750:
2747:
2743:
2739:
2736:
2733:
2730:
2727:
2707:
2687:
2675:
2672:
2659:
2656:
2653:
2650:
2647:
2642:
2638:
2617:
2597:
2577:
2574:
2571:
2566:
2562:
2558:
2555:
2552:
2549:
2546:
2542:
2538:
2535:
2530:
2526:
2501:
2481:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2420:
2400:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2346:
2342:
2339:
2336:
2333:
2330:
2310:
2290:
2270:
2250:
2226:
2206:
2194:
2191:
2179:
2175:
2169:
2164:
2160:
2156:
2151:
2147:
2143:
2140:
2137:
2133:
2130:
2125:
2121:
2117:
2114:
2111:
2108:
2101:
2097:
2091:
2087:
2081:
2076:
2072:
2067:
2061:
2053:
2047:
2044:
2041:
2038:
2035:
2013:
1993:
1990:
1987:
1964:
1961:
1958:
1938:
1935:
1932:
1912:
1909:
1906:
1886:
1866:
1846:
1819:
1816:
1808:
1807:
1796:
1793:
1790:
1787:
1782:
1778:
1774:
1771:
1768:
1765:
1762:
1757:
1753:
1749:
1745:
1741:
1738:
1735:
1732:
1729:
1726:
1721:
1718:
1715:
1711:
1707:
1704:
1701:
1698:
1693:
1689:
1685:
1682:
1679:
1676:
1673:
1668:
1665:
1662:
1658:
1634:
1614:
1591:
1588:
1585:
1582:
1579:
1574:
1570:
1549:
1525:
1522:
1519:
1514:
1510:
1489:
1469:
1445:
1442:
1439:
1436:
1433:
1428:
1425:
1422:
1418:
1405:
1404:
1389:
1386:
1383:
1378:
1374:
1368:
1365:
1362:
1359:
1356:
1351:
1348:
1345:
1341:
1334:
1331:
1328:
1325:
1322:
1319:
1314:
1311:
1308:
1304:
1292:
1279:
1259:
1239:
1222:Similarly for
1219:
1216:
1203:
1200:
1197:
1177:
1174:
1171:
1151:
1148:
1145:
1132:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1109:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1086:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1049:
1046:
1043:
1023:
1020:
1017:
997:
994:
991:
971:
968:
965:
949:
946:
945:
944:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
766:
746:
724:
721:
718:
715:
712:
709:
706:
686:
683:
680:
677:
674:
671:
659:
658:
643:
640:
637:
634:
631:
628:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
525:
521:
517:
513:
501:
488:
485:
482:
462:
446:
443:
398:
378:
343:
323:
303:
283:
263:
243:
219:
199:
175:
155:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
5618:
5607:
5604:
5602:
5599:
5598:
5596:
5584:
5583:
5578:
5574:
5573:
5559:
5553:
5549:
5548:
5540:
5533:
5528:
5521:
5516:
5508:
5502:
5498:
5491:
5483:
5481:0-12-598455-3
5477:
5473:
5469:
5463:
5459:
5444:
5441:
5439:
5436:
5434:
5431:
5429:
5426:
5425:
5419:
5330:
5327:
5280:
5277:
5185:
5165:
5094:
5091:
5088:
5082:
5070:
5056:
5005:
5002:
4979:
4968:
4962:
4954:
4902:
4893:
4855:
4820:
4737:
4702:
4692:
4672:
4644:
4639:
4626:
4620:
4607:
4604:
4598:
4588:
4587:
4586:
4572:
4564:
4538:
4535:
4529:
4499:
4472:
4448:
4442:
4436:
4430:
4422:
4409:
4399:
4398:
4397:
4395:
4369:
4366:
4363:
4360:
4349:
4342:
4339:
4336:
4333:
4322:
4316:
4311:
4305:
4297:
4283:
4282:
4281:
4279:
4276:, define the
4256:
4253:
4239:
4222:
4219:
4216:
4213:
4207:
4203:
4190:
4186:
4182:
4169:
4166:
4160:
4116:
4087:
4078:
4076:
4075:
4048:
4045:
3998:
3990:
3959:
3953:
3931:
3910:
3906:
3882:
3879:
3859:
3836:
3828:
3825:
3821:
3809:
3789:
3767:
3763:
3735:
3732:
3702:
3699:
3676:
3667:
3664:
3652:
3634:
3630:
3626:
3613:
3610:
3604:
3574:
3571:
3563:
3549:
3543:
3537:
3531:
3527:
3517:
3508:
3505:
3501:
3470:
3461:
3446:
3445:
3444:
3441:
3424:
3421:
3391:
3377:
3358:
3342:
3339:
3333:
3322:
3321:
3297:
3294:
3271:
3268:
3265:
3259:
3256:
3250:
3244:
3241:
3235:
3219:
3216:
3210:
3205:
3201:
3192:
3189:, called the
3165:
3152:
3149:
3143:
3111:
3088:
3085:
3041:
3014:
2984:
2981:
2971:
2951:
2931:
2917:
2909:
2901:
2897:
2888:
2884:
2880:
2874:
2866:
2862:
2852:
2838:
2830:
2814:
2794:
2774:
2751:
2748:
2745:
2737:
2734:
2731:
2725:
2705:
2685:
2671:
2657:
2654:
2648:
2640:
2636:
2615:
2595:
2572:
2564:
2560:
2556:
2550:
2547:
2544:
2536:
2528:
2524:
2515:
2499:
2479:
2459:
2456:
2450:
2447:
2444:
2438:
2418:
2398:
2375:
2372:
2369:
2363:
2360:
2354:
2351:
2348:
2340:
2337:
2334:
2328:
2308:
2288:
2268:
2248:
2240:
2224:
2204:
2190:
2177:
2173:
2167:
2162:
2158:
2149:
2145:
2141:
2138:
2131:
2123:
2119:
2115:
2112:
2106:
2099:
2095:
2089:
2085:
2079:
2074:
2070:
2065:
2051:
2045:
2042:
2039:
2036:
2033:
2025:
2011:
1991:
1988:
1985:
1977:
1962:
1959:
1956:
1936:
1933:
1930:
1910:
1907:
1904:
1884:
1864:
1844:
1836:
1829:
1828:joint density
1824:
1815:
1813:
1794:
1788:
1780:
1776:
1769:
1766:
1763:
1755:
1747:
1743:
1739:
1733:
1730:
1727:
1719:
1716:
1713:
1709:
1705:
1699:
1691:
1687:
1680:
1677:
1674:
1666:
1663:
1660:
1656:
1648:
1647:
1646:
1645:is given by:
1632:
1612:
1603:
1589:
1586:
1580:
1572:
1568:
1547:
1539:
1520:
1512:
1508:
1487:
1467:
1459:
1458:joint density
1440:
1437:
1434:
1426:
1423:
1420:
1416:
1384:
1376:
1372:
1363:
1360:
1357:
1349:
1346:
1343:
1339:
1332:
1326:
1323:
1320:
1312:
1309:
1306:
1302:
1294:
1293:
1291:
1277:
1257:
1237:
1229:
1225:
1215:
1201:
1198:
1195:
1175:
1172:
1169:
1149:
1146:
1143:
1129:
1126:
1123:
1120:
1117:
1114:
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92:
88:
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74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
5581:
5546:
5539:
5527:
5515:
5499:. Springer.
5496:
5490:
5471:
5462:
5158:has measure
5071:
4894:
4678:
4670:
4464:
4393:
4391:
4245:
4079:
4073:
3988:
3656:
3442:
3319:
2957:
2853:
2677:
2196:
2026:
1832:
1809:
1604:
1407:
1221:
1135:
951:
736:
661:
448:
437:conditional
435:
418:
368:
187:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
3193:, such that
2516:, it means
2512:, having a
2239:independent
1060:otherwise.
413:, then its
5595:Categories
5449:References
3287:for every
3054:-field in
2827:, it is a
2807:for given
2698:for given
2674:Properties
1536:gives the
1456:gives the
138:statistics
110:April 2013
80:newspapers
5454:Citations
5334:Ω
5331:∈
5328:ω
5284:Ω
5281:∈
5278:ω
5080:Ω
5006:∈
4738:⊂
4690:Ω
4645:∣
4627:
4608:∣
4599:
4539:∣
4530:
4500:⊆
4473:σ
4465:Given a
4437:
4410:
4364:∉
4361:ω
4337:∈
4334:ω
4306:ω
4257:∈
4223:⋅
4208:μ
4199:∞
4194:∞
4191:−
4187:∫
4170:∣
4088:σ
3970:→
3954:ω
3932:⋅
3907:μ
3886:Ω
3883:∈
3880:ω
3826:−
3764:μ
3755:, define
3736:∈
3674:→
3671:Ω
3614:∣
3605:
3575:∈
3538:
3521:Ω
3515:∅
3509:∣
3474:Ω
3468:∅
3428:Ω
3425:∈
3422:ω
3389:Ω
3359:ω
3343:∣
3340:⋅
3334:
3298:∈
3269:∩
3251:ω
3236:ω
3220:∣
3202:∫
3172:→
3169:Ω
3153:∣
3089:∈
3042:σ
3015:⊆
2969:Ω
2159:σ
2146:ρ
2142:−
2120:μ
2116:−
2107:ρ
2096:σ
2086:σ
2071:μ
2052:∼
2037:∣
1767:∣
1678:∣
1664:∣
1324:∣
1310:∣
899:∣
860:∩
803:∣
604:∩
562:∣
544:≜
535:∣
5579:(1995).
5534:, p. 439
5522:, p. 430
5470:(1993).
5422:See also
4995:for all
4396:itself:
4357:if
4330:if
3872:For any
3594:, then
3414:for all
3078:. Given
1500:, while
956:and let
5569:Sources
4485:-field
4100:-field
4074:regular
3320:regular
1949:in the
1818:Example
948:Example
423:moments
230:is the
94:scholar
5554:
5503:
5478:
5110:. Let
4011:given
3108:, the
2929:
2921:
2261:given
2055:
2049:
1625:given
1408:where
757:given
473:given
389:given
210:given
186:, the
96:
89:
82:
75:
67:
3689:be a
3651:below
3440:a.e.
3374:is a
2628:with
2431:with
1976:plane
409:is a
254:when
101:JSTOR
87:books
5552:ISBN
5501:ISBN
5476:ISBN
3657:Let
3323:if
2958:Let
2655:>
2608:and
2492:and
2457:>
2411:and
2237:are
1857:and
1587:>
1540:for
1480:and
777:is:
449:For
429:and
358:, a
354:are
334:and
166:and
136:and
73:news
5178:or
4939:if
4131:on
3991:of
3564:If
3378:on
1460:of
1270:of
1230:of
954:die
294:of
234:of
190:of
132:In
56:by
5597::
5418:.
4892:.
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4280::
4077:.
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3034:a
2950:.
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2670:.
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2113:70
2046:70
2024:.
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1911:70
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433:.
5560:.
5509:.
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5380:F
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5098:]
5095:1
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4983:)
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4959:|
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607:{
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586:(
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