3675:
4179:
3919:
3431:
Realistically, H will be dependent on L. That is, P(H = Hit) will take different values depending on whether L is red, yellow or green (and likewise for P(H = Not Hit)). A person is, for example, far more likely to be hit by a car when trying to cross while the lights for perpendicular
3443:
P(H = Hit), what is being sought is the probability that H = Hit in the situation in which the particular value of L is unknown and in which the pedestrian ignores the state of the light. In general, a pedestrian can be hit if the lights are red OR if the lights are yellow OR if the lights
3665:
The marginal probability P(H = Hit) is the sum 0.572 along the H = Hit row of this joint distribution table, as this is the probability of being hit when the lights are red OR yellow OR green. Similarly, the marginal probability that P(H = Not Hit) is the sum along the H =
84:
being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables. In many applications, an analysis may start with a given collection of random variables, then first extend the set by defining new ones (such as the sum of the original
3539:
To find the joint probability distribution, more data is required. For example, suppose P(L = red) = 0.2, P(L = yellow) = 0.1, and P(L = green) = 0.7. Multiplying each column in the conditional distribution by the probability of that column occurring
3413:
68:
are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is
1283:
3148:
3447:
Here is a table showing the conditional probabilities of being hit, depending on the state of the lights. (Note that the columns in this table must add up to 1 because the probability of being hit or not hit is 1 regardless of the state of the light.)
940:
85:
random variables) and finally reduce the number by placing interest in the marginal distribution of a subset (such as the sum). Several different analyses may be done, each treating a different subset of variables as the marginal distribution.
3678:
Many samples from a bivariate normal distribution. The marginal distributions are shown in red and blue. The marginal distribution of X is also approximated by creating a histogram of the X coordinates without consideration of the Y
3885:
1099:
1781:
3164:
4174:{\displaystyle f_{X_{i}}(x_{i})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f(x_{1},x_{2},\dots ,x_{n})dx_{1}dx_{2}\cdots dx_{i-1}dx_{i+1}\cdots dx_{n}.}
1073:
2211:
2330:
3424:
Suppose that the probability that a pedestrian will be hit by a car, while crossing the road at a pedestrian crossing, without paying attention to the traffic light, is to be computed. Let H be a
58:
of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a
2980:
326:
238:
3444:
are green. So, the answer for the marginal probability can be found by summing P(H | L) for all possible values of L, with each value of L weighted by its probability of occurring.
1989:
1486:
1406:
779:
4292:
1656:
2057:
The conditional distribution of a variable given another variable is the joint distribution of both variables divided by the marginal distribution of the other variable. That is,
1909:
1852:
3737:
3485:
2025:
3579:
1565:
1527:
974:
2071:
3540:
results in the joint probability distribution of H and L, given in the central 2×3 block of entries. (Note that the cells in this 2×3 block add up to 1).
4404:
3408:{\displaystyle p_{Y|X}(y_{1}|x_{4})=P(Y=y_{1}|X=x_{4})={\frac {P(X=x_{4},Y=y_{1})}{P(X=x_{4})}}={\frac {8/200}{70/200}}={\frac {8}{70}}={\frac {4}{35}}}
73:(that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been
1667:
2222:
763:. The values of the joint distribution are in the 3×4 rectangle; the values of the marginal distributions are along the right and bottom margins.
1278:{\displaystyle p_{X}(x)=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y=\int _{y}\delta {\big (}x-g(y){\big )}\,p_{Y}(y)\,\mathrm {d} y.}
3428:
taking one value from {Hit, Not Hit}. Let L (for traffic light) be a discrete random variable taking one value from {Red, Yellow, Green}.
4244:
4209:
2046:
968:
4385:
4321:
4189:
3695:
being interpreted as vectors. In particular, each summation or integration would be over all variables except those contained in
3432:
traffic are green than if they are red. In other words, for any given possible pair of values for H and L, one must consider the
1576:
3436:
of H and L to find the probability of that pair of events occurring together if the pedestrian ignores the state of the light.
1593:
4194:
3433:
3143:{\displaystyle p_{Y}(y_{1})=P_{Y}(Y=y_{1})=\sum _{i=1}^{4}P(x_{i},y_{1})={\frac {2}{200}}+{\frac {8}{200}}={\frac {10}{200}}}
2967:
of dataset of the relationship in a classroom of 200 students between the amount of time studied and the percentage correct
243:
155:
3160:
can be used to determine the probability that a student that studied 60 minutes or more obtains a scored of 20 or below:
1077:
Therefore, marginalization provides the rule for the transformation of the probability distribution of a random variable
3913:
1311:
1412:
1332:
1921:
4368:
4345:
3731:
3684:
3425:
3156:
122:
59:
55:
935:{\displaystyle p_{X}(x)=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y=\operatorname {E} _{Y}\;.}
3416:, meaning there is about a 11% probability of scoring 20 after having studied for at least 60 minutes.
1857:
1800:
4199:
4339:
4204:
1307:
746:
99:
31:
3455:
2050:, on the other hand, is the probability that an event occurs given that another specific event
4381:
4327:
4317:
1994:
1315:
134:
2054:
occurred. This means that the calculation for one variable is dependent on another variable.
3549:
1532:
1494:
3726:
2064:
1586:
1296:
106:
51:
956:, and then averaging this conditional probability over the distribution of all values of
3880:{\displaystyle p_{X_{i}}(k)=\sum p(x_{1},x_{2},\dots ,x_{i-1},k,x_{i+1},\dots ,x_{n});}
964:
773:
47:
2340:
Suppose there is data from a classroom of 200 students on the amount of time studied (
141:. Naturally, the converse is also true: the marginal distribution can be obtained for
4398:
2964:
81:
4314:
A modern introduction to probability and statistics : understanding why and how
62:, which gives the probabilities contingent upon the values of the other variables.
3674:
4245:"Marginal & Conditional Probability Distributions: Definition & Examples"
17:
2044:
is the probability of a single event occurring, independent of other events. A
35:
4331:
1068:{\displaystyle \operatorname {E} _{Y}=\int _{y}f(y)p_{Y}(y)\,\mathrm {d} y.}
2206:{\displaystyle p_{Y|X}(y|x)=P(Y=y\mid X=x)={\frac {P(X=x,Y=y)}{P_{X}(x)}}}
80:
The context here is that the theoretical studies being undertaken, or the
4268:
737:
Joint and marginal distributions of a pair of discrete random variables,
133:
are not taken into consideration. This can be calculated by summing the
4376:
Dekking, F. M.; Kraaikamp, C.; Lopuhaä, H. P.; Meester, L. E. (2005).
1579:
from the joint cumulative distribution function is easy. Recall that:
43:
1776:{\displaystyle F(x,y)=\int _{a}^{x}\int _{c}^{y}f(x',y')\,dy'dx'}
2976:
can be used to determine how many students scored 20 or below:
2325:{\displaystyle f_{Y|X}(y|x)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}}
3687:, formulae similar to those above apply with the symbols
2356:
are discrete random variables, the joint distribution of
948:
is computed by examining the conditional probability of
2364:
can be described by listing all the possible values of
1924:
3922:
3740:
3552:
3458:
3167:
2983:
2225:
2074:
1997:
1860:
1803:
1670:
1596:
1535:
1497:
1415:
1335:
1102:
977:
782:
246:
158:
321:{\displaystyle p_{Y}(y_{j})=\sum _{i}p(x_{i},y_{j})}
233:{\displaystyle p_{X}(x_{i})=\sum _{j}p(x_{i},y_{j})}
4378:
A modern introduction to probability and statistics
4229:Trumpler, Robert J. & Harold F. Weaver (1962).
4316:. Dekking, Michel, 1946-. London: Springer. 2005.
4173:
3879:
3573:
3479:
3407:
3142:
2324:
2205:
2031:Marginal distribution vs. conditional distribution
2019:
1983:
1903:
1846:
1775:
1650:
1559:
1521:
1480:
1400:
1277:
1067:
934:
320:
232:
117:, the marginal distribution of either variable –
1984:{\textstyle F_{X}(x)=\lim _{y\to \infty }F(x,y)}
1948:
1481:{\displaystyle f_{Y}(y)=\int _{a}^{b}f(x,y)\,dx}
1401:{\displaystyle f_{X}(x)=\int _{c}^{d}f(x,y)\,dy}
1238:
1213:
8:
928:
4162:
4140:
4121:
4105:
4092:
4076:
4057:
4044:
4028:
4020:
4007:
3999:
3989:
3981:
3971:
3963:
3947:
3932:
3927:
3921:
3865:
3840:
3815:
3796:
3783:
3750:
3745:
3739:
3551:
3457:
3395:
3382:
3368:
3355:
3349:
3334:
3307:
3288:
3269:
3257:
3242:
3236:
3208:
3199:
3193:
3176:
3172:
3166:
3130:
3117:
3104:
3092:
3079:
3063:
3052:
3036:
3017:
3001:
2988:
2982:
2344:) and the percentage of correct answers (
2304:
2271:
2264:
2250:
2234:
2230:
2224:
2185:
2146:
2099:
2083:
2079:
2073:
2002:
1996:
1951:
1929:
1923:
1865:
1859:
1808:
1802:
1750:
1716:
1711:
1701:
1696:
1669:
1651:{\displaystyle F(x,y)=P(X\leq x,Y\leq y)}
1595:
1571:Marginal cumulative distribution function
1534:
1496:
1471:
1447:
1442:
1420:
1414:
1391:
1367:
1362:
1340:
1334:
1264:
1263:
1248:
1243:
1237:
1236:
1212:
1211:
1202:
1187:
1186:
1171:
1166:
1139:
1129:
1107:
1101:
1054:
1053:
1038:
1016:
982:
976:
944:Intuitively, the marginal probability of
898:
882:
867:
866:
851:
846:
819:
809:
787:
781:
309:
296:
280:
264:
251:
245:
221:
208:
192:
176:
163:
157:
3673:
3542:
3450:
2384:
330:
4221:
145:by summing over the separate values of
4337:
4294:Marginal and conditional distributions
4363:Everitt, B. S.; Skrondal, A. (2010).
4233:. Dover Publications. pp. 32–33.
1288:Marginal probability density function
7:
3439:However, in trying to calculate the
963:This follows from the definition of
27:Aspect of probability and statistics
4405:Theory of probability distributions
1314:can be obtained by integrating the
969:law of the unconscious statistician
4365:Cambridge Dictionary of Statistics
4029:
4024:
4008:
4003:
3990:
3985:
3972:
3967:
1958:
1265:
1188:
1055:
979:
879:
868:
94:Marginal probability mass function
25:
4190:Compound probability distribution
745:, dependent, thus having nonzero
1918:is ∞, then this becomes a limit
1794:jointly take values on × then
1577:cumulative distribution function
137:distribution over all values of
1904:{\displaystyle F_{Y}(y)=F(b,y)}
1847:{\displaystyle F_{X}(x)=F(x,d)}
4195:Joint probability distribution
4082:
4037:
3953:
3940:
3871:
3776:
3764:
3758:
3568:
3556:
3474:
3462:
3434:joint probability distribution
3340:
3321:
3313:
3275:
3263:
3243:
3223:
3214:
3200:
3186:
3177:
3098:
3072:
3042:
3023:
3007:
2994:
2316:
2310:
2295:
2283:
2258:
2251:
2244:
2235:
2197:
2191:
2176:
2152:
2140:
2116:
2107:
2100:
2093:
2084:
2014:
2008:
1978:
1966:
1955:
1941:
1935:
1898:
1886:
1877:
1871:
1841:
1829:
1820:
1814:
1747:
1725:
1686:
1674:
1645:
1621:
1612:
1600:
1554:
1542:
1516:
1504:
1468:
1456:
1432:
1426:
1388:
1376:
1352:
1346:
1260:
1254:
1233:
1227:
1183:
1177:
1163:
1151:
1119:
1113:
1050:
1044:
1031:
1025:
1006:
1003:
997:
991:
925:
922:
910:
891:
863:
857:
843:
831:
799:
793:
315:
289:
270:
257:
227:
201:
182:
169:
1:
3151:, meaning 10 students or 5%.
3914:probability density function
3532:
3529:
3526:
3518:
3515:
3512:
2957:
2940:
2923:
2906:
2889:
2840:
2823:
2806:
2803:
1312:probability density function
1310:is known, then the marginal
1081:and another random variable
952:given a particular value of
772:can always be written as an
4269:"Exam P [FSU Math]"
3910:continuous random variables
2217:continuous random variables
1662:continuous random variables
4421:
4380:. London : Springer.
4369:Cambridge University Press
3685:multivariate distributions
3670:Multivariate distributions
3658:
3655:
3652:
3649:
3641:
3638:
3635:
3632:
3624:
3621:
3618:
3615:
3480:{\displaystyle P(H\mid L)}
3452:Conditional distribution:
3732:probability mass function
2422:
2413:
2404:
2399:
4210:Conditional distribution
3603:
3600:
3597:
3594:
3504:
3501:
3498:
3426:discrete random variable
3157:conditional distribution
2382:), as shown in Table.3.
2020:{\displaystyle F_{Y}(y)}
1326:and vice versa. That is
123:probability distribution
60:conditional distribution
56:probability distribution
3604:Marginal probability P(
2400:Time studied (minutes)
2047:conditional probability
4344:: CS1 maint: others (
4175:
3881:
3680:
3575:
3574:{\displaystyle P(H,L)}
3481:
3409:
3144:
3068:
2326:
2207:
2021:
1985:
1905:
1848:
1777:
1652:
1561:
1523:
1482:
1402:
1279:
1069:
936:
322:
234:
4231:Statistical Astronomy
4176:
3882:
3677:
3576:
3482:
3410:
3145:
3048:
2974:marginal distribution
2327:
2208:
2022:
1986:
1906:
1849:
1778:
1653:
1575:Finding the marginal
1562:
1560:{\displaystyle y\in }
1524:
1522:{\displaystyle x\in }
1483:
1403:
1280:
1070:
937:
323:
235:
121:for example – is the
40:marginal distribution
3920:
3912:, then the marginal
3738:
3730:, then the marginal
3550:
3544:Joint distribution:
3456:
3441:marginal probability
3165:
2981:
2223:
2072:
2042:marginal probability
1995:
1922:
1858:
1801:
1668:
1594:
1533:
1495:
1413:
1333:
1100:
975:
967:(after applying the
780:
770:marginal probability
244:
156:
4200:Marginal likelihood
4033:
4012:
3994:
3976:
3583:
3487:
2969:
1721:
1706:
1452:
1372:
765:
129:when the values of
4205:Wasserstein metric
4171:
4016:
3995:
3977:
3959:
3877:
3681:
3666:Not Hit row.
3571:
3543:
3477:
3451:
3420:Real-world example
3405:
3140:
2962:
2322:
2203:
2017:
1981:
1962:
1901:
1844:
1773:
1707:
1692:
1648:
1557:
1519:
1478:
1438:
1398:
1358:
1308:joint distribution
1275:
1065:
932:
747:mutual information
735:
318:
285:
230:
197:
100:joint distribution
66:Marginal variables
32:probability theory
3663:
3662:
3537:
3536:
3415:
3403:
3390:
3377:
3344:
3150:
3138:
3125:
3112:
2970:
2961:
2348:). Assuming that
2320:
2201:
1947:
1316:joint probability
766:
734:
276:
188:
135:joint probability
18:Marginalizing out
16:(Redirected from
4412:
4391:
4372:
4350:
4349:
4343:
4335:
4310:
4304:
4303:
4302:
4301:
4289:
4283:
4282:
4280:
4279:
4273:www.math.fsu.edu
4265:
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4167:
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4110:
4109:
4097:
4096:
4081:
4080:
4062:
4061:
4049:
4048:
4032:
4027:
4011:
4006:
3993:
3988:
3975:
3970:
3952:
3951:
3939:
3938:
3937:
3936:
3886:
3884:
3883:
3878:
3870:
3869:
3851:
3850:
3826:
3825:
3801:
3800:
3788:
3787:
3757:
3756:
3755:
3754:
3727:random variables
3584:
3582:
3580:
3578:
3577:
3572:
3488:
3486:
3484:
3483:
3478:
3414:
3412:
3411:
3406:
3404:
3396:
3391:
3383:
3378:
3376:
3372:
3363:
3359:
3350:
3345:
3343:
3339:
3338:
3316:
3312:
3311:
3293:
3292:
3270:
3262:
3261:
3246:
3241:
3240:
3213:
3212:
3203:
3198:
3197:
3185:
3184:
3180:
3161:
3149:
3147:
3146:
3141:
3139:
3131:
3126:
3118:
3113:
3105:
3097:
3096:
3084:
3083:
3067:
3062:
3041:
3040:
3022:
3021:
3006:
3005:
2993:
2992:
2977:
2968:
2955:
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2790:
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2739:
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2708:
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2513:
2511:
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2507:
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2490:
2488:
2487:
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2481:
2385:
2331:
2329:
2328:
2323:
2321:
2319:
2309:
2308:
2298:
2282:
2281:
2265:
2254:
2243:
2242:
2238:
2212:
2210:
2209:
2204:
2202:
2200:
2190:
2189:
2179:
2147:
2103:
2092:
2091:
2087:
2065:random variables
2026:
2024:
2023:
2018:
2007:
2006:
1990:
1988:
1987:
1982:
1961:
1934:
1933:
1910:
1908:
1907:
1902:
1870:
1869:
1853:
1851:
1850:
1845:
1813:
1812:
1782:
1780:
1779:
1774:
1772:
1761:
1746:
1735:
1720:
1715:
1705:
1700:
1657:
1655:
1654:
1649:
1587:random variables
1566:
1564:
1563:
1558:
1528:
1526:
1525:
1520:
1487:
1485:
1484:
1479:
1451:
1446:
1425:
1424:
1407:
1405:
1404:
1399:
1371:
1366:
1345:
1344:
1321:
1297:random variables
1284:
1282:
1281:
1276:
1268:
1253:
1252:
1242:
1241:
1217:
1216:
1207:
1206:
1191:
1176:
1175:
1150:
1149:
1134:
1133:
1112:
1111:
1095:
1074:
1072:
1071:
1066:
1058:
1043:
1042:
1021:
1020:
987:
986:
941:
939:
938:
933:
909:
908:
887:
886:
871:
856:
855:
830:
829:
814:
813:
792:
791:
764:
762:
731:
729:
728:
725:
722:
714:
712:
711:
708:
705:
697:
695:
694:
691:
688:
680:
678:
677:
674:
671:
663:
661:
660:
657:
654:
631:
629:
628:
625:
622:
605:
603:
602:
599:
596:
578:
576:
575:
572:
569:
561:
559:
558:
555:
552:
544:
542:
541:
538:
535:
527:
525:
524:
521:
518:
510:
508:
507:
504:
501:
483:
481:
480:
477:
474:
466:
464:
463:
460:
457:
449:
447:
446:
443:
440:
432:
430:
429:
426:
423:
415:
413:
412:
409:
406:
331:
327:
325:
324:
319:
314:
313:
301:
300:
284:
269:
268:
256:
255:
239:
237:
236:
231:
226:
225:
213:
212:
196:
181:
180:
168:
167:
148:
144:
140:
132:
128:
120:
116:
112:
107:random variables
75:marginalized out
52:random variables
21:
4420:
4419:
4415:
4414:
4413:
4411:
4410:
4409:
4395:
4394:
4388:
4375:
4362:
4359:
4354:
4353:
4336:
4324:
4312:
4311:
4307:
4299:
4297:
4291:
4290:
4286:
4277:
4275:
4267:
4266:
4262:
4253:
4251:
4243:
4242:
4238:
4228:
4227:
4223:
4218:
4186:
4158:
4136:
4117:
4101:
4088:
4072:
4053:
4040:
3943:
3928:
3923:
3918:
3917:
3906:
3900:
3893:
3861:
3836:
3811:
3792:
3779:
3746:
3741:
3736:
3735:
3721:
3715:
3708:
3702:That means, If
3672:
3592:
3589:
3548:
3547:
3545:
3496:
3493:
3454:
3453:
3422:
3364:
3351:
3330:
3317:
3303:
3284:
3271:
3253:
3232:
3204:
3189:
3168:
3163:
3162:
3088:
3075:
3032:
3013:
2997:
2984:
2979:
2978:
2963:
2950:
2947:
2944:
2943:
2941:
2933:
2930:
2927:
2926:
2924:
2916:
2913:
2910:
2909:
2907:
2899:
2896:
2893:
2892:
2890:
2881:
2867:
2864:
2861:
2860:
2858:
2850:
2847:
2844:
2843:
2841:
2833:
2830:
2827:
2826:
2824:
2816:
2813:
2810:
2809:
2807:
2800:
2785:
2782:
2779:
2778:
2776:
2768:
2765:
2762:
2761:
2759:
2751:
2748:
2745:
2744:
2742:
2734:
2731:
2728:
2727:
2725:
2718:
2703:
2700:
2697:
2696:
2694:
2686:
2683:
2680:
2679:
2677:
2669:
2666:
2663:
2662:
2660:
2652:
2649:
2646:
2645:
2643:
2635:
2632:
2629:
2628:
2626:
2622:
2607:
2604:
2601:
2600:
2598:
2587:
2584:
2581:
2580:
2578:
2570:
2567:
2564:
2563:
2561:
2553:
2550:
2547:
2546:
2544:
2540:
2525:
2522:
2519:
2518:
2516:
2508:
2505:
2502:
2501:
2499:
2485:
2482:
2479:
2478:
2476:
2472:
2457:
2446:
2437:
2428:
2419:
2409:
2397:
2392:
2380:
2373:
2338:
2300:
2299:
2267:
2266:
2226:
2221:
2220:
2181:
2180:
2148:
2075:
2070:
2069:
2038:
2033:
1998:
1993:
1992:
1991:. Likewise for
1925:
1920:
1919:
1861:
1856:
1855:
1804:
1799:
1798:
1765:
1754:
1739:
1728:
1666:
1665:
1592:
1591:
1573:
1531:
1530:
1493:
1492:
1416:
1411:
1410:
1336:
1331:
1330:
1319:
1290:
1244:
1198:
1167:
1135:
1125:
1103:
1098:
1097:
1082:
1034:
1012:
978:
973:
972:
894:
878:
847:
815:
805:
783:
778:
777:
749:
736:
726:
723:
720:
719:
717:
709:
706:
703:
702:
700:
692:
689:
686:
685:
683:
675:
672:
669:
668:
666:
658:
655:
652:
651:
649:
640:
626:
623:
620:
619:
617:
600:
597:
594:
593:
591:
588:
573:
570:
567:
566:
564:
556:
553:
550:
549:
547:
539:
536:
533:
532:
530:
522:
519:
516:
515:
513:
505:
502:
499:
498:
496:
493:
478:
475:
472:
471:
469:
461:
458:
455:
454:
452:
444:
441:
438:
437:
435:
427:
424:
421:
420:
418:
410:
407:
404:
403:
401:
398:
382:
375:
367:
359:
351:
343:
338:
305:
292:
260:
247:
242:
241:
217:
204:
172:
159:
154:
153:
146:
142:
138:
130:
126:
118:
114:
110:
96:
91:
28:
23:
22:
15:
12:
11:
5:
4418:
4416:
4408:
4407:
4397:
4396:
4393:
4392:
4386:
4373:
4358:
4355:
4352:
4351:
4322:
4305:
4284:
4260:
4236:
4220:
4219:
4217:
4214:
4213:
4212:
4207:
4202:
4197:
4192:
4185:
4182:
4170:
4165:
4161:
4157:
4154:
4149:
4146:
4143:
4139:
4135:
4130:
4127:
4124:
4120:
4116:
4113:
4108:
4104:
4100:
4095:
4091:
4087:
4084:
4079:
4075:
4071:
4068:
4065:
4060:
4056:
4052:
4047:
4043:
4039:
4036:
4031:
4026:
4023:
4019:
4015:
4010:
4005:
4002:
3998:
3992:
3987:
3984:
3980:
3974:
3969:
3966:
3962:
3958:
3955:
3950:
3946:
3942:
3935:
3931:
3926:
3904:
3898:
3891:
3876:
3873:
3868:
3864:
3860:
3857:
3854:
3849:
3846:
3843:
3839:
3835:
3832:
3829:
3824:
3821:
3818:
3814:
3810:
3807:
3804:
3799:
3795:
3791:
3786:
3782:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3753:
3749:
3744:
3719:
3713:
3706:
3671:
3668:
3661:
3660:
3657:
3654:
3651:
3648:
3644:
3643:
3640:
3637:
3634:
3631:
3627:
3626:
3623:
3620:
3617:
3614:
3610:
3609:
3602:
3599:
3596:
3593:
3590:
3587:
3570:
3567:
3564:
3561:
3558:
3555:
3535:
3534:
3531:
3528:
3525:
3521:
3520:
3517:
3514:
3511:
3507:
3506:
3503:
3500:
3497:
3494:
3491:
3476:
3473:
3470:
3467:
3464:
3461:
3421:
3418:
3402:
3399:
3394:
3389:
3386:
3381:
3375:
3371:
3367:
3362:
3358:
3354:
3348:
3342:
3337:
3333:
3329:
3326:
3323:
3320:
3315:
3310:
3306:
3302:
3299:
3296:
3291:
3287:
3283:
3280:
3277:
3274:
3268:
3265:
3260:
3256:
3252:
3249:
3245:
3239:
3235:
3231:
3228:
3225:
3222:
3219:
3216:
3211:
3207:
3202:
3196:
3192:
3188:
3183:
3179:
3175:
3171:
3137:
3134:
3129:
3124:
3121:
3116:
3111:
3108:
3103:
3100:
3095:
3091:
3087:
3082:
3078:
3074:
3071:
3066:
3061:
3058:
3055:
3051:
3047:
3044:
3039:
3035:
3031:
3028:
3025:
3020:
3016:
3012:
3009:
3004:
3000:
2996:
2991:
2987:
2960:
2959:
2956:
2939:
2922:
2905:
2888:
2879:
2874:
2873:
2856:
2839:
2822:
2805:
2802:
2798:
2792:
2791:
2774:
2757:
2740:
2723:
2720:
2716:
2710:
2709:
2692:
2675:
2658:
2641:
2624:
2620:
2614:
2613:
2596:
2593:
2576:
2559:
2542:
2538:
2532:
2531:
2514:
2497:
2494:
2491:
2474:
2470:
2464:
2463:
2453:
2448:
2444:
2439:
2435:
2430:
2426:
2421:
2417:
2412:
2410:
2405:
2402:
2401:
2398:
2393:
2388:
2378:
2371:
2337:
2334:
2333:
2332:
2318:
2315:
2312:
2307:
2303:
2297:
2294:
2291:
2288:
2285:
2280:
2277:
2274:
2270:
2263:
2260:
2257:
2253:
2249:
2246:
2241:
2237:
2233:
2229:
2213:
2199:
2196:
2193:
2188:
2184:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2102:
2098:
2095:
2090:
2086:
2082:
2078:
2037:
2034:
2032:
2029:
2016:
2013:
2010:
2005:
2001:
1980:
1977:
1974:
1971:
1968:
1965:
1960:
1957:
1954:
1950:
1946:
1943:
1940:
1937:
1932:
1928:
1912:
1911:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1868:
1864:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1811:
1807:
1784:
1783:
1771:
1768:
1764:
1760:
1757:
1753:
1749:
1745:
1742:
1738:
1734:
1731:
1727:
1724:
1719:
1714:
1710:
1704:
1699:
1695:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1658:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1572:
1569:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1489:
1488:
1477:
1474:
1470:
1467:
1464:
1461:
1458:
1455:
1450:
1445:
1441:
1437:
1434:
1431:
1428:
1423:
1419:
1408:
1397:
1394:
1390:
1387:
1384:
1381:
1378:
1375:
1370:
1365:
1361:
1357:
1354:
1351:
1348:
1343:
1339:
1318:distribution,
1289:
1286:
1274:
1271:
1267:
1262:
1259:
1256:
1251:
1247:
1240:
1235:
1232:
1229:
1226:
1223:
1220:
1215:
1210:
1205:
1201:
1197:
1194:
1190:
1185:
1182:
1179:
1174:
1170:
1165:
1162:
1159:
1156:
1153:
1148:
1145:
1142:
1138:
1132:
1128:
1124:
1121:
1118:
1115:
1110:
1106:
1064:
1061:
1057:
1052:
1049:
1046:
1041:
1037:
1033:
1030:
1027:
1024:
1019:
1015:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
985:
981:
965:expected value
931:
927:
924:
921:
918:
915:
912:
907:
904:
901:
897:
893:
890:
885:
881:
877:
874:
870:
865:
862:
859:
854:
850:
845:
842:
839:
836:
833:
828:
825:
822:
818:
812:
808:
804:
801:
798:
795:
790:
786:
774:expected value
733:
732:
715:
698:
681:
664:
647:
638:
633:
632:
615:
612:
609:
606:
589:
586:
580:
579:
562:
545:
528:
511:
494:
491:
485:
484:
467:
450:
433:
416:
399:
396:
390:
389:
380:
376:
373:
368:
365:
360:
357:
352:
349:
344:
339:
334:
329:
328:
317:
312:
308:
304:
299:
295:
291:
288:
283:
279:
275:
272:
267:
263:
259:
254:
250:
229:
224:
220:
216:
211:
207:
203:
200:
195:
191:
187:
184:
179:
175:
171:
166:
162:
98:Given a known
95:
92:
90:
87:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4417:
4406:
4403:
4402:
4400:
4389:
4387:9781852338961
4383:
4379:
4374:
4370:
4366:
4361:
4360:
4356:
4347:
4341:
4333:
4329:
4325:
4323:9781852338961
4319:
4315:
4309:
4306:
4296:
4295:
4288:
4285:
4274:
4270:
4264:
4261:
4250:
4246:
4240:
4237:
4232:
4225:
4222:
4215:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4187:
4183:
4181:
4168:
4163:
4159:
4155:
4152:
4147:
4144:
4141:
4137:
4133:
4128:
4125:
4122:
4118:
4114:
4111:
4106:
4102:
4098:
4093:
4089:
4085:
4077:
4073:
4069:
4066:
4063:
4058:
4054:
4050:
4045:
4041:
4034:
4021:
4017:
4013:
4000:
3996:
3982:
3978:
3964:
3960:
3956:
3948:
3944:
3933:
3929:
3924:
3915:
3911:
3907:
3897:
3890:
3874:
3866:
3862:
3858:
3855:
3852:
3847:
3844:
3841:
3837:
3833:
3830:
3827:
3822:
3819:
3816:
3812:
3808:
3805:
3802:
3797:
3793:
3789:
3784:
3780:
3773:
3770:
3767:
3761:
3751:
3747:
3742:
3733:
3729:
3728:
3722:
3712:
3705:
3700:
3698:
3694:
3690:
3686:
3676:
3669:
3667:
3646:
3645:
3629:
3628:
3612:
3611:
3607:
3586:
3585:
3565:
3562:
3559:
3553:
3541:
3523:
3522:
3509:
3508:
3490:
3489:
3471:
3468:
3465:
3459:
3449:
3445:
3442:
3437:
3435:
3429:
3427:
3419:
3417:
3400:
3397:
3392:
3387:
3384:
3379:
3373:
3369:
3365:
3360:
3356:
3352:
3346:
3335:
3331:
3327:
3324:
3318:
3308:
3304:
3300:
3297:
3294:
3289:
3285:
3281:
3278:
3272:
3266:
3258:
3254:
3250:
3247:
3237:
3233:
3229:
3226:
3220:
3217:
3209:
3205:
3194:
3190:
3181:
3173:
3169:
3159:
3158:
3152:
3135:
3132:
3127:
3122:
3119:
3114:
3109:
3106:
3101:
3093:
3089:
3085:
3080:
3076:
3069:
3064:
3059:
3056:
3053:
3049:
3045:
3037:
3033:
3029:
3026:
3018:
3014:
3010:
3002:
2998:
2989:
2985:
2975:
2966:
2965:Two-way table
2886:
2882:
2876:
2875:
2857:
2797:
2794:
2793:
2775:
2758:
2741:
2724:
2721:
2715:
2712:
2711:
2693:
2676:
2659:
2642:
2625:
2619:
2616:
2615:
2597:
2594:
2577:
2560:
2543:
2537:
2534:
2533:
2515:
2498:
2495:
2492:
2475:
2469:
2466:
2465:
2461:
2456:
2452:
2449:
2443:
2440:
2434:
2431:
2425:
2416:
2411:
2408:
2403:
2396:
2391:
2387:
2386:
2383:
2381:
2374:
2367:
2363:
2359:
2355:
2351:
2347:
2343:
2335:
2313:
2305:
2301:
2292:
2289:
2286:
2278:
2275:
2272:
2268:
2261:
2255:
2247:
2239:
2231:
2227:
2218:
2214:
2194:
2186:
2182:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2149:
2143:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2113:
2110:
2104:
2096:
2088:
2080:
2076:
2067:
2066:
2060:
2059:
2058:
2055:
2053:
2049:
2048:
2043:
2035:
2030:
2028:
2011:
2003:
1999:
1975:
1972:
1969:
1963:
1952:
1944:
1938:
1930:
1926:
1917:
1895:
1892:
1889:
1883:
1880:
1874:
1866:
1862:
1838:
1835:
1832:
1826:
1823:
1817:
1809:
1805:
1797:
1796:
1795:
1793:
1789:
1769:
1766:
1762:
1758:
1755:
1751:
1743:
1740:
1736:
1732:
1729:
1722:
1717:
1712:
1708:
1702:
1697:
1693:
1689:
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1680:
1677:
1671:
1663:
1659:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1618:
1615:
1609:
1606:
1603:
1597:
1589:
1588:
1582:
1581:
1580:
1578:
1570:
1568:
1551:
1548:
1545:
1539:
1536:
1513:
1510:
1507:
1501:
1498:
1475:
1472:
1465:
1462:
1459:
1453:
1448:
1443:
1439:
1435:
1429:
1421:
1417:
1409:
1395:
1392:
1385:
1382:
1379:
1373:
1368:
1363:
1359:
1355:
1349:
1341:
1337:
1329:
1328:
1327:
1325:
1317:
1313:
1309:
1305:
1301:
1298:
1295:
1287:
1285:
1272:
1269:
1257:
1249:
1245:
1230:
1224:
1221:
1218:
1208:
1203:
1199:
1195:
1192:
1180:
1172:
1168:
1160:
1157:
1154:
1146:
1143:
1140:
1136:
1130:
1126:
1122:
1116:
1108:
1104:
1093:
1089:
1085:
1080:
1075:
1062:
1059:
1047:
1039:
1035:
1028:
1022:
1017:
1013:
1009:
1000:
994:
988:
983:
970:
966:
961:
959:
955:
951:
947:
942:
929:
919:
916:
913:
905:
902:
899:
895:
888:
883:
875:
872:
860:
852:
848:
840:
837:
834:
826:
823:
820:
816:
810:
806:
802:
796:
788:
784:
775:
771:
760:
756:
752:
748:
744:
740:
716:
699:
682:
665:
648:
645:
641:
635:
634:
616:
613:
610:
607:
590:
585:
582:
581:
563:
546:
529:
512:
495:
490:
487:
486:
468:
451:
434:
417:
400:
395:
392:
391:
387:
383:
377:
372:
369:
364:
361:
356:
353:
348:
345:
342:
337:
333:
332:
310:
306:
302:
297:
293:
286:
281:
277:
273:
265:
261:
252:
248:
222:
218:
214:
209:
205:
198:
193:
189:
185:
177:
173:
164:
160:
152:
151:
150:
136:
124:
108:
105:
101:
93:
88:
86:
83:
82:data analysis
78:
76:
72:
71:marginalizing
67:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4377:
4364:
4357:Bibliography
4313:
4308:
4298:, retrieved
4293:
4287:
4276:. Retrieved
4272:
4263:
4252:. Retrieved
4248:
4239:
4230:
4224:
3909:
3902:
3895:
3888:
3724:
3717:
3710:
3703:
3701:
3696:
3692:
3688:
3682:
3679:coordinates.
3664:
3605:
3538:
3446:
3440:
3438:
3430:
3423:
3155:
3153:
2973:
2971:
2884:
2877:
2795:
2713:
2617:
2535:
2467:
2459:
2454:
2450:
2441:
2432:
2423:
2414:
2406:
2394:
2389:
2376:
2369:
2365:
2361:
2357:
2353:
2349:
2345:
2341:
2339:
2216:
2062:
2056:
2051:
2045:
2041:
2039:
1915:
1913:
1791:
1787:
1785:
1661:
1584:
1574:
1490:
1323:
1303:
1299:
1293:
1291:
1091:
1087:
1083:
1078:
1076:
962:
957:
953:
949:
945:
943:
769:
767:
758:
754:
750:
742:
738:
643:
636:
583:
488:
393:
385:
378:
370:
362:
354:
346:
340:
335:
103:
97:
79:
74:
70:
69:obtained by
65:
64:
39:
29:
2052:has already
4300:2019-11-16
4278:2019-11-16
4254:2019-11-16
4216:References
3916:should be
3734:should be
2036:Definition
1294:continuous
1292:Given two
89:Definition
48:collection
36:statistics
4340:cite book
4332:262680588
4249:Study.com
4153:⋯
4126:−
4112:⋯
4067:…
4030:∞
4025:∞
4022:−
4018:∫
4014:⋯
4009:∞
4004:∞
4001:−
3997:∫
3991:∞
3986:∞
3983:−
3979:∫
3973:∞
3968:∞
3965:−
3961:∫
3856:…
3820:−
3806:…
3771:∑
3725:discrete
3469:∣
3050:∑
2801:(80-100)
2447:(>60)
2407:% correct
2129:∣
2063:discrete
1959:∞
1956:→
1709:∫
1694:∫
1640:≤
1628:≤
1585:discrete
1540:∈
1502:∈
1440:∫
1360:∫
1222:−
1209:δ
1200:∫
1158:∣
1144:∣
1127:∫
1014:∫
989:
917:∣
903:∣
889:
838:∣
824:∣
807:∫
278:∑
190:∑
4399:Category
4184:See also
3613:Not Hit
3510:Not Hit
2719:(60-79)
2623:(41-59)
2541:(21-40)
2438:(41-60)
2429:(21-40)
1770:′
1759:′
1744:′
1733:′
104:discrete
3691:and/or
3598:Yellow
3581:
3546:
3502:Yellow
2954:
2942:
2937:
2925:
2920:
2908:
2903:
2891:
2871:
2859:
2854:
2842:
2837:
2825:
2820:
2808:
2789:
2777:
2772:
2760:
2755:
2743:
2738:
2726:
2707:
2695:
2690:
2678:
2673:
2661:
2656:
2644:
2639:
2627:
2611:
2599:
2591:
2579:
2574:
2562:
2557:
2545:
2529:
2517:
2512:
2500:
2489:
2477:
2473:(0-20)
2420:(0-20)
2336:Example
1322:, over
730:
718:
713:
701:
696:
684:
679:
667:
662:
650:
630:
618:
604:
592:
577:
565:
560:
548:
543:
531:
526:
514:
509:
497:
482:
470:
465:
453:
448:
436:
431:
419:
414:
402:
109:, say,
102:of two
54:is the
4384:
4330:
4320:
3647:Total
3642:0.572
3633:0.002
3625:0.428
3616:0.198
3601:Green
3505:Green
1529:, and
1491:where
1306:whose
240:, and
44:subset
38:, the
3639:0.56
3636:0.01
3622:0.14
3619:0.09
3527:0.01
3513:0.99
46:of a
42:of a
4382:ISBN
4346:link
4328:OCLC
4318:ISBN
3908:are
3723:are
3683:For
3656:0.7
3653:0.1
3650:0.2
3630:Hit
3595:Red
3533:0.8
3530:0.1
3524:Hit
3519:0.2
3516:0.9
3499:Red
3154:The
2972:The
2887:) →
2462:) ↓
2360:and
2352:and
2215:For
2061:For
2040:The
1854:and
1790:and
1660:For
1583:For
1302:and
741:and
646:) →
388:) ↓
113:and
34:and
3901:,…,
3887:if
3716:,…,
3374:200
3361:200
3136:200
3123:200
3110:200
2951:200
2934:200
2917:200
2900:200
2868:200
2851:200
2834:200
2817:200
2786:200
2769:200
2752:200
2735:200
2704:200
2687:200
2670:200
2653:200
2636:200
2608:200
2588:200
2571:200
2554:200
2526:200
2509:200
2486:200
1949:lim
1914:If
1786:If
125:of
50:of
30:In
4401::
4367:.
4342:}}
4338:{{
4326:.
4271:.
4247:.
3699:.
3659:1
3608:)
3401:35
3388:70
3366:70
3133:10
2958:1
2945:70
2928:86
2911:30
2894:14
2862:40
2845:20
2828:16
2804:0
2780:60
2763:10
2746:30
2729:20
2722:0
2698:70
2681:32
2664:32
2602:20
2595:0
2548:10
2520:10
2496:0
2493:0
2027:.
1664:,
1590:,
1567:.
1324:Y,
1096::
971:)
960:.
776::
768:A
757:;
727:32
721:32
710:32
693:32
676:32
659:32
653:16
627:32
614:0
601:32
574:32
568:15
557:32
540:32
523:32
506:32
479:32
462:32
445:32
428:32
411:32
149:.
77:.
4390:.
4371:.
4348:)
4334:.
4281:.
4257:.
4169:.
4164:n
4160:x
4156:d
4148:1
4145:+
4142:i
4138:x
4134:d
4129:1
4123:i
4119:x
4115:d
4107:2
4103:x
4099:d
4094:1
4090:x
4086:d
4083:)
4078:n
4074:x
4070:,
4064:,
4059:2
4055:x
4051:,
4046:1
4042:x
4038:(
4035:f
3957:=
3954:)
3949:i
3945:x
3941:(
3934:i
3930:X
3925:f
3905:n
3903:X
3899:2
3896:X
3894:,
3892:1
3889:X
3875:;
3872:)
3867:n
3863:x
3859:,
3853:,
3848:1
3845:+
3842:i
3838:x
3834:,
3831:k
3828:,
3823:1
3817:i
3813:x
3809:,
3803:,
3798:2
3794:x
3790:,
3785:1
3781:x
3777:(
3774:p
3768:=
3765:)
3762:k
3759:(
3752:i
3748:X
3743:p
3720:n
3718:X
3714:2
3711:X
3709:,
3707:1
3704:X
3697:X
3693:Y
3689:X
3606:H
3591:H
3588:L
3569:)
3566:L
3563:,
3560:H
3557:(
3554:P
3495:H
3492:L
3475:)
3472:L
3466:H
3463:(
3460:P
3398:4
3393:=
3385:8
3380:=
3370:/
3357:/
3353:8
3347:=
3341:)
3336:4
3332:x
3328:=
3325:X
3322:(
3319:P
3314:)
3309:1
3305:y
3301:=
3298:Y
3295:,
3290:4
3286:x
3282:=
3279:X
3276:(
3273:P
3267:=
3264:)
3259:4
3255:x
3251:=
3248:X
3244:|
3238:1
3234:y
3230:=
3227:Y
3224:(
3221:P
3218:=
3215:)
3210:4
3206:x
3201:|
3195:1
3191:y
3187:(
3182:X
3178:|
3174:Y
3170:p
3128:=
3120:8
3115:+
3107:2
3102:=
3099:)
3094:1
3090:y
3086:,
3081:i
3077:x
3073:(
3070:P
3065:4
3060:1
3057:=
3054:i
3046:=
3043:)
3038:1
3034:y
3030:=
3027:Y
3024:(
3019:Y
3015:P
3011:=
3008:)
3003:1
2999:y
2995:(
2990:Y
2986:p
2948:/
2931:/
2914:/
2897:/
2885:x
2883:(
2880:X
2878:p
2865:/
2848:/
2831:/
2814:/
2811:4
2799:5
2796:y
2783:/
2766:/
2749:/
2732:/
2717:4
2714:y
2701:/
2684:/
2667:/
2650:/
2647:4
2633:/
2630:2
2621:3
2618:y
2605:/
2585:/
2582:8
2568:/
2565:2
2551:/
2539:2
2536:y
2523:/
2506:/
2503:8
2483:/
2480:2
2471:1
2468:y
2460:y
2458:(
2455:Y
2451:p
2445:4
2442:x
2436:3
2433:x
2427:2
2424:x
2418:1
2415:x
2395:Y
2390:X
2379:j
2377:y
2375:,
2372:i
2370:x
2368:(
2366:p
2362:Y
2358:X
2354:Y
2350:X
2346:Y
2342:X
2317:)
2314:x
2311:(
2306:X
2302:f
2296:)
2293:y
2290:,
2287:x
2284:(
2279:Y
2276:,
2273:X
2269:f
2262:=
2259:)
2256:x
2252:|
2248:y
2245:(
2240:X
2236:|
2232:Y
2228:f
2219:,
2198:)
2195:x
2192:(
2187:X
2183:P
2177:)
2174:y
2171:=
2168:Y
2165:,
2162:x
2159:=
2156:X
2153:(
2150:P
2144:=
2141:)
2138:x
2135:=
2132:X
2126:y
2123:=
2120:Y
2117:(
2114:P
2111:=
2108:)
2105:x
2101:|
2097:y
2094:(
2089:X
2085:|
2081:Y
2077:p
2068:,
2015:)
2012:y
2009:(
2004:Y
2000:F
1979:)
1976:y
1973:,
1970:x
1967:(
1964:F
1953:y
1945:=
1942:)
1939:x
1936:(
1931:X
1927:F
1916:d
1899:)
1896:y
1893:,
1890:b
1887:(
1884:F
1881:=
1878:)
1875:y
1872:(
1867:Y
1863:F
1842:)
1839:d
1836:,
1833:x
1830:(
1827:F
1824:=
1821:)
1818:x
1815:(
1810:X
1806:F
1792:Y
1788:X
1767:x
1763:d
1756:y
1752:d
1748:)
1741:y
1737:,
1730:x
1726:(
1723:f
1718:y
1713:c
1703:x
1698:a
1690:=
1687:)
1684:y
1681:,
1678:x
1675:(
1672:F
1646:)
1643:y
1637:Y
1634:,
1631:x
1625:X
1622:(
1619:P
1616:=
1613:)
1610:y
1607:,
1604:x
1601:(
1598:F
1555:]
1552:d
1549:,
1546:c
1543:[
1537:y
1517:]
1514:b
1511:,
1508:a
1505:[
1499:x
1476:x
1473:d
1469:)
1466:y
1463:,
1460:x
1457:(
1454:f
1449:b
1444:a
1436:=
1433:)
1430:y
1427:(
1422:Y
1418:f
1396:y
1393:d
1389:)
1386:y
1383:,
1380:x
1377:(
1374:f
1369:d
1364:c
1356:=
1353:)
1350:x
1347:(
1342:X
1338:f
1320:f
1304:Y
1300:X
1273:.
1270:y
1266:d
1261:)
1258:y
1255:(
1250:Y
1246:p
1239:)
1234:)
1231:y
1228:(
1225:g
1219:x
1214:(
1204:y
1196:=
1193:y
1189:d
1184:)
1181:y
1178:(
1173:Y
1169:p
1164:)
1161:y
1155:x
1152:(
1147:Y
1141:X
1137:p
1131:y
1123:=
1120:)
1117:x
1114:(
1109:X
1105:p
1094:)
1092:Y
1090:(
1088:g
1086:=
1084:X
1079:Y
1063:.
1060:y
1056:d
1051:)
1048:y
1045:(
1040:Y
1036:p
1032:)
1029:y
1026:(
1023:f
1018:y
1010:=
1007:]
1004:)
1001:Y
998:(
995:f
992:[
984:Y
980:E
958:Y
954:Y
950:X
946:X
930:.
926:]
923:)
920:y
914:x
911:(
906:Y
900:X
896:p
892:[
884:Y
880:E
876:=
873:y
869:d
864:)
861:y
858:(
853:Y
849:p
844:)
841:y
835:x
832:(
827:Y
821:X
817:p
811:y
803:=
800:)
797:x
794:(
789:X
785:p
761:)
759:Y
755:X
753:(
751:I
743:Y
739:X
724:/
707:/
704:4
690:/
687:4
673:/
670:8
656:/
644:x
642:(
639:X
637:p
624:/
621:9
611:0
608:0
598:/
595:9
587:3
584:y
571:/
554:/
551:3
537:/
534:3
520:/
517:6
503:/
500:3
492:2
489:y
476:/
473:8
459:/
456:1
442:/
439:1
425:/
422:2
408:/
405:4
397:1
394:y
386:y
384:(
381:Y
379:p
374:4
371:x
366:3
363:x
358:2
355:x
350:1
347:x
341:Y
336:X
316:)
311:j
307:y
303:,
298:i
294:x
290:(
287:p
282:i
274:=
271:)
266:j
262:y
258:(
253:Y
249:p
228:)
223:j
219:y
215:,
210:i
206:x
202:(
199:p
194:j
186:=
183:)
178:i
174:x
170:(
165:X
161:p
147:X
143:Y
139:Y
131:Y
127:X
119:X
115:Y
111:X
20:)
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