2286:
31:
2787:
2201:
1993:
2435:. Here the multiple random variables are the numbers of successes in each of the categories after a given number of trials, and each non-zero probability mass gives the probability of a certain combination of numbers of successes in the various categories.
2532:
2424:
If the discrete distribution has two or more categories one of which may occur, whether or not these categories have a natural ordering, when there is only a single trial (draw) this is a categorical distribution.
2058:
1311:
2205:
Binomial distribution, models the number of successes when someone draws n times with replacement. Each draw or experiment is independent, with two possible outcomes. The associated probability mass function is
1892:
2737:
805:
2534:
Despite the infinite number of possible outcomes, the total probability mass is 1/2 + 1/4 + 1/8 + ⯠= 1, satisfying the unit total probability requirement for a probability distribution.
2281:
896:
254:
967:
589:
400:
2548:
Two or more discrete random variables have a joint probability mass function, which gives the probability of each possible combination of realizations for the random variables.
2371:
629:
156:
1105:
442:
216:
685:
2447:
1496:
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1461:
1816:
1720:
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1559:
724:
314:
1171:
922:
1752:
1679:
1415:
991:
343:
2411:
2391:
2053:
2033:
2013:
1836:
1775:
1606:
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1519:
1388:
1356:
1333:
1145:
1125:
1055:
1031:
1011:
764:
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653:
540:
516:
492:
464:
274:
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2723:
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2444:
The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomesâall the positive integers:
1176:
446:
Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes
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1839:
2706:
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67:
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2413:
denotes the number of necessary coin tosses. Other distributions that can be modeled using a probability mass function are the
2300:
An example of the binomial distribution is the probability of getting exactly one 6 when someone rolls a fair die three times.
2196:{\displaystyle p_{X}(x)={\begin{cases}{\frac {1}{2}},&x=0,\\{\frac {1}{2}},&x=1,\\0,&x\notin \{0,1\}.\end{cases}}}
2543:
2428:
2055:
assigning 0 to the category "tails" and 1 to the category "heads". Since the coin is fair, the probability mass function is
1988:{\displaystyle p_{X}(x)={\begin{cases}p,&{\text{if }}x{\text{ is 1}}\\1-p,&{\text{if }}x{\text{ is 0}}\end{cases}}}
1034:
353:
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1363:
519:
86:
75:
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Geometric distribution describes the number of trials needed to get one success. Its probability mass function is
810:
2209:
1889:, is used to model an experiment with only two possible outcomes. The two outcomes are often encoded as 1 and 0.
766:
whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section)
90:
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2432:
2418:
2414:
495:
109:
59:
34:
The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.
224:
930:
89:(PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be
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is one. Consequently, the probability mass function is zero for all but a countable number of values of
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115:
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164:
79:
2089:
1923:
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1367:
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277:
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Probability, Markov Chains, Queues, and
Simulation: The Mathematical Basis of Performance Modeling
1466:
348:
The probabilities associated with all (hypothetical) values must be non-negative and sum up to 1,
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39:
1424:
30:
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112:, and provides the possible values and their associated probabilities. It is the function
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can be seen as a special case of two more general measure theoretic constructions: the
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2018:
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259:
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2527:{\displaystyle {\text{Pr}}(X=i)={\frac {1}{2^{i}}}\qquad {\text{for }}i=1,2,3,\dots }
970:
632:
2607:
A modern introduction to probability and statistics : understanding why and how
2565:
7.2 - Probability Mass
Functions | STAT 414 - PennState - Eberly College of Science
2564:
96:
The value of the random variable having the largest probability mass is called the
1584:
The discontinuity of probability mass functions is related to the fact that the
1418:
17:
1842:
is the process of converting a continuous random variable into a discrete one.
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43:
2624:
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2671:
1522:
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2015:
is the sample space of all outcomes of a single toss of a fair coin, and
1995:
An example of the
Bernoulli distribution is tossing a coin. Suppose that
66:. The probability mass function is often the primary means of defining a
1681:
is certain (it is true in 100% of the occurrences); on the contrary,
1306:{\displaystyle P(X=b)=P(X^{-1}(b))=X_{*}(P)(b)=\int _{b}fd\mu =f(b),}
2297:
have an equal chance of appearing on top when the die stops rolling.
62:
is exactly equal to some value. Sometimes it is also known as the
2284:
29:
2294:
2290:
108:
Probability mass function is the probability distribution of a
2719:
2417:(also known as the generalized Bernoulli distribution) and the
2373:.An example is tossing a coin until the first "heads" appears.
1358:, it may be convenient to assign numerical values to them (or
945:
613:
567:
2431:, and of its probability mass function, is provided by the
2189:
1981:
1338:
When there is a natural order among the potential outcomes
1033:
with respect to the counting measure, if it exists, is the
474:
A probability mass function of a discrete random variable
635:
is discrete, so in particular contains singleton sets of
1588:
of a discrete random variable is also discontinuous. If
2309:
2212:
1754:
is always impossible. This statement isn't true for a
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to the non-negative reals. As a consequence, for any
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denotes the probability of the outcome "heads", and
1869:
There are three major distributions associated, the
2654:
Engineering optimization : theory and practice
2609:. Dekking, Michel, 1946-. London: Springer. 2005.
2526:
2405:
2385:
2365:
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2195:
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58:) is a function that gives the probability that a
687:is discrete provided its image is countable. The
1525:subset on which the probability mass function
2731:
2229:
2216:
800:{\displaystyle f_{X}\colon B\to \mathbb {R} }
8:
2276:{\textstyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}
2180:
2168:
891:{\displaystyle f_{X}(b)=P(X^{-1}(b))=P(X=b)}
746:in this contextâis a probability measure on
2689:Johnson, N. L.; Kotz, S.; Kemp, A. (1993).
2582:. Princeton University Press. p. 105.
1057:(with respect to the counting measure), so
85:A probability mass function differs from a
2738:
2724:
2716:
27:Discrete-variable probability distribution
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1366:) and to consider also values not in the
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126:
125:
117:
93:over an interval to yield a probability.
1335:is in fact a probability mass function.
249:{\displaystyle -\infty <x<\infty }
2557:
962:{\displaystyle (B,{\mathcal {B}},\mu )}
631:is a measurable space whose underlying
2630:
70:, and such functions exist for either
655:. In this setting, a random variable
64:discrete probability density function
7:
2601:
2599:
1608:is a discrete random variable, then
993:. The probability density function
584:{\displaystyle (A,{\mathcal {A}},P)}
546:. We make this more precise below.
2289:The probability mass function of a
973:equipped with the counting measure
395:{\displaystyle \sum _{x}p_{X}(x)=1}
2890:Types of probability distributions
2429:multivariate discrete distribution
2366:{\textstyle p_{X}(k)=(1-p)^{k-1}p}
2220:
2035:is the random variable defined on
624:{\displaystyle (B,{\mathcal {B}})}
243:
231:
151:{\displaystyle p:\mathbb {R} \to }
25:
2693:Univariate Discrete Distributions
2656:(3rd ed.). New York: Wiley.
68:discrete probability distribution
2785:
2771:cumulative distribution function
1586:cumulative distribution function
2858:probability-generating function
2697:(2nd ed.). Wiley. p.
2491:
1100:{\displaystyle f=dX_{*}P/d\mu }
437:{\displaystyle p_{X}(x)\geq 0.}
211:{\displaystyle p_{X}(x)=P(X=x)}
2544:Joint probability distribution
2468:
2456:
2345:
2332:
2326:
2320:
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1906:
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1438:
1362:-tuples in case of a discrete
1297:
1291:
1260:
1254:
1251:
1245:
1229:
1226:
1220:
1204:
1195:
1183:
1037:of the pushforward measure of
956:
934:
885:
873:
864:
861:
855:
839:
830:
824:
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713:
707:
680:{\displaystyle X\colon A\to B}
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1:
470:Measure theoretic formulation
76:multivariate random variables
2765:probability density function
2576:Stewart, William J. (2011).
1722:means that the casual event
1649:means that the casual event
1491:{\displaystyle x\notin X(S)}
1364:multivariate random variable
726:âcalled the distribution of
520:probability density function
87:probability density function
2906:
2847:moment-generating function
2652:Rao, Singiresu S. (1996).
2541:
1849:
1756:continuous random variable
1456:{\displaystyle f_{X}(x)=0}
316:can also be simplified as
2842:
2794:
2783:
2759:probability mass function
2754:
2748:probability distributions
2293:. All the numbers on the
48:probability mass function
2433:multinomial distribution
2419:multinomial distribution
2415:categorical distribution
1885:Bernoulli distribution:
1811:{\displaystyle P(X=x)=0}
1715:{\displaystyle P(X=x)=0}
1642:{\displaystyle P(X=x)=1}
1554:{\displaystyle f_{X}(x)}
1498:as shown in the figure.
1035:RadonâNikodym derivative
719:{\displaystyle X_{*}(P)}
309:{\displaystyle p_{X}(x)}
110:discrete random variable
60:discrete random variable
2853:characteristic function
1417:may be defined for all
2637:: CS1 maint: others (
2528:
2407:
2387:
2367:
2298:
2277:
2197:
2049:
2029:
2009:
1989:
1879:geometric distribution
1871:Bernoulli distribution
1860:Geometric distribution
1852:Bernoulli distribution
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1167:
1166:{\displaystyle b\in B}
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918:
917:{\displaystyle b\in B}
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35:
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2010:
1990:
1875:binomial distribution
1856:Binomial distribution
1833:
1813:
1772:
1749:
1747:{\displaystyle (X=x)}
1717:
1676:
1674:{\displaystyle (X=x)}
1644:
1603:
1576:
1556:
1516:
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1458:
1412:
1410:{\displaystyle f_{X}}
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1111:
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1017:
997:
986:{\displaystyle \mu }
977:
931:
902:
811:
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750:
730:
694:
659:
639:
599:
553:
542:with respect to the
526:
502:
478:
450:
406:
354:
338:{\displaystyle p(x)}
320:
284:
260:
225:
165:
116:
52:probability function
1315:demonstrating that
1127:is a function from
689:pushforward measure
278:probability measure
2819:standard deviation
2524:
2403:
2383:
2363:
2299:
2273:
2193:
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2005:
1985:
1980:
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508:
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392:
366:
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266:
246:
208:
148:
56:frequency function
50:(sometimes called
36:
2877:
2876:
2777:quantile function
2616:978-1-85233-896-1
2589:978-1-4008-3281-1
2538:Multivariate case
2495:
2489:
2454:
2406:{\displaystyle k}
2386:{\displaystyle p}
2227:
2131:
2100:
2048:{\displaystyle S}
2028:{\displaystyle X}
2008:{\displaystyle S}
1976:
1968:
1945:
1937:
1831:{\displaystyle x}
1818:for any possible
1770:{\displaystyle X}
1601:{\displaystyle X}
1574:{\displaystyle x}
1514:{\displaystyle X}
1383:{\displaystyle X}
1351:{\displaystyle x}
1328:{\displaystyle f}
1140:{\displaystyle B}
1120:{\displaystyle f}
1050:{\displaystyle X}
1026:{\displaystyle X}
1006:{\displaystyle f}
927:Now suppose that
759:{\displaystyle B}
739:{\displaystyle X}
648:{\displaystyle B}
593:probability space
535:{\displaystyle X}
511:{\displaystyle X}
487:{\displaystyle X}
459:{\displaystyle x}
357:
269:{\displaystyle P}
104:Formal definition
16:(Redirected from
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2427:An example of a
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544:counting measure
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21:
18:Probability mass
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2781:
2750:
2744:
2709:
2688:
2685:
2683:Further reading
2680:
2679:
2664:
2651:
2650:
2646:
2629:
2617:
2605:
2604:
2597:
2590:
2575:
2574:
2570:
2563:
2559:
2554:
2546:
2540:
2479:
2446:
2445:
2442:
2395:
2394:
2375:
2374:
2344:
2310:
2305:
2304:
2257:
2235:
2214:
2208:
2207:
2187:
2186:
2160:
2151:
2150:
2136:
2120:
2119:
2105:
2085:
2062:
2057:
2056:
2037:
2036:
2017:
2016:
1997:
1996:
1979:
1978:
1963:
1948:
1947:
1932:
1919:
1896:
1891:
1890:
1867:
1862:
1850:Main articles:
1848:
1820:
1819:
1779:
1778:
1759:
1758:
1724:
1723:
1683:
1682:
1651:
1650:
1610:
1609:
1590:
1589:
1563:
1562:
1532:
1527:
1526:
1503:
1502:
1465:
1464:
1428:
1423:
1422:
1397:
1392:
1391:
1372:
1371:
1340:
1339:
1317:
1316:
1266:
1235:
1207:
1175:
1174:
1149:
1148:
1129:
1128:
1109:
1108:
1073:
1059:
1058:
1039:
1038:
1015:
1014:
995:
994:
975:
974:
929:
928:
900:
899:
842:
814:
809:
808:
773:
768:
767:
748:
747:
728:
727:
697:
692:
691:
657:
656:
637:
636:
597:
596:
551:
550:
524:
523:
500:
499:
476:
475:
472:
448:
447:
409:
404:
403:
367:
352:
351:
318:
317:
287:
282:
281:
258:
257:
223:
222:
219:
168:
163:
162:
114:
113:
106:
28:
23:
22:
15:
12:
11:
5:
2903:
2901:
2893:
2892:
2882:
2881:
2875:
2874:
2872:
2871:
2866:
2861:
2855:
2850:
2843:
2840:
2839:
2837:
2836:
2831:
2826:
2821:
2816:
2811:
2806:
2804:central moment
2801:
2795:
2792:
2791:
2784:
2782:
2780:
2779:
2774:
2768:
2762:
2755:
2752:
2751:
2745:
2743:
2742:
2735:
2728:
2720:
2714:
2713:
2707:
2684:
2681:
2678:
2677:
2662:
2644:
2615:
2595:
2588:
2568:
2556:
2555:
2553:
2550:
2542:Main article:
2539:
2536:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2486:
2482:
2478:
2473:
2470:
2467:
2464:
2461:
2458:
2441:
2438:
2437:
2436:
2425:
2422:
2402:
2382:
2362:
2357:
2354:
2351:
2347:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2317:
2313:
2301:
2270:
2267:
2264:
2260:
2256:
2253:
2250:
2247:
2242:
2238:
2231:
2226:
2223:
2218:
2203:
2190:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2159:
2156:
2153:
2152:
2149:
2146:
2143:
2140:
2137:
2135:
2130:
2127:
2122:
2121:
2118:
2115:
2112:
2109:
2106:
2104:
2099:
2096:
2091:
2090:
2088:
2083:
2080:
2077:
2074:
2069:
2065:
2044:
2024:
2004:
1982:
1972:
1964:
1962:
1959:
1956:
1953:
1950:
1949:
1941:
1933:
1931:
1928:
1925:
1924:
1922:
1917:
1914:
1911:
1908:
1903:
1899:
1866:
1863:
1847:
1844:
1840:Discretization
1827:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1766:
1743:
1740:
1737:
1734:
1731:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1670:
1667:
1664:
1661:
1658:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1597:
1570:
1550:
1547:
1544:
1539:
1535:
1510:
1487:
1484:
1481:
1478:
1475:
1472:
1452:
1449:
1446:
1443:
1440:
1435:
1431:
1404:
1400:
1379:
1347:
1324:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1273:
1269:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1242:
1238:
1234:
1231:
1228:
1225:
1222:
1217:
1214:
1210:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1162:
1159:
1156:
1136:
1116:
1096:
1093:
1089:
1085:
1080:
1076:
1072:
1069:
1066:
1046:
1022:
1002:
982:
958:
955:
952:
947:
942:
939:
936:
913:
910:
907:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
852:
849:
845:
841:
838:
835:
832:
829:
826:
821:
817:
795:
791:
788:
785:
780:
776:
755:
735:
715:
712:
709:
704:
700:
676:
673:
670:
667:
664:
644:
620:
615:
610:
607:
604:
580:
577:
574:
569:
564:
561:
558:
531:
507:
483:
471:
468:
455:
433:
430:
427:
424:
421:
416:
412:
391:
388:
385:
382:
379:
374:
370:
364:
360:
334:
331:
328:
325:
305:
302:
299:
294:
290:
265:
245:
242:
239:
236:
233:
230:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
175:
171:
160:
147:
144:
141:
138:
135:
132:
128:
124:
121:
105:
102:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2902:
2891:
2888:
2887:
2885:
2870:
2867:
2865:
2862:
2859:
2856:
2854:
2851:
2848:
2845:
2844:
2841:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2796:
2793:
2788:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2756:
2753:
2749:
2741:
2736:
2734:
2729:
2727:
2722:
2721:
2718:
2710:
2708:0-471-54897-9
2704:
2700:
2695:
2694:
2687:
2686:
2682:
2673:
2669:
2665:
2663:0-471-55034-5
2659:
2655:
2648:
2645:
2640:
2634:
2626:
2622:
2618:
2612:
2608:
2602:
2600:
2596:
2591:
2585:
2581:
2580:
2572:
2569:
2566:
2561:
2558:
2551:
2549:
2545:
2537:
2535:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2484:
2480:
2476:
2471:
2465:
2462:
2459:
2439:
2434:
2430:
2426:
2423:
2420:
2416:
2400:
2380:
2360:
2355:
2352:
2349:
2341:
2338:
2335:
2329:
2323:
2315:
2311:
2302:
2296:
2292:
2287:
2268:
2265:
2262:
2254:
2251:
2248:
2240:
2236:
2224:
2221:
2204:
2183:
2177:
2174:
2171:
2165:
2162:
2157:
2154:
2147:
2144:
2141:
2138:
2133:
2128:
2125:
2116:
2113:
2110:
2107:
2102:
2097:
2094:
2086:
2081:
2075:
2067:
2063:
2042:
2022:
2002:
1970:
1960:
1957:
1954:
1951:
1939:
1929:
1926:
1920:
1915:
1909:
1901:
1897:
1888:
1884:
1883:
1882:
1880:
1876:
1872:
1864:
1861:
1857:
1853:
1845:
1843:
1841:
1825:
1805:
1802:
1796:
1793:
1790:
1784:
1764:
1757:
1738:
1735:
1732:
1709:
1706:
1700:
1697:
1694:
1688:
1665:
1662:
1659:
1636:
1633:
1627:
1624:
1621:
1615:
1595:
1587:
1582:
1568:
1545:
1537:
1533:
1524:
1508:
1501:The image of
1499:
1482:
1476:
1473:
1470:
1450:
1447:
1441:
1433:
1429:
1420:
1402:
1398:
1377:
1369:
1365:
1361:
1345:
1336:
1322:
1313:
1300:
1294:
1288:
1285:
1282:
1279:
1276:
1271:
1267:
1263:
1257:
1248:
1240:
1236:
1232:
1223:
1215:
1212:
1208:
1201:
1198:
1192:
1189:
1186:
1180:
1160:
1157:
1154:
1134:
1114:
1094:
1091:
1087:
1083:
1078:
1074:
1070:
1067:
1064:
1044:
1036:
1020:
1000:
980:
972:
971:measure space
953:
950:
940:
937:
925:
911:
908:
905:
882:
879:
876:
870:
867:
858:
850:
847:
843:
836:
833:
827:
819:
815:
786:
783:
778:
774:
753:
733:
710:
702:
698:
690:
674:
668:
665:
662:
642:
634:
608:
605:
594:
575:
572:
562:
559:
549:Suppose that
547:
545:
529:
521:
505:
497:
481:
469:
467:
453:
444:
431:
428:
422:
414:
410:
389:
386:
380:
372:
368:
362:
358:
349:
346:
329:
323:
300:
292:
288:
279:
263:
240:
237:
234:
228:
218:
202:
199:
196:
190:
187:
181:
173:
169:
159:
142:
139:
136:
122:
119:
111:
103:
101:
99:
94:
92:
88:
83:
82:is discrete.
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
32:
19:
2758:
2692:
2653:
2647:
2606:
2578:
2571:
2560:
2547:
2443:
1886:
1868:
1777:, for which
1583:
1500:
1419:real numbers
1359:
1337:
1314:
926:
548:
496:distribution
473:
445:
350:
347:
220:
161:
107:
95:
84:
63:
55:
51:
47:
37:
1390:. That is,
158:defined by
40:probability
2799:raw moment
2746:Theory of
2552:References
1975: is 0
1944: is 1
91:integrated
44:statistics
2869:combinant
2633:cite book
2625:262680588
2522:…
2494:for
2353:−
2339:−
2266:−
2252:−
2166:∉
1955:−
1523:countable
1474:∉
1283:μ
1268:∫
1241:∗
1213:−
1158:∈
1095:μ
1079:∗
981:μ
954:μ
909:∈
898:for each
848:−
790:→
784::
703:∗
672:→
666::
633:Ď-algebra
595:and that
429:≥
359:∑
244:∞
232:∞
229:−
131:→
2884:Category
2864:cumulant
2834:L-moment
2829:kurtosis
2824:skewness
2814:variance
2672:62080932
2440:Infinite
2291:fair die
1967:if
1936:if
1877:and the
1846:Examples
1463:for all
1173:we have
518:and the
256:, where
1887:ber(p)
2705:
2670:
2660:
2623:
2613:
2586:
1873:, the
1865:Finite
1858:, and
1521:has a
807:since
80:domain
78:whose
72:scalar
2860:(pgf)
2849:(mgf)
2773:(cdf)
2767:(pdf)
2761:(pmf)
1368:image
969:is a
591:is a
276:is a
2809:mean
2703:ISBN
2668:OCLC
2658:ISBN
2639:link
2621:OCLC
2611:ISBN
2584:ISBN
1421:and
1107:and
402:and
241:<
235:<
221:for
98:mode
46:, a
42:and
2295:die
1370:of
1013:of
522:of
498:of
74:or
54:or
38:In
2886::
2701:.
2699:36
2666:.
2635:}}
2631:{{
2619:.
2598:^
2453:Pr
2283:.
1881:.
1854:,
1838:.
1581:.
924:.
466:.
432:0.
345:.
280:.
100:.
2739:e
2732:t
2725:v
2711:.
2674:.
2641:)
2627:.
2592:.
2519:,
2516:3
2513:,
2510:2
2507:,
2504:1
2501:=
2498:i
2485:i
2481:2
2477:1
2472:=
2469:)
2466:i
2463:=
2460:X
2457:(
2421:.
2401:k
2381:p
2361:p
2356:1
2350:k
2346:)
2342:p
2336:1
2333:(
2330:=
2327:)
2324:k
2321:(
2316:X
2312:p
2269:k
2263:n
2259:)
2255:p
2249:1
2246:(
2241:k
2237:p
2230:)
2225:k
2222:n
2217:(
2184:.
2181:}
2178:1
2175:,
2172:0
2169:{
2163:x
2158:,
2155:0
2148:,
2145:1
2142:=
2139:x
2134:,
2129:2
2126:1
2117:,
2114:0
2111:=
2108:x
2103:,
2098:2
2095:1
2087:{
2082:=
2079:)
2076:x
2073:(
2068:X
2064:p
2043:S
2023:X
2003:S
1971:x
1961:,
1958:p
1952:1
1940:x
1930:,
1927:p
1921:{
1916:=
1913:)
1910:x
1907:(
1902:X
1898:p
1826:x
1806:0
1803:=
1800:)
1797:x
1794:=
1791:X
1788:(
1785:P
1765:X
1742:)
1739:x
1736:=
1733:X
1730:(
1710:0
1707:=
1704:)
1701:x
1698:=
1695:X
1692:(
1689:P
1669:)
1666:x
1663:=
1660:X
1657:(
1637:1
1634:=
1631:)
1628:x
1625:=
1622:X
1619:(
1616:P
1596:X
1569:x
1549:)
1546:x
1543:(
1538:X
1534:f
1509:X
1486:)
1483:S
1480:(
1477:X
1471:x
1451:0
1448:=
1445:)
1442:x
1439:(
1434:X
1430:f
1403:X
1399:f
1378:X
1360:n
1346:x
1323:f
1301:,
1298:)
1295:b
1292:(
1289:f
1286:=
1280:d
1277:f
1272:b
1264:=
1261:)
1258:b
1255:(
1252:)
1249:P
1246:(
1237:X
1233:=
1230:)
1227:)
1224:b
1221:(
1216:1
1209:X
1205:(
1202:P
1199:=
1196:)
1193:b
1190:=
1187:X
1184:(
1181:P
1161:B
1155:b
1135:B
1115:f
1092:d
1088:/
1084:P
1075:X
1071:d
1068:=
1065:f
1045:X
1021:X
1001:f
957:)
951:,
946:B
941:,
938:B
935:(
912:B
906:b
886:)
883:b
880:=
877:X
874:(
871:P
868:=
865:)
862:)
859:b
856:(
851:1
844:X
840:(
837:P
834:=
831:)
828:b
825:(
820:X
816:f
794:R
787:B
779:X
775:f
754:B
734:X
714:)
711:P
708:(
699:X
675:B
669:A
663:X
643:B
619:)
614:B
609:,
606:B
603:(
579:)
576:P
573:,
568:A
563:,
560:A
557:(
530:X
506:X
482:X
454:x
426:)
423:x
420:(
415:X
411:p
390:1
387:=
384:)
381:x
378:(
373:X
369:p
363:x
333:)
330:x
327:(
324:p
304:)
301:x
298:(
293:X
289:p
264:P
238:x
206:)
203:x
200:=
197:X
194:(
191:P
188:=
185:)
182:x
179:(
174:X
170:p
146:]
143:1
140:,
137:0
134:[
127:R
123::
120:p
20:)
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