?_’’’’?6}tlp{B® éʁ†&’ƒ—MrEdMicrosoft DrawZ&©MrEdŸŒ,,|˜ƒ’’…¤ Systemler (^0) beim Zugriff auf’Ą€‚’€†’’€€’€†’’€€’Œ€’’€€&MrEd…ƒ‚ ‡ ƒƒ…‚’‚‚ …… ……ƒ%…÷€…€€€Š€€€€€€Ą‚’…’’’Š’’’’’’‚…4ƒ5Ę…C Ģƒƒƒ(ƒ‚  „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€ ˆ„€ˆ€„ˆˆ€ˆ„"„Hˆ‰€ˆˆ"„Hˆ‚€ˆ„"„Hˆ‰€ˆˆ"„Hˆ‚€ˆ„"„Hˆ‰€ˆˆˆ„Hˆ‚€ˆƒ„Hˆˆ€ˆˆˆ„Hˆ‚€ˆ‚„Hˆ„€ˆˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€š ‚-…÷€…€€€Š€€€€€Ą‘ĄÜĄ¦Źš’ūš  ¤€‚’…’’’Š’’’’’’‚…4ƒ5…š…'’’…'’’…'’’ƒ’’ˆ’’’’’/&;)z4Vņ‹F FuéŌ’’ė’’’’|CONTEXT.|CTXOMAPi |FONTQü|KWBTREE² |KWDATAt |KWMAP” |SYSTEM^.|TOPIC….|TTLBTREEį|bm0|bm1¼|bm10ŗ|bm11?|bm12Ń|bm13s |bm14"|bm15Ž&|bm16D+|bm2C |bm3Š |bm4U|bm5õ|bm6—|bm7)|bm8¦|bm9-‰Fš‰VņĒFō鄐 Ą}(‹F FtŋFÄ^&‰‹Fō&‰G‹Fš+Fų-&‰Gė¦‹F F t?’vņ’vššŗƗƒÄFš‹Vņ@‰Fü‰VžRP’v’všŚkŪ—™RP’vž’vü’v ’v šü±—ƒÄ Ž 0&Ē¢ ė„’Fō‹FōÄ^ų&9Gér’’v’v’vņ’všš¬5s– ĄéV’’vņ’vššŗ€–ƒÄ@Fš’vņ’vš’v’všŚk5–Fšė®U‹ģƒģWV’v’v’v ’v š†nՖ‹š‰Vņ Šuø’’ėrŽFņ&‹|&‹D‰FDŒĀ@@‹Č‰Vö‰všė2’v’v’vöVš6~— Ą|'’vöVšŗM—ƒÄ@šŽFö&‹‰F‹ĘŒĀ@@‹š‹ĒO ĄʋF Ft +všƒīÄ^&‰7‹F^_‹å]ŹU‹ģƒģ’v’v’v’vš†n.™‰Fų‰Vś ŠuŽ 0&”¢ ‹å]ŹŽ 0&Ē¢ ‹Vś ‰Fš‰VņĒFō鄐 Ą}(‹F FtŋFÄ^&‰‹Fō&‰G‹Fš+Fų-&‰Gė¦‹F F t?’vņ’vššŗ‡™ƒÄFš‹Vņ@‰Fü‰VžRP’v’všŚkŸ™™RP’vž’vü’v ’v šüu™ƒÄ Ž 0&Ē¢ ė„’Fō‹FōÄ^ų&9Gér’’v’v’vņ’všš67˜ ĄéV’’vņ’vššŗD˜ƒÄ@Fš’vņ’vš’v’všŚkł—Fšė®U‹ģƒģJWV’v’v øPšh"ܙ‹ųƒ’’u Ž0&”˜^éįW+ĄPPøPš%ó™‰Fų‰Vś Ņ|=‡~lp{FØĀ‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€ˆ†ˆ€ˆ‡€ˆ€ˆˆ€ˆ‡€ˆ€ˆˆ€ˆˆ€ˆ€ˆˆ€ˆˆ€ˆ€ˆˆ€ˆˆ€€ˆ€ˆˆ€ˆŽ€ˆˆˆ€ˆˆ€ˆˆˆ€ˆˆ€ˆˆˆ€ˆˆ€ˆ€ˆˆ€ˆ€ˆ€ˆ†€ˆˆˆ†€ˆˆ€ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€š ‚„lp{FØĪ‡ źĘ‚ ‡ ‚ ‚’‚…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ˆ…€ˆˆ€ˆˆˆˆˆ€ˆˆ€ˆˆ€ˆ†ˆˆ€ˆ†ˆˆ€ˆŠ€€ˆ€ˆˆˆ£€€ˆˆ€ˆˆ€€ˆˆˆˆ€€ˆˆˆ€’€ˆ€ˆˆ€ˆ€ˆˆ†€ˆˆ†€ˆˆˆ†€ˆˆˆ‚€ ˆ‚€ ˆ‚€š ‚…|lp{B¾ éʁ†&’ƒ—MrEdMicrosoft DrawZ&©MrEdŸŒXX|˜ƒ’’…ƒ=‰ System’Ą€‚’€†’’€€’€†’’€€’Œ€’’€€&MrEd…ƒ‚ ‡ ƒƒ…‚’‚‚ …… ……ƒ%…÷€…€€€Š€€€€€€Ą‚’…’’’Š’’’’’’‚…4ƒ5Ę…C Ģƒƒƒ(ƒ‚  „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ˆƒ€ˆ‰€ˆˆ€æ戅€ˆˆ ū‡ˆˆ€ˆ€扰ˆˆ€ˆ€ū ū‰šˆˆ€ˆæ°æ‰ˆ€ˆ ūū ū’ˆ€ˆææ°ˆ€ˆ ū†ˆ€ˆæ†ˆ€ˆ€ū‡šˆˆ€ˆ€昰ˆˆ€ˆˆ ūšūūˆˆ€ˆˆ€æ戂€ˆƒ€ˆ‚€ ˆ‚€ ˆ‚€š ‚-…÷€…€€€Š€€€€€Ą‘ĄÜĄ¦Źš’ūš  ¤€‚’…’’’Š’’’’’’‚…4ƒ5…š…'’’…'’’’Ą€‚’€†’’€€ —lp{FØō‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‰€ˆˆ€ˆ€pˆ”€ˆˆ€ˆˆ€ˆˆwxˆˆ€ˆˆˆ‰€ˆˆˆ‚€ˆ„ˆ‚€ˆ„xˆ‚€ˆ„xˆ‚€ˆ„ˆ‚€ˆ‰xˆˆ€ˆ”xˆˆ€ˆˆ‰ˆˆˆ€ˆˆ‰ˆˆ‘ˆˆ€ˆ†ˆˆ€ ˆ‚€ ˆ‚€ ˆ‚€š ‚¢™lp{FØų‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆƒ€ˆ‡ˆˆ€ˆ’ˆš€ˆ€ˆ’Ńpˆ€ˆ’’šx‡ˆˆ€ˆ’’ˆēpˆˆ€ˆ’’ˆ‡€ˆˆ€ˆ’’臀ˆˆ€ˆ’’ī‡pˆˆ€ˆ’’šx‡ˆˆ€ˆ’‚ˆ„€ˆ’šˆ„€ˆ’‚ˆ„€ˆ’‚ˆ„€ˆ’ˆƒ€ˆˆ‚€ ˆ‚€ ˆ‚€š ‚’‰lp{FØŲ‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ„€ˆˆ‡ˆ€ˆ€Ģ‰ˆ ˆ€ˆ€Ģ‰ˆ ˆ€ˆ€Ģ‰ˆ ˆ€ˆ€Ģ‡ ˆ€ˆ€Ģ‡ˆ€ˆ€ĢˆĢˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆˆ€ˆ€Ąī‡ˆ€ˆ€„ˆ€ ˆ‚€ ˆ‚€ ˆ‚€š ‚}tlp{B® éʁ†&’ƒ—MrEdMicrosoft DrawZ&©MrEdŸŒ,,|˜ƒ’’…¤ Systemler (^0) beim Zugriff auf’Ą€‚’€†’’€€’€†’’€€’Œ€’’€€&MrEd…ƒ‚ ‡ ƒƒ…‚’‚‚ …… ……ƒ%…÷€…€€€Š€€€€€€Ą‚’…’’’Š’’’’’’‚…4ƒ5Ę…C Ģƒƒƒ(ƒ‚  „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€ ˆ„€ˆ€„ˆˆ€ˆ„"„Hˆ‰€ˆˆ"„Hˆ‚€ˆ„"„Hˆ‰€ˆˆ"„Hˆ‚€ˆ„"„Hˆ‰€ˆˆˆ„Hˆ‚€ˆƒ„Hˆˆ€ˆˆˆ„Hˆ‚€ˆ‚„Hˆ„€ˆˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€š ‚-…÷€…€€€Š€€€€€Ą‘ĄÜĄ¦Źš’ūš  ¤€‚’…’’’Š’’’’’’‚…4ƒ5…š…'’’…'’’…'’’ƒ’’ˆ’’’’’‡~lp{FØĀ‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€ˆ†ˆ€ˆ‡€ˆ€ˆˆ€ˆ‡€ˆ€ˆˆ€ˆˆ€ˆ€ˆˆ€ˆˆ€ˆ€ˆˆ€ˆˆ€€ˆ€ˆˆ€ˆŽ€ˆˆˆ€ˆˆ€ˆˆˆ€ˆˆ€ˆˆˆ€ˆˆ€ˆ€ˆˆ€ˆ€ˆ€ˆ†€ˆˆˆ†€ˆˆ€ˆ‚€ ˆ‚€ ˆ‚€ ˆ‚€š ‚„lp{FØĪ‡ źĘ‚ ‡ ‚ ‚’‚…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ˆ…€ˆˆ€ˆˆˆˆˆ€ˆˆ€ˆˆ€ˆ†ˆˆ€ˆ†ˆˆ€ˆŠ€€ˆ€ˆˆˆ£€€ˆˆ€ˆˆ€€ˆˆˆˆ€€ˆˆˆ€’€ˆ€ˆˆ€ˆ€ˆˆ†€ˆˆ†€ˆˆˆ†€ˆˆˆ‚€ ˆ‚€ ˆ‚€š ‚…|lp{B¾ éʁ†&’ƒ—MrEdMicrosoft DrawZ&©MrEdŸŒXX|˜ƒ’’…ƒ=‰ System’Ą€‚’€†’’€€’€†’’€€’Œ€’’€€&MrEd…ƒ‚ ‡ ƒƒ…‚’‚‚ …… ……ƒ%…÷€…€€€Š€€€€€€Ą‚’…’’’Š’’’’’’‚…4ƒ5Ę…C Ģƒƒƒ(ƒ‚  „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‚€ˆƒ€ˆ‰€ˆˆ€æ戅€ˆˆ ū‡ˆˆ€ˆ€扰ˆˆ€ˆ€ū ū‰šˆˆ€ˆæ°æ‰ˆ€ˆ ūū ū’ˆ€ˆææ°ˆ€ˆ ū†ˆ€ˆæ†ˆ€ˆ€ū‡šˆˆ€ˆ€昰ˆˆ€ˆˆ ūšūūˆˆ€ˆˆ€æ戂€ˆƒ€ˆ‚€ ˆ‚€ ˆ‚€š ‚-…÷€…€€€Š€€€€€Ą‘ĄÜĄ¦Źš’ūš  ¤€‚’…’’’Š’’’’’’‚…4ƒ5…š…'’’…'’’’Ą€‚’€†’’€€’‰lp{FØŲ‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ„€ˆˆ‡ˆ€ˆ€Ģ‰ˆ ˆ€ˆ€Ģ‰ˆ ˆ€ˆ€Ģ‰ˆ ˆ€ˆ€Ģ‡ ˆ€ˆ€Ģ‡ˆ€ˆ€ĢˆĢˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆ ˆ€ˆ€Ąīˆˆ€ˆ€Ąī‡ˆ€ˆ€„ˆ€ ˆ‚€ ˆ‚€ ˆ‚€š ‚¢™lp{FØų‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆƒ€ˆ‡ˆˆ€ˆ’ˆš€ˆ€ˆ’Ńpˆ€ˆ’’šx‡ˆˆ€ˆ’’ˆēpˆˆ€ˆ’’ˆ‡€ˆˆ€ˆ’’臀ˆˆ€ˆ’’ī‡pˆˆ€ˆ’’šx‡ˆˆ€ˆ’‚ˆ„€ˆ’šˆ„€ˆ’‚ˆ„€ˆ’‚ˆ„€ˆ’ˆƒ€ˆˆ‚€ ˆ‚€ ˆ‚€š ‚ —lp{FØō‡ źĘ‚ ‡ ‡ ’’‡€…Ę…C Ģƒƒ(ƒ‚ „€€„€€€ˆ€€€€€Ą„’’„’’’ˆ’’’’’‚š ‚ ˆ‚€ ˆ‚€ ˆ‰€ˆˆ€ˆ€pˆ”€ˆˆ€ˆˆ€ˆˆwxˆˆ€ˆˆˆ‰€ˆˆˆ‚€ˆ„ˆ‚€ˆ„xˆ‚€ˆ„xˆ‚€ˆ„ˆ‚€ˆ‰xˆˆ€ˆ”xˆˆ€ˆˆ‰ˆˆˆ€ˆˆ‰ˆˆ‘ˆˆ€ˆ†ˆˆ€ ˆ‚€ ˆ‚€ ˆ‚€š ‚{rlp”Ą Ŗ w‚ …‚’‚….ƒ1‚ ‡  ą†&’ƒ‰ą’Ö’Ąv “& MathType …ś…-‡œ‡…ś…-‡Ü`‡ÜØ„ū€žƒ–Times New RomanĄ¦…- ‡2 `…hĄ ‡2 J…s– ‡2 bÄ…nĄ„ū€ž‚ŽSymbol…-…š ‡2 `(…=Ó„ū€ž‚’Times New Roman…-…š ‡2 Jt…3Ą ‡2 Jd‡49ĄĄ„ū ’‚’Times New Roman…-…š ‡2 “¤…1p ‡2 ī§…3p„ū€ž‚’Times New Roman…-…š ‡2 J….` ‡2 J…*Ą †& ’…ū‚¼"System…-…šŖrix Matrix ändernˆˆ€ˆˆ ūšūūˆˆ€ˆˆ€æ戂€ˆˆ‚€ ˆ‚€ ˆ€ ‚-…÷€…€€€Š€€€€€Ą‘ĄÜĄ¦Źš’ū  ¤€‚’…’’’Š’’’’’‚…4ƒ5…š&’Ą€‚’€†’’€€’anzwortanzzeich dateinamevorlŗ€^ÉĖŽF&ƒ| ’u&ƒ| ’t øhŗ€^ÉFō)Fų”Ŗ FöFö‰Fś’v FōPhšj!šŅö‹Fš ‹šÄ~ō¹ ó„‹^š‹‡°‰FüPšĖĄ Ąué!’6&j š>æ鿉6&Vƒ>j’včåöMM‹å]MŹøEU‹ģŽŲƒģ ¶­lp”Ą  w‚ …‚’‚….ƒ1‚ ‡  ą†&’ƒ‰ą’Ö’Ąv “& MathType …ś…-‡œ‡…ś…-‡Ü`‡ÜØ„ū€žƒ–Times New RomanĄ¦…- ‡2 `…hĄ ‡2 J…s– ‡2 bÄ…nĄ„ū€ž‚ŽSymbol…-…š ‡2 `(…=Ó„ū€ž‚’Times New Roman…-…š ‡2 Jt…3Ą ‡2 Jd‡49ĄĄ„ū ’‚’Times New Roman…-…š ‡2 “¤…1p ‡2 ī§…3p„ū€ž‚’Times New Roman…-…š ‡2 J….` ‡2 J…*Ą †& ’…ū‚¼"System…-…šŖrix Matrix ändernˆˆ€ˆˆ ūšūūˆˆ€ˆˆ€æ戂€ˆ’LOBALFROMSTREAMU OLECREATELINKMONIKERCOMMONPREFIXWITH/OLEUERYCREATEFROMDATAOLEDESTROYENUDESCRIPTOR*REGISTERDRAGDRƒO’‚…4ƒ5…š$źLLOCDUMPALLOBJECTSpOLECONVRTISTORAGETOOLESTREAMEX= _IID_DEBUG›OLECREATEDEFAULTHANDLER WRITECLASTM WRITECLASĮTGOLECREATEEMBEDDINGHELPER< OL Ąué!’6&j š>æ鿉6&Vƒ> _#D‚恟LACEACTIVEOBJECTOLELOCKRUNNIlpĒ ’Ąč ,‚ …‚’‚….ƒ1‚ ‡  € †&’ƒ‰ą’Ö’` v “& MathType …ś…-‡Ü`‡ÜR „ū€žƒ–Times New Roman…- ‡2 `…hĄ ‡2 JŒ…x© ‡2 J…x© ‡2 h…li„ū€ž‚ŽSymbol…-…š ‡2 `(…=Ó ‡2 Jņ…-Ó„ū€ž‚–Times New Roman…-…š ‡2 JR…_Ą ‡2 JZ‹max)©Ą ‡2 Jä…_Ą ‡2 Jģ ‹min)iĄ †& ’…ū‚¼"System…-…š(’‡2 J….` ‡2 J’’’’ĮLOBALFROMSTREAMU OLECREATELINKMONIKERCOMMONPREFIXWITH/OLE’’’’’ RTISTORAGETOOLESTREAMEX= _IID_"‚’‚ĄF'l½40ĢĶĆĶ’’’’ ; ’’’’G1ę’’’’’’’’Gt)Main Index- t% €€°Œ€‚’Index(Gœ$ €€€‚‚’It)D X€’€ąćtĆŻ€‰€‚ć`s倉€‚ć3ō€‰‚’About Generators How to use Information about Calculation ; œd1u’’’’’’’’d äHow to Use<) % €.€°Œ€‚’ How to use Win-Rand(dČ$ €€€‚‚’™ äƒ Ō€3€ČćR›wā€‰‚ćæj«‰€‚ć¢ Ķh€‰€‚ćĶžĢt€‰€‚ćü‘Ģ€‰€‚ćGC€‰€‚ćqs󀉀‚ć4`!€‰€‚’User Interface Menu File Menu Distributions Menu Options Menu Help Selecting a Generator Calculating Window Keys ?Č#1g’’’’’’’’#YUser Interface6äY% €"€°Œ€‚’User Interface6Ų#^ Š€±€€‚‚‚‚‚ćæj«‰€‚ć¢ Ķh€‰€‚ćĶžĢt€‰€‚ćü‘Ģ€‰€‚€‚‚’Once you started Win-Rand from the program manager, a Window (which we call the MainWindow of the program) has been created:It has 4 Menuitems:1. File 2. Distributions 3. Options 4. Help : YÉ1’’’’’’’’ÉŲMenu File9+ &€€°Œ€€€‚’Menu File&É(# €€€‚’ˆS°5 :€¦€ȂV€€€ €€ €‚’The menu File contains only the entry Exit. It is just to quit the program.((Ų$ €€€‚‚’C°1q’’’’’’’’]Menu DistributionsBŲ]+ &€.€°Œ€€€‚’Menu Distributions&ƒ# €€€‚’C]Ę& €:€˜‚8€ ‚’Selecting the distributionÆwƒu8 >€ļ€˜Ö€€ €€ €€ €‚’Here you have the choice between more than thirty discrete or continuous distributions. In most cases this activates automatically the dialog window of the attached generator. For distributions where at least two generators are offered you can select your favourite sampling routine (e.g. binomial distribution, choose one of the generators btpe, bruec or bprsc).(ʝ$ €€€‚‚’= uŚ1’’’’’’’’Ś f Menu Options< + &€"€°Œ€€€‚’Menu Options(Ś> % €€˜ņ€‚’Ÿ > a €?€ր€ €€€€€€€€€€€€€€ €€ €€ €‚’The menu Options allows for the following useful settings: Colors, Default Settings and Printer Setup. By selecting Colors you get into another submenu, where you can define the colors of the Empirical Bars, the Theoretical Curve, the max. Difference and Residue Classes.By selecting the item Default Settings you get a dialog in which you are able to confirm or change the default values of the input parameters Seed and Sample Size. You can also determine your own standard for output by marking at least one of the four boxes Graphics, Moments or Quantiles, Time and Save. Save enables to store the sampled values on a file having ASCII format. Ė> @ 7 <€—€ր €€ €€ €€‚’Colors and Default Settings are saved in the file. winrand.ini and may be changed at will by using any text editor. After selecting the item Printer Setup you can determine the printer device.&> f # €€€‚’: @   1b’’’’’’’’  Ł AMenu Help9f Ł + &€€°Œ€€€‚’Menu Help ™  ī@p ®€3€€‚€ €€€€€€€€€€€€€€€€€€€€€€€€‚’In the menu Help the following items are available: Contents, About Calculation, Generators, How to use Win_Rand and About. Activating Contents leads to an Index containing some keywords you can look for. The item About Calculation gives you information about calculation under the key words Graphics, Moments, Quantiles. Generators yields a list of all generators implemented. There you can find short descriptions ofŁ ī@f the methods and references of the underlying papers (see the example below). How to use Win_Rand gives a list of 8 keywords. Clicking About activates a window containing information on the actual version of Win_Rand.(Ł A$ €€€‚‚’; ī@QA1’’’’’’’’QAŽAoDGenerators=AŽA% €0€°Œ€‚’Selecting a Generator&QA“A# €€€‚’DŽAųA* $€4€˜‚8€ƒ€ ‚’Selecting a GeneratorĪŠ“AĘCD V€€°Ö€€ €€ €€ €€ €€ €‚’After selecting a generator (here binomial generator bprsc) a dialog appears, where you can select the starting value of Seed, the input value of Sample Size and the values of the Parameters. You can also choose the output configuration by marking the corresponding checkboxes (Graphics, Moments or Quantiles, Time and Save).Your settings are accepted by activating the OK-Button.©‚ųAoD' €€°°Ö€‚’For discrete distributions an additional dialog appears in which the suggested number of classes for histograms may be changed.CĘC²D1x’’’’’’’’²DģD3MCalculating Window:oDģD% €*€°Œ€‚’Calculating Window&²DE# €€€‚’7ģDIF% €%€Ȁ‚’The calculation begins once the input values have been fixed, and output is created according the configuration defined before. Note that different sets of random numbers may be generated simultanously; the only exception is during the calculation of timings (see below).³EüH2 2€€˜€€ €€€‚‚’The theoretical curve is plotted and one can observe the generation process of the empirical values as growing sticks. After completion the graphics is represented in colors that have been set in the menu Colors (see 3. Options).If the choice has been restricted to the items Moments (or Quantiles) and Save then the small window includes a bar carrying the percentage of the task already completed. The selection of Time shuts down the mouse during calculations to make the timing as precise as possible. (If you want to compare the timings of two generators, be sure that your system runs under the same conditions for both cases).DąIF@Kd –€Ļ€°€†"€†"€†"€†"€†"€†"€†"€‚’On the right hand side of the small window there are the following 6 or (in case of SVGA Graphics) 7 buttons: . With those buttons (i) one can determine the kind of display on the small window, (ii) one can calculate output values that havent been calculated before, (iii) one can delete the small window or create an enlarged window in which all informations are visualized together. These informations can also be printed. The buttons are selfexplanatory:óUüH3Mž ¹€˜ø‚Ę€†"€ƒ€€‚†"€ƒ€€‚†"€ ƒ€€‚†"€ ƒ€€‚†"€ ƒ€€‚†"€ ƒ€€‚†"€ ƒ€€‚’View or calculate graphicsView or calculate moments (or quantiles)View or calculate noncritical regions (only in case of SVGA Graphics)Calculate timeSave the output on fileOpen large window containing all informationDelete small window (if the corresponding large window is open, it will be deleted too!)5@KhM1’’’’’’’’ hM”MäNKeys,3M”M% €€°Œ€‚’KeysP)hMäN' €S€€‚‚‚‚’If you have more calculating windows you can scroll down by using the scrollbar or by using the arrow, page down, page up home and end keys. Arrow down, up scrolls by one small window; page down, up scrolls by two small windows; home brings you to the first small window, end to the lastone.N”M2O1É’’’’’’’’ 2OxO€Information about CalculationF!äNxO% €B€°Œ€‚’ Information about Calculation&2OžO# €€€‚’r+xO€G ^€V€ąćiĶ8؀‰€‚ćQcų€‰€‚ćö„¬Å€‰€‚’Graphics Moments žO€äNQuantiles JžOf€1’’’’’’’’ f€”€v‚Information about Moments. €”€$ €€Œ€‚’Momentsāŗf€v‚( €u€€‚‚‚‚‚’Calculation of power sums of deviations about the meanThe algorithm AS 52 (APPL.STATIST. (1972) VOL.21, NO.2) adds a new value, x, when calculating a mean, and sums of powers of deviations. K indicates the powers required and must equal 1,2,3 or 4. On exit, S1 is the new value of the mean, the power sums, as indicated by k, are in S2, S3 and S4. N is the current number of observations. It must be set to zero before the first entry.L”€Ā‚1Ø’’’’’’’’ Ā‚ó‚j…Information about Quantiles1 v‚ó‚% €€°Œ€‚’QuantilesšĀ‚ …' €į€€‚‚‚‚’The P^2 Algorithm for Dynamic Calculation of Quantiles without storing Observations.A heuristic algorithm is implemented for dynamic calculation of empirical quantiles. The estimates are produced dynamically as the observations are generated. The observations are not stored therefore, the algorithm has a very small and fixed storage requirement regardless of the number of observations. This makes it ideal for implementing in this program. Reference: Raj JAIN, Imrich CHLAMTAC `;ó‚j…% €v€€ƒ‚‚’Communications of the ACM October 1985, Vol.28, No.10 K …µ…1į’’’’’’’’ µ…÷…§‡Information about GraphicsBj…÷…% €:€°Œ€‚’Information about Graphicsż½µ…ō†@ N€€€‚‚‚†"€‚‚‚‚‚†"€‚‚‚’Discrete distribution:Binomial (npq > 9), Poisson (mue > 9), Hypergeom. (n(1-M/N)M/N > 9): else : no closed formContinuous distribution:Normal: others: l = [10log10n]3÷…'‡. ,€ €‚Ę€†"€‚’ €Uō†§‡+ &€Ŗ€€‚ƒƒ‚ƒƒ‚‚‚’h .. cell widthl .. no. of cellsn .. sample size Ns .. standard deviation.A'‡č‡1u’’’’’’’’č‡!ˆeŒAbout Generators9§‡!ˆ% €(€°Œ€‚’ About Generators(č‡Iˆ$ €€€‚‚’O!ˆ˜ˆ3 6€8€‚©ć`zM€‰ƒćh,‰‚’Beta bbc Beta bsprc )gIˆĮŠĀ RĻ€„©įć¼·‹$€‰ƒć†§‰‚ć< źŽ‰ƒć¾Ė_ˆ‰‚ćoö“ˆ‰ƒć°rŲ‰‚ć@±ū‰ƒć³¦õ扂ćµŖų‰ƒć”ś~o‰‚ćģ†²¼‰ƒć"Ļq‰‚ć#ՉƒćĮ³=‰‚ć`”hʉƒćPd˜‰‚ćĪѩʉƒćÜwčP‰‚ć³^‰ƒćIf­į‰‚ćo†®ó‰ƒć9ķŗŲ‰‚’Binomial bprsc Binomial bruec Binomial btpec Burr Burr2 Cauchy Chi Exponential Exponential Power Extreme Value IExtreme Value II Gamma gds Gamma gll General Poisson Generalized inversed Gauss Geometric Hyperbolic Hypergeometric h2pec Hypergeometric hprsc Hypergeometric hruec Johnson Lambda ¤ž˜ˆeŒ¦ ż€„©įćtöC€‰ƒć­Į( ‰‚ćœĆ)n‰ƒćxbÉĖ‰‚ćł‹5Ā‰ƒćę^‰‚ćj‰‰ƒć ēV‰‚ćQŁįŁ‰ƒćtQD|‰‚ćĪ“­¦‰ƒćń;鉉‚ć[¼ī‰ƒć?°Ÿ)‰‚ćÕųüD‰ƒćiu]‰‚ćäZĀG‰ƒćYµG‰‚’Laplace Logarithmic Logarithmic Negative Binomial Normal nal Normal nsc Poisson pd Poisson pprsc Poisson pruec Slash Stable Student-t tpol Student-t trouo Triangular Uniform Von Mises Weibull Zeta S"ĮŠøŒ1›’’’’’’’’øŒģŒbĄGeometric Distribution - Inversion4 eŒģŒ( €€°Œ‚©€‚’Geometric9 øŒ%, &€€€‚‚‚‚‚‚‚‚‚’Geometric Distribution - InversionGenerating discrete random variates by Inversion is usually carried out by some search method. However, in the case of the Geometric distribution a direct transformation is possible, because of the special parallel to the continuous Exponential distribution Exp(t):X - Exp(t): G(x)=1-exp(-tx) Geo(p): pk=G(k+1)-G(k)=exp(-tk)*(1-exp(-t))p=1-exp(-t)A random number of the Geometric distribution Geo(p) is obtained by K=(long int)X, where X is from Exp(t) with parameter t=-log(1-p).1ģŒbĄ, &€ €€ćÕųüD‰‚‚‚’FUNCTION : - geo samples a random number from the Geometric distribution with parameter 0 < p < 1. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned lon%bĄeŒg integer *seed. Implemented by R.Kremer 1990, revised by P.Busswald, July 1992 !!W&%¹Ą1÷’’’’’’’’¹ĄģĄ²ÅLogarithmic - Inversion/Transformation3bĄģĄ% €€°Œ€‚’Logarithmicˆa¹ĄtĆ' €Ć€€‚‚‚‚’Logarithmic Distribution - Inversion/TransformationThe method is based on the following: A random variable X from the Logarithmic distribution has the property that X for fixed,Y=y is a Geometric variable with P(X=x|Y=y)=(1-y)*y^(x-1) (*),where Y has distribution function F(y)=ln(1-y)/ln(1-p).So first random numbers Y are generated by simple Inversion, then K=(long int) (1+ln(U)/ln(Y)) is a Geometric random number,and because of (*) a Logarithmic one. To speed up the algorithm, squeezes are used as well as the fact that many of the random numbers are 1 or 2 (depending on, special circumstances). >ģĄ²Å/ ,€€€‚‚‚‚ćÕųüD‰‚‚’On an IBM/PC 486 optimal performance is achieved, if for p<.97,simple inversion is used and otherwise the transformation. FUNCTION : - lsk samples a random number from the Logarithmic distribution with parameter 0 < p < 1 . REFERENCE : - A.W. Kemp (1981): Efficient generation of logarithmically distributed pseudo-random variables, Applied Statistics 30, 249-253. SUBPROGRAM : - drand(seed) .(0,1)-Uniform generator with unsigned long integer *seed. Implemented by R.Kremer 1990, revised by P.Busswald, July 1992,!!DtĆöÅ1é’’’’’’’’öÅ-ĘåČPoisson - Inversion7²Å-Ę( €€°Œ€€‚’Poisson pd ģöÅHČ/ ,€Ł€€‚‚‚‚‚ćÕųüD‰‚’Poisson Distribution - Tabulated Inversion combined with Acceptance ComplementFUNCTION : - pd samples a random number from the Poisson distribution with parameter my > 0. Tabulated Inversion for my < 10. Acceptance Complement for my >= 10.REFERENCE : - J.H. Ahrens, U. Dieter (1982): Computer generation of Poisson deviates from modified normal distributions ACM Trans. Math. Software 8, 163-179.SUBPROGRAMS: - drand(seed) (0,1)-Uniform generator with unsigned long integer *seed. x-ĘåČ% €š€€‚‚‚’NORMAL(seed): Normal generator N(0,1).Implemented by R. Kremer, August 1990, Revised by E. Stadlober, April 1992 !!V%HČ;É1!’’’’’’’’;ÉpÉ`ĶPoisson - Ratio of Uniforms/Inversion5åČpÉ% € €°Œ€‚’Poisson pruec=;É­Ė' €-€€‚‚‚‚’Poisson Distribution - Ratio of Uniforms/InversionFor parameter my < 5.5 the Inversion method is applied: The random numbers are generated via sequential search, starting at the lowest index k=0. The cumulative probabilities,are avoided by using the technique of chop-down. For my >= 5.5 Ratio of Uniforms is employed: A table mountain hat function h(x) with optimal scale parameter s for fixed location parameter a = my+1/2 is used.FUNCTION : - pruec samples a random number from the Poisson distribution with parameter my > 0.³…pÉ`Ķ. *€ €€‚‚ćÕųüD‰‚‚‚’REFERENCE : - E. Stadlober (1989): Sampling from Poisson binomial and hypergeometric distributions: ratio of uniforms as a simple and fast alternative, Bericht 303, Math. Stat. Sektion, Forschungsgesellschaft Joanneum, Graz. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by R.Kremer 1990, revised by P.Busswald, July 1992,!!X'­ĖøĶ1Ä’’’’’’’’øĶķĶōPoisson - Patchwork Rejection/Inversion5`ĶķĶ% € €°Œ€‚’Poisson pprscÜøĶ ' €¹€€‚‚‚‚’Poisson Distribution - Patchwork Rejection/InversionFor parameter my < 10 Tabulated Inversion is applied. For my >= 10 Patchwork Rejection is employed:The area below the histogram function f(x) is rearranged in its body by certain point reflections. Within a large center, interval variates are sampled efficiently by rejection from uniform hats. Rectangular immediate acceptance regions speed up the generation. The remaining tails are covered by exponential functions.ķĶ `Ķč¹ķĶō/ ,€s€€‚‚‚ćÕųüD‰‚‚‚’FUNCTION : - pprsc samples a random number from the Poisson distribution with parameter my > 0.REFERENCE : - H. Zechner (1994): Efficient sampling from continuous and discrete unimodal distributions, Doctoral Dissertation, 156 pp., Technical University Graz, Austria. SUBPROGRAM : - drand(seed) (0,1)-Uniform generator with, unsigned long integer *seed. Implemented by H. Zechner, January 1994, Revised by F. Niederl, July 1994 !!Z) N1Ī’’’’’’’’N„!Binomial - Acceptance Rejection/Inversion6ō„% €"€°Œ€‚’Binomial btpec÷N¢' €ļ€€‚‚‚‚’Binomial-Distribution - Acceptance Rejection/Inversion.For min(n*p,n*(1-p)) < 10 the Inversion method is applied: The random numbers are generated via sequential search, starting at the lowest index k=0. The cumulative probabilities,are avoided by using the technique of chop-down. For min(n*p,n*(1-p)) >= 10 Acceptance Rejection is used: The algorithm is based on a hat-function which is uniform in the centre region and exponential in the tails. A triangular immediate acceptance region in the centre speeds up the generation of binomial variates. If candidate K is near the mode, f(K) is computed recursively, starting at the mode m. If p <= .5 the algorithm is applied to parameters n, p. Otherwise p is replaced by 1-p, and K is replaced by n - K. š„¾, &€į€€‚‚ćÕųüD‰‚’FUNCTION : - btpec samples a random number from the binomial, distribution with parameters n and p and is valid for n*min(p,1-p) > 0. REFERENCE : - V. Kachitvichyanukul, B.W. Schmeiser (1988): Binomial random variate generation, Communications of the ACM 31, 216-222. SUBPROGRAMS: - fc(k) ... Correction term of the Stirling, approximation for log(k!) (series in 1/k or, table values for small k) with long int k, - drand(seed) ... (0,1)-Uniform generator with unsigned long int *seed. c?¢!$ €~€€‚‚’Implemented by H. Zechner and P. Busswald, September 1992!!W&¾x1ž’’’’’’’’x®Binomial - Ratio of Uniforms/Inversion6!®% €"€°Œ€‚’Binomial bruecŻ·x‹ & €o€€‚‚‚’Binomial Distribution - Ratio of Uniforms/Inversion.For min(n*p,n*(1-p)) < 5 the Inversion method is applied: The random numbers are generated via sequential search, starting at the lowest index k=0. The cumulative probabilities are avoided by using the technique of chop-down. For min(n*p,n*(1-p)) >=5 Ratio of Uniforms is employed: A table mountain hat function h(x) with optimal scale parameter s for fixed location parameter a = mu+1/2 is used. If the candidate K is near the mode and K>29 (respectively, n-K>29) f(K) is computed recursively starting at the mode m. The basic algorithm is valid for n*p > 0, n > =1, p <= 0.5. For p > 0.5, p is replaced by 1-p and K is replaced by n-K. 'ś®² - (€õ€€‚‚‚ćÕųüD‰‚’FUNCTION : - bruec samples a random number from the Binomial, distribution with parameters n and p. It is valid for n * min(p,1-p) > 0.REFERENCE : - E. Stadlober (1989): Sampling from Poisson binomial and hypergeometric distributions: ratio of uniforms as a simple and fast alternative, Bericht 303, Math. Stat. Sektion, Forschungsgesellschaft Joanneum, Graz. SUBPROGRAMS: - flogfak(k) ... log(k!) with long integer k, - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. iE‹ $ €Š€€‚‚’Implemented by R.Kremer 1990, revised by P.Busswald, July 1992,!!Y(² t1% ’’’’’’’’tŖFBinomial - Patchwork Rejection/Inversion6Ŗ% €"€°Œ€‚’Binomial bprsc?tõA& €3€€‚‚‚’Binomial Distribution - Patchwork Rejection/Inversion For l=min(n*p,n*(1-p))< 10 Inversion method bmdu is applied: The random numbers are generated via modal down-up search, starting at the mode m. The cumulative probabilities are avoided by using the technique of chop-down. For min(n*p,n*(1-p)) >=10ŖõA Patchwork Rejection method bprs is employed: The area below the histogram function f(x) is rearranged in its body by certain point reflections. Within a large center interval variates are sampled efficiently by rejection from uniform hats. Rectangular immediate acceptance regions speed up the generation. The remaining tails are covered by exponential functions. The basic algorithms are valid for n*p > 0, n > =1, p <= 0.5. For p > 0.5, p is replaced by 1-p and K is replaced by n-K.“fŖˆE- (€Ķ€€‚‚‚ćÕųüD‰‚’FUNCTION : - bprsc samples a random number from the Binomial distribution with parameters n and p. It is valid for n * min(p,1-p) > 0. REFERENCE : - C.D. Kemp (1986): A modal method for generating binomial variables, Commun. Statistics, Theory & Methods 15, 805-813. - H. Zechner (1994): Efficient sampling from continuous and discrete unimodal distributions, Doctoral Dissertation, 156 pp, Technical, University Graz, Austria. SUBPROGRAMS: - fc(k) ... Correction term of the Stirling approximation for log(k!) (series in 1/k or table values for small k) with long int k, - flogfak(k) ... log(k!) with long integer k, - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. - bmdu(seed,n,p) ... Binomial generator for l<10, - bprs(seed,n,p) ... Binomial generator for l>=10, with unsigned long integer *seed, long integer n, double p. yRõAF' €¤€€ƒƒƒƒ‚’Implemented by H. Zechner, January 1994, Revised by F. Niederl, July 1994!!l;ˆEmF1y ’’’’’’’’mFØF €Generalized Poisson - Inversion/Ratio of Uniforms/Rejection;FØF% €,€°Œ€‚’Generalized PoissonŽmF¬K& €½ €€‚‚‚’Generalized Poisson Distribution - Inversion/Ratio of Uniforms/ Rejection.The Inversion method gpdinv is applied in the two regions (lm < lambda < 0, my > 0) and (0 <= lambda <= .65, 0 < my < 2),where lm = max{-1,-my/n} and n(>=4) is n=max{k|my+k*lambda}. The validity of parameter combinations (my,lambda) may be checked by the external function checkinv. The random numbers are generated via sequential search started at the lowest index k=k1. The cumulative probabilities are, avoided by using the technique of chop-down. The Ratio of Uniforms method gpdrou of Busswald applies within the three regions (0 < lambda < myu, 2 <= my <= 120), (0 < lambda < mym, 120 < my <= 250) and (0 < lambda < myl, 250 < my) where myu=my/(my+1), myl=my/(my+2) and mum=(myu+myl)/2. A table mountain hat h(x) with simple approximations of the parameters a (location) and s (scale) is used In the set-up step the mode m has to be searched by starting at a lower bound of the mode. For some parameter combinations this search may be expensive. The remaining parts of the parameter space are covered by the rejection method gpcd of Devroye. It uses a two-part envelope: A geometric hat within the left interval [0,t] and a certain polynomial hat otherwise. ńØFÄM' €ć€€‚‚‚‚’FUNCTION : - gpoic samples a random number from the Generalized Poisson distribution with the two parameters my > 0 and lambda < 1. REFERENCE : - P. Busswald (1993): Effiziente Zufallszahlen-, erzeugung aus der Zetaverteilung und der, verallgemeinerten Poissonverteilung, Diplomarbeit, 152 pp., Technische Universitaet Graz, Austria., - L. Devroye (1989): Random variate generators, for the Poisson-Poisson and related distributions, Computational Statistics & Data Analysis 8, 247-278. ,ž¬K €. *€ż€€ćÕųüD‰ƒƒƒƒ‚’SUBPROGRAMS: - flogfak(k) ... log(k!) with long integer k - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed - gpinv(seed,my,lambda) ... Gen.Poiss. generator for two parameter regions - gpdrou(seed,my,lambda) ... Gen.Poiss. generator for three parameter regions - gpdc(seed,my,lambda) ... Gen.Poiss. generator for remaining parameter regions with unsigned long *seed, double my, double lambda, Implemented by P. Busswald, January 1993, Revised by F. Niederl, July 1994 !!ÄM €FT#ÄM`€1”’’’’’’’’`€™€ „Negative Binomial - Compound method9 €™€% €(€°Œ€‚’Negative Binomial]/`€ö‚. *€_€€‚‚‚‚ćÕųüD‰‚’Negative Binomial Distribution - Compound methodFUNCTION : - nbp samples a random number from the Negative Binomial distribution with parameters r (no. of failures given) and p (probability of success), valid for r > 0, 0 < p < 1., If G from Gamma(r) then K from Poiss(pG/(1-p)), is NB(r,p)--distributed. REFERENCE : - J.H. Ahrens, U. Dieter (1974): Computer methods for sampling from gamma, beta, Poisson and, binomial distributions, Computing 12, 223--246.SUBPROGRAMS: - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed ļ™€ „$ €ß€€‚’ - Gamma(seed,a) ... Gamma generator for a > 0, with unsigned long integer *seed, double a, - Poisson(seed,a) ...Poisson generator for a > 0, with unsigned long integer *seed, double a., Implemented by E. Stadlober, September 1991 !!Y(ö‚b„1’’’’’’’’b„Ž„k‰Zeta Distribution - Acceptance Rejection, „Ž„% €€°Œ€‚’ZetaX2b„ę‡& €e€€‚‚‚’Zeta Distribution - Acceptance RejectionTo sample from the Zeta distribution with parameters ro and pk, it suffices to sample variates X from the distribution with density function f(x)=B*{[x+0.5]+pk}^(-(1+ro)) ( x > .5 ), and then to deliver K=[X+0.5].1/B=Sum[(j+pk)^-(ro+1)] (j=1,2,...) converges for ro >= .5 . Variates X are generated by rejection using density function g(x)=ro*(c+0.5)^ro*(c+x)^-(ro+1). Integer overflow is possible, when ro is small (ro <= .5) and pk large. In this case a new sample is generated. If ro and pk satisfy the inequality ro > .14 + pk*1.85e-8 + .02*ln(pk) the percentage of overflow is less than 1%, so that the, result is reliable. If either ro > 100 or k > 10000 numerical problems in computing the theoretical moments arise, therefore ro<=100 and,k<=10000 are recommended. …YŽ„k‰, &€³€€‚‚ćÕųüD‰‚’FUNCTION : - zet samples a random number from the Zeta distribution with parameters ro > 0 and pk >= 0. REFERENCE : - J. Dagpunar (1988): Principles of Random Variate Generation, Clarendon Press, Oxford. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by P. Busswald, September 1992!!m<ę‡Ų‰1Ō’’’’’’’’Ų‰Š·Hypergeometric Distribution - Acceptance Rejection/Inversion<k‰Š% €.€°Œ€‚’Hypergeometric h2pecļŲ‰(% €ß€€‚‚’Hypergeometric Distribution - Acceptance Rejection/InversionThe basic algorithm works for parameters 1 <= n <= M <= N/2. Otherwise parameters are redefined in the set-up step and the random number K is adapted before delivering. For l = m - max(0,n-N+M) < 10 Inversion is applied: The random numbers are generated via sequential search starting at the lowest index k=0. The cumulative probabilities are avoided by using the technique of chop-down. For l >= 10 Acceptance Rejection is employed: The algorithm is based on a hat-function which is uniform in the centre region and exponential in the tails. Squeezes (lower and upper bounds) are used. If m <= 100 or candidate K is near the mode m, f(K) is computed recursively starting at the mode m. )üŠQ- (€ł€€‚‚‚ćÕųüD‰‚’FUNCTION : - h2pec samples a random number from the Hypergeometric distribution with parameters N (number of red and black balls), M (number, of red balls) and n (number of trials), valid for N >= 2, M,n <= N. REFERENCE : - V. Kachitvichyanukul, B.W. Schmeiser (1985): Computer generation of hypergeometric random variates, J.Statist.Comput.Simul. 22, 127-145. SUBPROGRAMS: - flogfak(k) ... log(k!) with long integer k - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. fC(·# €†€€‚’Implemented by R.Kremer 1990, revised by P.Busswald, July 1992!!],Q Ą1’’’’’’’’ Ą]Ą3ĘHypergeometric - Ratio o· Ą·f Uniforms/Inversion=·]Ą% €0€°Œ€‚’ Hypergeometric hruecņĶ ĄOĆ% €›€€‚‚’Hypergeometric Distribution - Ratio of Uniforms/InversionThe basic algorithm works for parameters 1 <= n <= M <= N/2. Otherwise parameters are redefined in the set-up step and the random number K is adapted before delivering. For l = m - max(0,n-N+M) < 5 Inversion is applied: The random numbers are generated via sequential search starting at the lowest index k=0. The cumulative probabilities are avoided by using the technique of chop-down. For l >=5 Ratio of Uniforms is employed: A table mountain hat function h(x) with optimal scale, parameter s for fixed location parameter a = my+1/2 is used. If the mode m <= 20 and the candidate K is near the mode, f(K) is computed recursively starting at the mode m. vI]ĄÅÅ- (€“€€‚‚‚ćÕųüD‰‚’FUNCTION : - hruec samples a random number from the Hypergeometric distribution with parameters, N (number of red and black balls), M (number, of red balls) and n (number of trials), valid for N >= 2, M,n <= N. REFERENCE : - E. Stadlober (1989): Sampling from Poisson, binomial and hypergeometric distributions: ratio of uniforms as a simple and fast alternative, Bericht 303, Math. Stat. Sektion, Forschungsgesellschaft Joanneum, Graz. SUBPROGRAMS: - flogfak(k) ... log(k!) with long integer k - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. nHOĆ3Ę& €€€ƒƒƒ‚’Implemented by R.Kremer 1990, revised by P.Busswald, July 1992,!! l;ÅÅŸĘ1ų’’’’’’’’ŸĘŪĘ¢ĶHypergeometric Distribution - Patchwork Rejection/Inversion<3ĘŪĘ% €.€°Œ€‚’Hypergeometric hprschCŸĘCŹ% €‡€€‚‚’Hypergeometric Distribution - Patchwork Rejection/InversionThe basic algorithms work for parameters 1 <= n <= M <= N/2. Otherwise parameters are redefined in the set-up step and the random number K is adapted before delivering. For l = m-max(0,n-N+M) < 10 Inversion method hmdu is applied: The random numbers are generated via modal down-up search, starting at the mode m. The cumulative probabilities, are avoided by using the technique of chop-down. For l >= 10 the Patchwork Rejection method hprs is employed: The area below the histogram function f(x) in its body is rearranged by certain point reflections. Within a large center interval variates are sampled efficiently by rejection from uniform hats. Rectangular immediate acceptance regions speed up the generation. The remaining tails are covered by exponential functions.ģĄŪĘ/Ķ, &€€€‚‚ćÕųüD‰‚’ FUNCTION : - hprsc samples a random number from the Hypergeometric distribution with parameters N (number of red and black balls), M (number of red balls) and n (number of trials), valid for N >= 2, M,n <= N.REFERENCE : - H. Zechner (1994): Efficient sampling from continuous and discrete unimodal distributions, Doctoral Dissertation, 156 pp., Technical University Graz, Austria., SUBPROGRAMS: - flogfak(k) ... log(k!) with long integer k - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. - hmdu(seed,N,M,n) ... Hypergeometric generator for l<10 - hprs(seed,N,M,n) ... Hypergeometric generator, for l>=10 with unsigned long integer *seed, long integer N, M, n..sPCŹ¢Ķ# € €€‚’ Implemented by H. Zechner, January 1994, Revised by F. Niederl, July 1994!!P/ĶņĶ1V’’’’’’’’ņĶ ĪEĻCauchy Distribution - Inversion. ¢Ķ Ī% €€°Œ€‚’Cauchy%÷ņĶEĻ. *€ļ€€‚‚‚ćÕųüD‰‚‚’Cauchy Distribution - InversionFUNCTION : - cin samples a random number from the standard Cauchy distribution C(0,1).SUBPROGRAMS: - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by R. Kremer, 1990!!U$ ĪšĻ1,’’’’’’’’šĻĶĻExponential Distribution - Inversion3EĻĶĻ% €€°Œ€‚’Exponential7šĻ/ ,€€€‚‚‚ćÕųüD‰‚ƒ‚’ExpĶĻEĻonential Distribution - Inversion FUNCTION : - eln samples a random number from the standard, Exponential distribution Exp(0,1). SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by R. Kremer 1990 !!Y(ĶĻi1h’’’’’’’’i ×Extreme value I Distribution - Inversion7 % €$€°Œ€‚’Extreme Value I7i×0 .€€€‚‚‚ćÕųüD‰ƒƒƒ‚’Extreme value I Distribution - Inversion FUNCTION : - ev_i samples a random number from the Extreme Value I distribution.SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, August 1992!! Q  (1}’’’’’’’’ (W£Laplace Distribution - Inversion/ ×W% €€°Œ€‚’LaplaceL(£0 .€9€€‚‚‚ćÕųüD‰ƒƒƒ‚’Laplace (Double Exponential) Distribution - InversionFUNCTION : - lapin samples a random number from the standard Laplace distribution Lap(0,1).SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by W. Hoermann, April 92!!R!Wõ1e’’’’’’’’!õ)]Logistic Distribution - Inversion4£)% €€°Œ€‚’ Logarithmic4õ]0 .€ €€‚‚‚ćÕųüD‰ƒƒƒ‚’Logistic Distribution - InversionFUNCTION : - login samples a random number from the standard Logistic distribution Log(0,1).SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by R. Kremer, 1990 !! S")°1d’’’’’’’’"°ā Normal Distribution - Alias Method2 ]ā% €€°Œ€‚’Normal nal¤°† % €’€€‚‚’Normal Distribution - Alias Method The signless normal density, defined in [0,infinity) is majorized by a step function h_1(x) in the center (0<=x<4). The tail (x>=4) is majorized by a function h_2(x) proportional to (4x-15)^-2. Sampling from h_1(x) is done by the Alias method, using three tables of 128 one-byte integers each independent of the intended precision. Sampling from h_2(x) is carried out by Inversion. A normal variate x (-4=4) is obtained as x=sign(u)(3.75+.25/u) if the rejection test is successful. Note that algorithm nal is not portable. For the alias part an intrinsic uniform generator is used. Logical and shift operations lead to a loss of a few bits. ]ā 2 2€»€€‚‚‚ćÕųüD‰‚ƒƒƒ‚‚’FUNCTION : - nal samples a random number from the standard Normal distribution N(0,1).REFERENCE : - J.H. Ahrens, U. Dieter (1989):, An alias method for sampling from the normal distribution, Computing 42, 159-170.SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by R. Kremer, 1990!!g6† | 1ė’’’’’’’’#| ® hNormal Distribution - Sine-Cosine or Box/Muller Method2  ® % €€°Œ€‚’Normal nscą| ¶( €Į€€‚‚‚‚‚’Normal Distribution - Sine-Cosine or Box/Muller MethodThis method is based on the transformation: X=sqrt(-2lnU)cos(2pi*V), Y=sqrt(-2lnU)sin(2pi*V), which converts two independent (0,1)-Uniforms U and V to two independent standard Normal variates X and Y. FUNCTION : - nsc samples a random number from the standard Normal distribution N(0,1).REFERENCE : - G.E.P. Box, M.E. Muller (1958): A note on the generation of random normal deviates, Annals Math. Statist. 29, 610-611.²‡® h+ $€€€ćÕųüD‰‚‚’SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by W. Hoermann, April 1992!!I¶±1’’’’’’’’$±ŽBASlash Distribution - Z/U-hŽ% €€°Œ€‚’SlashX*±BA. *€U€€ŽBAh‚‚‚ćÕųüD‰‚‚’Slash Distribution - Z/U (Z from N(0,1), U from U(0,1))FUNCTION : sla samples a random number from the Slash distributionSUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed - NORMAL(seed) ... Normal generator N(0,1). Implemented by R. Kremer, 1990 !!T#Ž–A1’’’’’’’’%–AČACTriangular Distribution - Inversion2 BAČA% €€°Œ€‚’TriangularN–AC/ ,€?€€‚‚‚ćÕųüD‰‚ƒ‚’Triangular Distribution - Inversion: X = +-(1-sqrt(U))FUNCTION : - tra samples a random number from the standard Triangular distribution in (-1,1)SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, August 1992!!,Z)ČApC1^’’’’’’’’&pCCŹEBurr II; VII; X Distributions - Inversion-CC% €€°Œ€‚’Burr1ŪpC¤E, &€·€€‚‚‚‚‚ƒƒƒ‚’Burr II, VII, VIII, X Distributions - InversionFUNCTION : - burr1 samples a random number from one of the Burr II, VII, VIII, X distributions with parameter r > 0, where the no. of the distribution is indicated by a pointer variable. REFERENCE : - L. Devroye (1986): Non-Uniform Random Variate, Generation, Springer Verlag, New York.SUBPROGRAM : - drand(seed) ... (0,1)-uniform generator with unsigned long integer *seed.Implemented by F. Niederl, August 1992!! &CŹE# €€€‚’j9¤E4F1ū’’’’’’’’'4F`F*HChi Distribution - Ratio of Uniforms with shift at mode m,ŹE`F% €€°Œ€‚’Chi Ź™4F*H1 0€3€€‚‚‚‚ćÕųüD‰‚ƒ‚‚’Chi Distribution - Ratio of Uniforms with shift at mode m FUNCTION : - chru samples a random number from the Chi distribution with parameter a > 1. REFERENCE : - J.F. Monahan (1987): An algorithm for generating chi random variables, ACM Trans., Math. Software 13, 168-172. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by R. Kremer, 1990 !!f5`FH1~’’’’’’’’(HÉHKExponential Power Distribution - Acceptance Rejection9*HÉH% €(€°Œ€‚’Exponential Power'öHšJ1 0€ķ€€‚‚‚ćÕųüD‰‚ƒƒƒ‚’Exponential Power Distribution - Acceptance RejectionFUNCTION : - epd samples a random number from the Exponential Power distribution with parameter tau >= 1 by using the non-universal rejection method for logconcave densities. REFERENCE : - L. Devroye (1986): Non-Uniform Random Variate Generation, Springer Verlag, New York., SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by K. Lehner, 1990, Revised by F. Niederl, August 1992 !! &ÉHK# €€€‚’Z)šJpK1ƒ’’’’’’’’)pKØKśLExtreme Value II Distribution - Inversion8KØK% €&€°Œ€‚’Extreme Value IIR!pKśL1 0€C€€‚‚‚ćÕųüD‰‚ƒƒƒ‚’Extreme Value II Distribution - InversionFUNCTION : - ev_ii samples a random number from the Extreme Value II distribution with parameter a > 0. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, August 1992 !!~MØKxM1&’’’’’’’’*xM©M’€Gamma Distribution - Acceptance Rejection combined with Acceptance Complement1 śL©M% €€°Œ€‚’Gamma gdsK#xM €( €G€€‚‚‚‚‚’Gamma Distribution - Acceptance Rejection combined with Acceptance Complement FUNCTION : - gds samples a random number from the standard gamma distribution with parameter a > 0. Acceptance Rejection gs for a < 1, Acceptance Complement gd for a >= 1. REFERENCES : - J.H. Ahrens, U. Dieter (1974): Computer methods for sampling from gamma, beta, Poisson and binomial distributions, Computing 12, 223-246., - J.H. Ahrens, U. Dieter (1982): Generating gamma, variates by a modified rejection technique, Communications of the ACM 25, 47-54. ©M €śLóČ©M’€+ $€‘€€ćÕųüD‰‚‚’SUBPROGRAMS: - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed - NORMAL(seed) ... Normal generator N(0,1). Implemented by G. Thallinger and E. Stadlober, August 1990!!wF €v1 ’’’’’’’’+v§ƒGamma Distribution - Acceptance Rejection with log-logistic; envelopes1 ’€§% €€°Œ€‚’Gamma gllŚ«vƒ/ ,€W€€‚‚‚‚ćÕųüD‰‚‚’Gamma Distribution - Acceptance Rejection with log-logistic envelopes FUNCTION : - gll produces a sample from the standard gamma distribution with parameter a > 0. REFERENCE : - R.C.H. Cheng (1977): The generation of gamma, variables with non-integral shape parameter, Appl. Statist. 26, 71-75.SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by R. Kremer, 1990!![*§܃1ü’’’’’’’’,܃„Ž…Student-t Distribution - Ratio of Uniforms7ƒ„% €$€°Œ€‚’Student-t trouoĖ™܃Ž…2 2€3€€‚‚‚‚ćÕųüD‰‚ƒƒƒ‚’Student-t Distribution - Ratio of UniformsFUNCTION : - trouo samples a random number from the Student-t distribution with parameter a >= 1. REFERENCE : - A.J. Kinderman, J.F. Monahan (1980): New methods for generating Student's t and gamma variables, Computing 25, 369-377.SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by R. Kremer, 1990!! V%„4†1č’’’’’’’’-4†j†!‰Student-t Distribution - Polar Method6Ž…j†% €"€°Œ€‚’Student-t tpol ā4†tˆ( €Å€€‚‚‚‚‚’Student-t Distribution - Polar Method. The polar method of Box/Muller for generating Normal variates is adapted to the Student-t distribution. The two generated variates are not independent and the expected no. of uniforms per variate is 2.5464.FUNCTION : - tpol samples a random number from the Student-t distribution with parameter a > 0. REFERENCE : - R.W. Bailey (1994): Polar generation of random variates with the t-distribution, Mathematics of Computation 62, 779-781.­j†!‰, &€€€ćÕųüD‰‚‚‚’SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by F. Niederl, 1994!!^-tˆ‰1C’’’’’’’’.‰°‰Ā‹Von Mises Distribution - Acceptance Rejection1 !‰°‰% €€°Œ€‚’von Misesą‰Ā‹2 2€Į€€‚‚‚‚ćÕųüD‰‚ƒƒƒ‚’Von Mises Distribution - Acceptance Rejection FUNCTION : - mwc samples a random number from the von Mises distribution ( -Pi <= x <= Pi) with parameter, k > 0 via rejection from the wrapped Cauchy distribution.REFERENCE : - D.J. Best, N.I. Fisher (1979): Efficient simulation of the von Mises distribution, Appl. Statist. 28, 152-157.SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, August 1992!! Q °‰Œ1h’’’’’’’’/ŒBŒyWeibull Distribution - Inversion/ Ā‹BŒ% €€°Œ€‚’Weibull7 Œy. *€€€‚‚‚ćÕųüD‰‚‚’Weibull Distribution - Inversion FUNCTION : - win samples a random number from the Weibull distribution with parameter c > 0. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, August 1992!!k:BŒä1«’’’’’’’’0䍎›ĄBeta Distribution - Acceptance Rejection with log-logistic1 yŽ% €€°Œ€‚’Beta bbbcīä=Ą. *€Ż€€‚‚‚‚ćÕųüD‰‚’Beta Distribution - Acceptance Rejection with log-logistic hats for the Beta Prime DistributionFUNCTION : - bbbc samples a random number from the beta distribution with parameters a,b (a > 0, b > 0). It combines algorithms bb (a > 1, b > 1) and bc (a<1 or b<1). REFERENCE : - R.C.H. Cheng (1978): Generating beta variates with nonintegral shape parameters, Communications of the ACM 21, 317-322.SUBPROGRAM : - drand(seed) ... (0,1)-UniformŽ=Ąy generator with unsigned long integer *seed. ^8Ž›Ą& €p€€ƒƒƒ‚’Implemented by E. Stadlober and R. Kremer, 1990!! m<=ĄĮ1Ü’’’’’’’’1Į:ĮåĒBeta Distribution - Stratified Rejection/Patchwork Rejection2 ›Ą:Į% €€°Œ€‚’Beta bsprcōĻĮ.Ä% €Ÿ€€‚‚’Beta Distribution - Stratified Rejection/Patchwork RejectionFor parameters a < 1, b < 1 and a < 1 < b or b < 1 < a, the stratified rejection methods b00 and b01 of Sakasegawa are used. Both procedures employ suitable two-part power functions, from which samples can be obtained by inversion. If a > 1, b > 1 (unimodal case) the patchwork rejection method b1prs of Zechner/Stadlober is utilized: The area below the density function f(x) in its body is rearranged by certain point reflections. Within a large center, interval variates are sampled efficiently by rejection from uniform hats. Rectangular immediate acceptance regions speed up the generation. The remaining tails are covered by exponential functions. "ū:ĮPĘ' €÷€€‚‚‚‚’If (a-1)(b-1) = 0 sampling is done by inversion if either a or b are not equal to one. If a = b = 1 a uniform random, variate is delivered.FUNCTION : - bsprc samples a random variate from the beta distribution with parameters a > 0, b > 0.REFERENCES : - H. Sakasegawa (1983): Stratified rejection and squeeze method for generating beta random numbers, Ann. Inst. Statist. Math. 35 B, 291-302., - H. Zechner, E. Stadlober (1993): Generating beta variates via patchwork rejection, Computing 50, 1-18.•g.ÄåĒ. *€Ļ€€ćÕųüD‰‚ƒƒƒ‚’SUBPROGRAMS: - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. - b00(seed,a,b) ... Beta generator for a<1, b<1, - b01(seed,a,b) ... Beta generator for a<11, b>1, with unsigned long integer *seed, double a, b. Implemented by H. Zechner and F. Niederl, July 1994!! f5PĘKČ1Z’’’’’’’’2KČxČ”ŹBurr III; IV; V; VI; IX; XII Distribution - Inversion-åĒxČ% €€°Œ€‚’Burr2)ųKČ”Ź1 0€ń€€‚‚‚ćÕųüD‰‚ƒƒƒ‚’Burr III, IV, V, VI, IX, XII Distribution - Inversion FUNCTION : - burr2 samples a random number from one of the Burr III, IV, V, VI, IX, XII distributions with parameters r > 0 and k > 0, where the no. of the distribution is indicated by a pointer variable. REFERENCE : - L. Devroye (1986): Non-Uniform Random Variate, Generation, Springer Verlag, New York., SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, August 1992!! n=xČĖ1ƒ’’’’’’’’3ĖHĖšĪGeneralized Inverse Gaussion Distribution - Ratio of Uniforms9”ŹHĖ% €(€°Œ€‚’Generalized Gauss{TĖĆĶ' €©€€‚‚‚‚’Generalized Inverse Gaussian Distribution - Ratio of UniformsFUNCTION : - gigru samples a random number from the reparameterized Generalized Inverse Gaussian distribution with parameters l > 0 and b > 0, using Ratio of Uniforms method without shift, for l <= 1 and b <= 1 and shift at mode m, otherwise. REFERENCES : - J.S. Dagpunar (1989): An easily implemented, generalized inverse Gaussian generator, Commun. Statist. Simul. 18(2), 703-710., - K. Lehner (1989): Erzeugung von Zufallszahlen fuer zwei exotische Verteilungen, Diplomarbeit,, 107 pp., Techn. Universitaet Graz, Austria. שHĖšĪ. *€S€€ćÕųüD‰‚ƒƒƒ‚’SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by K. Lehner, 1990, Revised by F. Niederl, August 1992!! rAĆĶ Ļ1’’’’’’’’4 Ļ;ĻöJohnson Distribution - Transformation of one Normal random number/ šĪ;Ļ% €€°Œ€‚’Johnson> Ļ…. *€!€€‚‚‚‚ćÕųüD‰‚’Johnson Distribution - Transformation of one Normal random, numberFUNCTION : - john samples a random number from the Johnson S(B), S(L), or S(U) dist;Ļ…šĪribution with parameters m and s > 0, where the type of the distribution is indicated by a pointer variable.REFERENCES : - Johnson, N.L. (1949): Systems of frequency curves generated by methods of translation, Biometrika 36, 146-176. SUBPROGRAMS: - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed, - NORMAL(seed) ... Normal generator N(0,1). qK;Ļö& €–€€ƒƒƒ‚’Implemented by R. Kremer, 1990, Revised by F. Niederl, August 1992!! P…F1’’’’’’’’5FtDLambda Distribution - Inversion. öt% €€°Œ€‚’LambdaŠžFD2 2€=€€‚‚‚‚ćÕųüD‰‚ƒƒƒ‚’Lambda Distribution - InversionFUNCTION : - lamin samples a random number from the Lambda distribution with parameter l3 and l4.REFERENCES : - J.S. Ramberg, B:W. Schmeiser (1974): An approximate method for generating asymmetric, variables, Communications ACM 17, 78-82. SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed.Implemented by F. Niederl, August 1992!! _.t£1E’’’’’’’’6£ŃåStable Distribution - Integral Representations. DŃ% €€°Œ€‚’StableB£) €3€€‚‚‚‚‚‚’Stable Distribution - Integral Representations The method is based upon an integral representation of the form g(U)*X^(-(1-a)/a) where X is exponentially distributed and g(U) is a function of a uniform (0,1) random variate U.FUNCTION : - stab samples a random number from the Stable distribution with parameters 0 < a <= 2 and -1 <= d <= 1. REFERENCES : - J.M. Chambers et al. (1976): A method for simulating stable random variables, J. Amer. Statist. Assoc. 71, 340-344., Correction: J. Amer. Statist. Assoc. 82, (1987),, 704.Ņ¤Ńå. *€I€€ćÕųüD‰‚ƒƒƒ‚’SUBPROGRAM : - drand(seed) .. (0,1)-Uniform generator, unsigned long integer *seed. Implemented by R. Kremer, 1990, Revised by F. Niederl, August 1992!! b1G1n’’’’’’’’7Gy¶ Hyperbolic Distribution - Non Universal Rejection2 åy% €€°Œ€‚’HyperbolicēG 0 .€Ļ€€‚‚‚‚ćÕųüD‰‚ƒ‚’Hyperbolic Distribution - Non-Universal RejectionFUNCTION : - hyplc samples a random number from the hyperbolic distribution with parameters a and b, valid for a > 0 and |b| < a using the non-universal rejection method for log-concave densities.REFERENCE : - L. Devroye (1986): Non-Uniform Random Variate, Generation, Springer Verlag, New York.SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with unsigned long integer *seed. Implemented by F. Niederl, November 1992!!&y¶ # €€€‚’J  1“’’’’’’’’8 0 ’ (0;1)Uniform Distribution0 ¶ 0 % €€°Œ€‚’Uniform Ś 2 ( €µ€€‚‚‚ƒ‚’(0,1) Uniform Distribution FUNCTION : - drand samples a random number from the (0,1), uniform distribution by the multiplicative, congruential method of the form z(i+1) = a * z(i) (mod m), a=663608941 (=0X278DDE6DL), m=2**32. - Before the first call of the generator an integer, value z(0)=4*k+1 where 0 < 4*k+1 < m, has to be initialized. ( e.g *seed = 1; u=drand(seed); ), Then z(i) assumes all different values 0<4*k+1€€‚‚€‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚’12 ’’’’1’’’’’’’’9’’’’’’’’’’’’Ø( HelvĻĶɑ••••“©Tms RmnøøøĀĒĆSymbol Courier"#$%&'()*+,-Times New Roman>?@aArialhijklmnopqrstuMS Serif_`abcdefghiMS Sans Serifxyz{|}Times„…†‡ˆ‰‰‹ŒŽ‘HelveticaœžŸ ”¢£¤„System­®Æ°±²³“µ¶·ø¹Courier NewęēčéźėģķCourier 10cpiüżžßąįCourier 5cpiļšńņóōõCourier 6cpi Courier 8.5cpiCourier 12cpi,-./01Courier 15cpi@ABCDECourier 17cpiTUVWXYCourier 20cpiHIJKLMCourier PSFš‹F FtSanserif 5cpiv’všSanserif 6cpiFų‹VśRSanserif 8.5cpi F Sanserif 10cpiv ’v Sanserif 12cpi‹FņÄSanserif 15cpi+ĄéSanserif 17cpiüŒFž&Sanserif 20cpitÄ^Roman 5cpi&‰GøŽRoman 6cpi^ü&‹G‰FšRoman 8.5cpišd5s ĄtRoman 10cpi&Ē’’ŽśRoman 12cpi‹G,&‹W.&Roman 15cpiG HPSčRoman 17cpi‹Vö ‰FRoman 20cpi’vś’vų’vRoman PSx“osFų’vś’Prestige 2.5cpiFņÄ^Prestige 5cpi’vų’vžPrestige 6cpi‹F FPrestige 8.5cpi’vPrestige 10cpiFīRP’Prestige 12cpiģ‹F Prestige 15cpi’v šüPrestige 17cpiFšÄ^Prestige 20cpi‹Fņ&Prestige PS]ŹU‹ģƒCourier 2.5cpi øSanserif 2.5cpi,uRoman 2.5cpi/&”¢ éĘSanserif PS.&‰D0&‰TFixedsysŽĄ&‹L IQPVTerminal ŠtĮ‰všÄ^&ModernVö‹ų‰Vī‹F ScriptPW’vņV+ÉQ‹ņšxRomanVW’v ’v šüśtƒSmall FontsvšøPšxMS LineDraw$‹FīPW’vMS Dialog™RPVW’v’vMS Dialog Light£¢ ^MS SystemExģ’v’vWingdingsRPFüP’vFences‹å]Ź U‹ģƒģMT ExtraÄ^&‹‰Fę‹FAardvarkžŽĀ‹Ų&ƒ Arabia/&£¢ é&‹G.AvalonSššd-w=Ą@tBahamasÄ^&Ē’’ŽśBahamasHeavyG,&‹W.&BahamasLightÄ^ü&‹G Banff¬ų‰Fš‰Vņ ŠtĆÄ^BangkokV‰Fģ‰Vīƒ~Bodnoff‹G‰Fę=’’téBrooklynFģ‹Vī‰Fč‰VźCasablanca&‹G‹Č™;VCasperOpenFaceī&‹G CenturionOld‰Fę‹FģCommonBulletsź‹ĮH™‰CottageÄ^&9ut&‹GCupertino;Fģwa‰Nö&‹DawnCastleė]PÄ^ü&‹Dixieland‰Fš‰Vņ ŠuErieš&‹G™FģVīé"’ErieBlackSč ÷‰Fš‰ErieLight±žĒFö‹FšFranceFų‰Vś‹Fö™;VīFrankensteinų’vž’vüFrankfurtGothicvų’vFrankfurtGothicHeavFreeport&‰‹Fų+Fš-Fujiyama&‰G‹F‹V+Fujiyama2&‰W‹Fź FFujiyamaExtraBoldFujiyamaLight’u &‹Gatineau&‹|&’t(&’GeographicSymbolsŒFGreekMathSymbolsš*ĆHomewardBoundGPøIrelandx=u Ņt šøJupiterėšøƍuŽüKoala’’ė‹Ē^_‹å]ŹLincolnĄ‰Fö‰Fś‹F‹VLinusü‰VžÄ^&‹G™+ĀMemorandum‹G‰Fņ’vžMonospacedPšx“ x‰FMotoržRP’v’všŚkīwMusicalSymbols;FųMystical™Äv‹Č&‹DNebraska‹Fś‹Ś™+ČŚƒNewBrunswickFś ROttawa RPšü·tƒÄ PalmSprings&)D&‹DParadiseGÄv&‹D+FParagon‰G&€OÄv &Penguin‹Äv &‰DÄ^PenguinLightÄ^&‰GPosse ’u!Ä^&‰Gé®Presidentü‹Fö‰Fś’NņProseAntique™RPQuantum‰Vź Šu øŽRenfrewÄ^ &‹G Ä^č‰^Southern’v’všFmzStampī’vģšĀ3+zŽü/&SwitzerlandBlackGSwitzerlandCondBlacSwitzerlandCondLighSwitzerlandCondenseSwitzerlandInseratÄSwitzerlandLight&‹GSwitzerlandNarrow’NSwitzerland øPšx“Technical‰Fō;Fų~ĢÄ^Timpani ‹Č&‹D-‰FTimpaniHeavy‹Ś™+ČŚToronto‹F‹VFōUSABlackRPšü©yƒÄUSALight‰G‹FšÄv&)UmbrellaHHÄ^&‰G‹FUnicornFņ÷ŲÄ^&‰G&Vogue&ĘG^‹å]ŹZurichCalligraphicFAlgerian’v’v SøArial Rounded MT BoBookman Old StyleŽBraggadocioPÄ^&’G Britannic BoldF։VŲBrush Script MTVÄ^Century Gothic&’w0‹Colonna MTFźRPšü|Desdemona,&‹W.‰Fö‰VFootlight MT Light&ImpactVāŽĀ‹Ų‹Nś&‰&Kino MT‹\+^܃ėŽĀ‹Wide LatinÄ^&‹wŽĀMatura MT Script CaPlaybillÄ ’vų’vöšĀ3Roman PXvÜ&‰@ ‹FśÄ^Sans Serif 10cpiSans Serif 12cpi2‰~Sans Serif 17cpi+ĄSans Serif 20cpi ŠuSans Serif 5cpi։^üSans Serif 6cpiözéSans Serif PS’v’vSans Serif PXNä‹^ęƒBorlandTE^ņÄ^&‹G ‰BorlandTEi&+G÷n &+ %% -$ó‡2IJ„ó‡Čž R 8€¹š÷ ;,v†v†v†v†v†,,v†ÄūE‚cŠą‹ƒą‹ąŒAR;Æ €,ƒ²„æ†ó‡Č*žŠ8€v†vˆ‡‹KņøÆĖ€~„ń†œŠ¹š÷ øƒ2Ä€v…€~„ń†œŠ¹š÷ øƒ2Ä€v…ą‹ąŒAR;Æ €,ƒ²„æ†ó‡Č*žŠ 8€v†vˆ‡‹KņøÆąŒ²„æ†ņó‡Čvˆøƒ2ĖūUøÄ€v…ņĖ€~„ń†œŠ¹š÷ øƒ€v…ą‹ąŒA€,æ†Šv†K‡‹3‰K€AņZ R;8€v…~„ń†œŠ*Äń†šøƒ€ƒvˆœŠ¹÷ øƒø€;Æ ņ8€*€€Æ žŠ2/&;)i24d.rtf 53 F3A’’J€J’’’’Acceptance ComplementAcceptance RejectionAlias Method Beta$Binomial,Box/Muller Method8Burr II<Burr III@Burr IVDBurr IXHBurr VLBurr VIPBurr VIITBurr XXBurr XII\Calculating`Calculating WindowlCalculation InfospCauchytChixCompound method|Continuos Distributions€Discrete DistributionsčDistributions) ExponentialÄExponential PowerČExtreme value IĢExtreme Value IIŠFileŌGammaŲGeneralized Inverse Gaussian DistributionąGeneralized PoissonäGeneratorsčGeometricģGlobal IndexšGraphicsōHow to UseųHyperbolicüHypergeometricIntegral Representations InversionJohnsonhKeyslLambdapLaplacetLogarithmicxLogistic|Menus€Menus, Distributions„Menus, HelpˆMenus, OptionsŒMomentsNegative Binomial”Normal˜Patchwork Rejection Patchwort Rejection¤PoissonØPolar Method“QuantilesøRatio of Uniforms¼RejectionŲSelecting a GeneratorsģSine-Cosine MethodšSlashōStableųStratified RejectionüStudent-tTriangularTrransformation Uniform DistributionUser InterfaceVon MisesWeibullZeta  ™‹ųƒ’’u Ž0&”˜^éįW+ĄPPøPš%ó™‰Fų‰Vś Ņ|=X’’’’|bm0|bm1¼|bm10ŗ|bm11?|bm12Ń|bm13s |bm14"|bm15Ž&|bm16D+|bm2C |bm3Š |bm4U|bm5õ|bm6—|bm7)|bm8¦|bm9-ŒĀ@@‹š‹ĒO ĄʋF Ft +všƒīÄ^&‰7‹F^_‹å]ŹU‹ģƒģ’v’v’v’vš†nj—‰Fų‰Vś ŠuŽ 0&”¢ ‹å]ŹŽ 0&Ē¢ ‹Vś ‰Fš‰VņĒFō鄐 Ą}(‹F FtŋFÄ^&‰‹Fō&‰G‹Fš+Fų-&‰Gė¦‹F F t?’vņ’vššŗƗƒÄFš‹Vņ@‰Fü‰VžRP’v’všŚkŪ—™RP’vž’vü’v ’v šü±—ƒÄ Ž 0&Ē¢/&;)Lz60&Ē˜^+Ą™’’:&2’’Main IndexUHow to Use User InterfaceņMenu FileZMenu Distributions Menu OptionsMenu Help€GeneratorsE‚Calculating Window3‰KeyscŠInformation about CalculationInformation about MomentsÄInformation about QuantilesūInformation about Graphics2About GeneratorsĖGeometric Distribution - Inversion€Logarithmic - Inversion/Transformation~„Poisson - Inversionń†Poisson - Ratio of Uniforms/InversionœŠPoisson - Patchwork Rejection/Inversion¹Binomial - Acceptance Rejection/InversionšBinomial - Ratio of Uniforms/Inversion÷ Binomial - Patchwork Rejection/InversionøƒGeneralized Poisson - Inversion/Ratio of Uniforms/RejectionNegative Binomial - Compound method2Zeta Distribution - Acceptance RejectionÄHypergeometric Distribution - Acceptance Rejection/Inversion Hypergeometric - Ratio of Uniforms/Inversionv…Hypergeometric Distribution - Patchwork Rejection/Inversioną‹Cauchy Distribution - InversionąŒExponential Distribution - InversionExtreme value I Distribution - InversionLaplace Distribution - Inversion?Logistic Distribution - InversionRNormal Distribution - Alias Method;Normal Distribution - Sine-Cosine or Box/Muller MethodÆ Slash Distribution - Z/U€Triangular Distribution - Inversion,Burr II; VII; X Distributions - InversionƒChi Distribution - Ratio of Uniforms with shift at mode m²„Exponential Power Distribution - Acceptance Rejectionæ†Extreme Value II Distribution - Inversionó‡Gamma Distribution - Acceptance Rejection combined with Acceptance ComplementČGamma Distribution - Acceptance Rejection with log-logistic; envelopesStudent-t Distribution - Ratio of Uniforms*Student-t Distribution - Polar MethodžVon Mises Distribution - Acceptance RejectionŠWeibull Distribution - Inversion Beta Distribution - Acceptance Rejection with log-logistic8€Beta Distribution - Stratified Rejection/Patchwork Rejection’v’v軞‹ų‰Vź Štä‹Fź‰~š‰Fņ‹F ‹V‰Fģ‰†’’v†Burr III; IV; V; VI; IX; XII Distribution - InversionvˆGeneralized Inverse Gaussion Distribution - Ratio of Uniforms‡‹Johnson Distribution - Transformation of one Normal random numberKLambda Distribution - InversionņStable Distribution - Integral RepresentationsøHyperbolic Distribution - Non Universal RejectionÆ(0;1)Uniform Distribution³nversion/Transformation~„Poisson - Inversionń†Poisson - Ratio of Uniforms/InversionœŠPoisson - Patchwork Rejection/Inversion¹Binomial - Acceptance Rejection/InversionšBinomial - Ratio of Uniforms/Inversion÷ Binomial - Patchwork Rejection/InversionøƒGeneralized Poisson - Inversion/Ratio of Uniforms/RejectionNegative Binomial - Compound method2Zeta Distribution - Acceptance RejectionÄHypergeometric Distribution - Acceptance Rejection/Inversion Hypergeometric - Ratio of Uniforms/Inversionv…Hypergeometric Distribution - Patchwork Rejection/Inversioną‹Cauchy Distribution - InversionąŒExponential Distribution - InversionExtreme value I Distribution - InversionLaplace Distribution - Inversion?Logistic Distribution - InversionRNormal Distribution - Alias Method;Normal Distribution - Sine-Cosine or Box/Muller MethodÆ Slash Distribution - Z/U€Triangular Distribution - Inversion,Burr II; VII; X Distributions - InversionƒChi Distribution - Ratio of Uniforms with shift at mode m²„Exponential Power Distribution - Acceptance Rejectionæ†Extreme Value II Distribution - Inversionó‡Gamma Distribution - Acceptance Rejection combined with Acceptance ComplementČGamma Distribution - Acceptance Rejection with log-logistic; envelopesStudent-t Distribution - Ratio of Uniforms*Student-t Distribution - Polar MethodžVon Mises Distribution - Acceptance RejectionŠWeibull Distribution - Inversion Beta Distribution - Acceptance Rejection with log-logistic8€Beta Distribution - Stratified Rejection/Patchwork Rejection’v’v軞‹ų‰Vź Štä‹Fź‰~š‰Fņ‹F ‹V‰Fģ‰ōv†T’’’’8mcTš-#éR’ē ÷<ļL÷<’’v~cT’’’’~cTvŲ-#é~Ä^ģ ÷<ōL÷<’’H9cT’’’’H~cT’’-#0900Ī ÷<ÖL÷<’’`„cTD’’’’ī_cT n_ms€k_msgNegative Binomial nbp_msg nal_msg Normal nsc nsc_msg pd_msgPoisson pprsc pprsc_msg pruec_msg Slash sla_msg stab_msgStudent-t tpol tpol_msgˆ trouo_msg Triangular tra_msg drand_msg Von Mises ˆ mwc_msg win_msgZeta zet_msg«ˆˆxˆˆˆ€ˆˆˆxˆˆˆ€ˆˆˆˆˆ-#ł÷<R÷<’’ƒcTÓ’’’’JicTˆ‘-#d_msŽ÷<ęO÷<’’ācTÆ’’’’āfcT i  *se¼€cTGamma€€€Distributions® *seJohnsonրś€cT#‚€-Continuos Distributions’’Acceptance Rejection!…CAcceptance Complement% *se€cT¢€cTրcTś€cTcT€ˆ€€€Distributions’’ ’’’’’’’YO ˆGamma Distribution - Acceptance Rejection combined with Acceptance Complement-#ˆ„0÷<8Q÷<’’‚cT°’’’’2T€cT-#ˆ”D÷<LQ÷<’’R‡cT°’’’’ācTˆ†-# ˆ‚€I÷<QQ÷<’’°00 *se¼€cTbcT†‚cT0000#0000Continuos Distributions00%\~hc’’’’’’’’4’’’’’’’’’’’’290MC8000Johnson Distribution - Transformation of one Normal random number-#÷<Š÷<ŲR÷<’’LƒcTŌ’’’’&€cT-#ä÷<ģR÷<’’“…cTŌ’’’’ƒcT-#cTé÷<ńR÷<’’Ō³¾’’ „%€°ŒInformation about Moments -#‰€ˆR÷<ZD÷<’’„cT4’’’’ˆ-#ˆ‰€f÷<nD÷<’’2„cT4’’’’փcT‚€-#xˆk÷<sD÷<’’l†cT4’’’’„cTxˆ-#ˆ÷÷<’N÷<’’X…cTE’’’’¤~cT€ ‚œ„cT bfbfGeometric88!f0fbDiscrete Distributionsf‚œ„cT²„cTī„cT0088 8008Inversion88%8880’’’’’’’’’’’’’’’’’’’’f000-$0000Geometric Distribution - Inver-#-#ū ÷<J÷<’’ȆcT’’’’ƒcT -#Ž…cT·PHypergeometric’!X…cTDi-# DisB÷<JS÷<’’ŅcTÕ’’’’LƒcTž… drand_msg] generator with unsigned long integer *seed, - NORMAL(/&;)L49s &‹&‹T‰F’’909’’’’¾Ė_ˆ,oö“ˆv†ń;é‰*YµG2"Ļqó‡Pd˜ĖĪ“­¦ņ†§šiĶ8Øūģ†²¼æ†ł‹5ĀRö„¬ÅÄ`”hĘvˆĪŃ©ĘøxbÉĖü‘Ģ#ÕČ°rŲą‹9ķŗŲKQŁįŁń†tĆŻ2< źŽ¹If­į€R›wā ³¦õ汌`såUqsóE‚o†®ó‡‹@±ūƒ4`!3‰µŖų²„h,8€iu]ž­Į( A[¼īæj«ņj‰~„3ōcŠQcų¼·‹$÷ ?°Ÿ)€Į³=øƒGC€töCÕųüDÆäZĀGŠm}ŪG`zM ÜwčPÄ ēVœŠę^;³^v…¢ ĶhZœĆ)n€”ś~oĶžĢt tQD|Æ ‹ģ’v’v+ĄPPPč`’‹å]ŹU‹ģWV‹vŽF&ƒ|’u}&’t&’t&ŠD$<Ą%PŒĒšh"pĮŽĒ&‰D=’’uŽ0&”˜^Ž0&£Ō +ĄėLŽF&’t+ĄPPøPš%™Į Ņ|=s Ž0&ĒŌ ėĻŽF&’t+ĄPPPš%ĒĄŽ0&ĒŌ ø^_‹å]ŹU‹ģƒģWV‹F‹V‰Fņ‰VōŽ0&ĒŌ ƒ~}&ĒŌ ø’’™é=Ä^ņ&‹G &‹W‹Č‹ņFV&;W |)&;Gv!&‹G&‹W +Į։F‰V Ņ | Ąu+ĄėøÄ^ņ&öGt&‹G‰Fö+Ą‰Fś‰Fųė3&‹&‹W‰Fī‰VšRPč³ž ĄtÄ^ī&‹G‰FöÄ^ņ&‹G&‹W‰Fų‰Vś’vö&‹G &‹W‹Č‹ŚFųVś ƒŅRP+ĄP‹ń‹ūš%ņĀ+šśvų~śƒž÷uƒ’’t+Ž0&ƒ>˜^uŽ0&ĒŌ 钐&”˜^Ž0&£Ō é’ž’vö’v ’v ’v’všä"<Į‰Fü‰VžŽ0&”˜^Ž0&£Ō Ņ|‹FüÄ^ņ&G &W‹Fü‹Vž^_‹å]Ź U‹ģƒģVÅv ‹F ĄtHtHtĒFü’’ĒFž’’ė#‹F‹V ė‹D ‹Tė‹D‹T FV ‰Fü‰Vžƒ~ž|‹Fü‹Vž‰D ‰Tø›ĆŽĄ&ĒŌ ėĒFü’’ĒFž’’ø­ĆŽĄ&ĒŌ ‹Fü‹Vž¹3²ŽŁ^‹å]Ź Ž0&”Ō ĖU‹ģƒģ˜^u鐞&”˜^Ž"0&£Ō 遞ƒ~ģt‹F܋VŽ‰Fč‰Vź~švąŒŠŽĄ„„„„’vī‹FŠ‹VŅ‰F܉VŽRP+ÉQš%*Ę;FÜu£;VŽuž’vīFąPø™RPšä"Ē=u† Ņu‚‹Fä‹Vę‰FŠ‰VŅ=’’u;Št‹FŠ‹VŅ9V|9Fréw’‹FŠ‹VŅ‰Fü‰Vž‹F‹V‰Fä‰VęĒFĪ‹F܋VŽFąVā;FtéZž;VtéRž‹Fą‹VāFųVś‹F܋VŽ‰F‰Vƒ~ģué,ž~ąvšŒŠŽĄ„„„„‹Fč‹Vź‰F܉VŽ=’’u;Štøė+Ą‰FĪéžĒFä’’ĒFę’’éHž‹F‹VFųVś;FŠu\;VŅuW’vīRP+ĄPš%<Ē;FŠt锾;V