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MacBinary | 1993-03-07 | 8.8 KB | [ TEXT/KAHL]
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MacBinary II, inited, Sun Mar 7 16:08:36 1993, modified Sun Mar 7 16:08:36 1993, creator Think C, type ASCII, 8241 bytes "Binomial.c" , at 0x20b1 428 bytes resource
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id metadata key value macFileType [ TEXT] macFileCreator [ KAHL]
hex view +--------+-------------------------+-------------------------+--------+--------+ |00000000| 00 0a 42 69 6e 6f 6d 69 | 61 6c 2e 63 00 00 00 00 |..Binomi|al.c....| |00000010| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000020| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000030| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000040| 00 54 45 58 54 4b 41 48 | 4c 01 00 00 00 00 00 00 |.TEXTKAH|L.......| |00000050| 00 00 00 00 00 20 31 00 | 00 01 ac a7 c0 14 d4 a7 |..... 1.|........| |00000060| c0 14 d4 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 36 1b 00 00 |........|....6...| |00000080| 2f 2a 0d 42 69 6e 6f 6d | 69 61 6c 2e 63 0d 43 6f |/*.Binom|ial.c.Co| |00000090| 70 79 72 69 67 68 74 20 | 28 63 29 20 31 39 39 30 |pyright |(c) 1990| |000000a0| 2c 31 39 39 31 2c 31 39 | 39 32 20 44 65 6e 69 73 |,1991,19|92 Denis| |000000b0| 20 47 2e 20 50 65 6c 6c | 69 0d 0d 56 61 72 69 6f | G. Pell|i..Vario| |000000c0| 75 73 20 72 6f 75 74 69 | 6e 65 73 20 74 68 61 74 |us routi|nes that| |000000d0| 20 64 65 61 6c 20 77 69 | 74 68 20 42 69 6e 6f 6d | deal wi|th Binom| |000000e0| 69 61 6c 20 73 74 61 74 | 69 73 74 69 63 73 2c 20 |ial stat|istics, | |000000f0| 69 6e 63 6c 75 64 69 6e | 67 20 67 65 6e 65 72 61 |includin|g genera| |00000100| 74 69 6e 67 20 72 61 6e | 64 6f 6d 0d 73 61 6d 70 |ting ran|dom.samp| |00000110| 6c 65 73 20 61 6e 64 20 | 63 6f 6d 70 75 74 69 6e |les and |computin| |00000120| 67 20 63 6f 6e 66 69 64 | 65 6e 63 65 20 69 6e 74 |g confid|ence int| |00000130| 65 72 76 61 6c 73 2e 0d | 0d 54 68 69 73 20 66 69 |ervals..|.This fi| |00000140| 6c 65 20 63 6f 6e 74 61 | 69 6e 73 20 74 77 6f 20 |le conta|ins two | |00000150| 67 65 6e 65 72 61 74 69 | 6f 6e 73 20 6f 66 20 72 |generati|ons of r| |00000160| 6f 75 74 69 6e 65 73 20 | 74 68 61 74 20 67 65 6e |outines |that gen| |00000170| 65 72 61 74 65 20 63 6f | 6e 66 69 64 65 6e 63 65 |erate co|nfidence| |00000180| 0d 69 6e 74 65 72 76 61 | 6c 73 2e 20 54 68 65 20 |.interva|ls. The | |00000190| 6f 72 69 67 69 6e 61 6c | 20 72 6f 75 74 69 6e 65 |original| routine| |000001a0| 73 20 42 69 6e 6f 6d 69 | 61 6c 55 70 70 65 72 42 |s Binomi|alUpperB| |000001b0| 6f 75 6e 64 20 61 6e 64 | 20 42 69 6e 6f 6d 69 61 |ound and| Binomia| |000001c0| 6c 4c 6f 77 65 72 42 6f | 75 6e 64 20 61 72 65 0d |lLowerBo|und are.| |000001d0| 6f 6c 64 2c 20 62 75 74 | 20 61 72 65 20 6d 6f 72 |old, but| are mor| |000001e0| 65 20 72 6f 62 75 73 74 | 2e 20 54 68 65 20 6e 65 |e robust|. The ne| |000001f0| 77 20 72 6f 75 74 69 6e | 65 73 20 61 72 65 20 62 |w routin|es are b| |00000200| 61 73 65 64 20 6f 6e 20 | 61 20 4e 75 6d 65 72 69 |ased on |a Numeri| |00000210| 63 61 6c 20 52 65 63 69 | 70 65 73 20 69 6e 20 43 |cal Reci|pes in C| |00000220| 0d 72 6f 75 74 69 6e 65 | 20 74 6f 20 63 6f 6d 70 |.routine| to comp| |00000230| 75 74 65 20 74 68 65 20 | 69 6e 63 6f 6d 70 6c 65 |ute the |incomple| |00000240| 74 65 20 62 65 74 61 20 | 66 75 6e 63 74 69 6f 6e |te beta |function| |00000250| 2e 20 54 68 65 20 6e 65 | 77 20 72 6f 75 74 69 6e |. The ne|w routin| |00000260| 65 73 20 61 64 64 20 6c | 69 74 74 6c 65 20 74 6f |es add l|ittle to| |00000270| 0d 74 68 65 20 6f 6c 64 | 20 72 6f 75 74 69 6e 65 |.the old| routine| |00000280| 73 2c 20 61 6e 64 20 72 | 65 71 75 69 72 65 20 74 |s, and r|equire t| |00000290| 68 65 20 4e 75 6d 65 72 | 69 63 61 6c 20 52 65 63 |he Numer|ical Rec| |000002a0| 69 70 65 73 20 69 6e 20 | 43 2c 20 73 6f 20 49 20 |ipes in |C, so I | |000002b0| 63 6f 6d 6d 65 6e 74 65 | 64 20 74 68 65 6d 0d 6f |commente|d them.o| |000002c0| 75 74 2e 20 54 68 65 20 | 6e 65 77 20 72 6f 75 74 |ut. The |new rout| |000002d0| 69 6e 65 73 20 64 6f 20 | 68 61 76 65 20 74 68 65 |ines do |have the| |000002e0| 20 61 64 76 61 6e 74 61 | 67 65 20 6f 66 20 61 6c | advanta|ge of al| |000002f0| 6c 6f 77 69 6e 67 20 61 | 6e 20 61 72 62 69 74 72 |lowing a|n arbitr| |00000300| 61 72 79 20 63 6f 6e 66 | 69 64 65 6e 63 65 2c 0d |ary conf|idence,.| |00000310| 77 68 65 72 65 61 73 20 | 74 68 65 20 6f 6c 64 20 |whereas |the old | |00000320| 72 6f 75 74 69 6e 65 73 | 20 77 65 72 65 20 68 61 |routines| were ha| |00000330| 72 64 20 63 6f 64 65 64 | 20 74 6f 20 70 72 6f 64 |rd coded| to prod| |00000340| 75 63 65 20 61 20 39 35 | 25 20 63 6f 6e 66 69 64 |uce a 95|% confid| |00000350| 65 6e 63 65 20 69 6e 74 | 65 72 76 61 6c 2e 0d 0d |ence int|erval...| |00000360| 48 49 53 54 4f 52 59 3a | 0d 31 2f 35 2f 39 31 20 |HISTORY:|.1/5/91 | |00000370| 41 64 64 65 64 20 6e 65 | 77 20 62 69 6e 6f 6d 69 |Added ne|w binomi| |00000380| 61 6c 20 72 6f 75 74 69 | 6e 65 73 2e 20 54 68 65 |al routi|nes. The| |00000390| 73 65 20 72 6f 75 74 69 | 6e 65 73 20 6d 61 79 20 |se routi|nes may | |000003a0| 62 65 20 0d 09 09 63 6f | 6d 6d 65 6e 74 65 64 20 |be ...co|mmented | |000003b0| 6f 75 74 20 62 79 20 73 | 65 74 74 69 6e 67 20 4e |out by s|etting N| |000003c0| 45 57 5f 42 49 4e 4f 4d | 49 41 4c 20 74 6f 20 7a |EW_BINOM|IAL to z| |000003d0| 65 72 6f 2e 0d 31 2f 37 | 2f 39 31 20 52 65 77 72 |ero..1/7|/91 Rewr| |000003e0| 6f 74 65 20 42 69 6e 6f | 6d 69 61 6c 55 70 70 65 |ote Bino|mialUppe| |000003f0| 72 42 6f 75 6e 64 20 26 | 20 42 69 6e 6f 6d 69 61 |rBound &| Binomia| |00000400| 6c 4c 6f 77 65 72 42 6f | 75 6e 64 20 74 6f 20 61 |lLowerBo|und to a| |00000410| 6c 6c 6f 77 20 73 70 65 | 63 69 66 69 63 61 74 69 |llow spe|cificati| |00000420| 6f 6e 20 6f 66 20 74 68 | 65 0d 09 09 64 65 73 69 |on of th|e...desi| |00000430| 72 65 64 20 63 6f 6e 66 | 69 64 65 6e 63 65 20 6f |red conf|idence o| |00000440| 66 20 74 68 65 20 69 6e | 74 65 72 76 61 6c 2e 0d |f the in|terval..| |00000450| 34 2f 32 34 2f 39 32 09 | 64 67 70 09 41 64 64 65 |4/24/92.|dgp.Adde| |00000460| 64 20 42 69 6e 6f 6d 69 | 61 6c 53 61 6d 70 6c 65 |d Binomi|alSample| |00000470| 51 75 69 63 6b 6c 79 28 | 29 2e 0d 34 2f 32 37 2f |Quickly(|)..4/27/| |00000480| 39 32 09 64 67 70 09 4f | 6f 70 73 2e 20 49 20 6a |92.dgp.O|ops. I j| |00000490| 75 73 74 20 6e 6f 74 69 | 63 65 64 20 74 68 61 74 |ust noti|ced that| |000004a0| 20 74 68 69 73 20 66 69 | 6c 65 20 77 61 73 20 61 | this fi|le was a| |000004b0| 6c 77 61 79 73 20 69 6e | 63 6c 75 64 69 6e 67 20 |lways in|cluding | |000004c0| 6e 72 2e 68 2c 20 77 68 | 69 63 68 20 79 6f 75 27 |nr.h, wh|ich you'| |000004d0| 64 0d 09 09 09 6f 6e 6c | 79 20 68 61 76 65 20 69 |d....onl|y have i| |000004e0| 66 20 79 6f 75 20 6f 77 | 6e 20 74 68 65 20 4e 75 |f you ow|n the Nu| |000004f0| 6d 65 72 69 63 61 6c 20 | 52 65 63 69 70 65 73 2e |merical |Recipes.| |00000500| 20 49 20 6d 6f 76 65 64 | 20 74 68 65 20 69 6e 63 | I moved| the inc| |00000510| 6c 75 64 65 20 73 74 61 | 74 65 6d 65 6e 74 0d 09 |lude sta|tement..| |00000520| 09 09 64 6f 77 6e 20 69 | 6e 73 69 64 65 20 74 68 |..down i|nside th| |00000530| 65 20 63 6f 6e 64 69 74 | 69 6f 6e 61 6c 20 62 65 |e condit|ional be| |00000540| 6c 6f 77 2c 20 77 68 65 | 72 65 20 69 74 20 62 65 |low, whe|re it be| |00000550| 6c 6f 6e 67 73 2e 20 0d | 2a 2f 0d 23 69 6e 63 6c |longs. .|*/.#incl| |00000560| 75 64 65 20 22 56 69 64 | 65 6f 54 6f 6f 6c 62 6f |ude "Vid|eoToolbo| |00000570| 78 2e 68 22 0d 23 69 6e | 63 6c 75 64 65 20 3c 6d |x.h".#in|clude <m| |00000580| 61 74 68 2e 68 3e 0d 0d | 23 64 65 66 69 6e 65 20 |ath.h>..|#define | |00000590| 4e 45 57 5f 42 49 4e 4f | 4d 49 41 4c 20 30 0d 0d |NEW_BINO|MIAL 0..| |000005a0| 6c 6f 6e 67 20 42 69 6e | 6f 6d 69 61 6c 53 61 6d |long Bin|omialSam| |000005b0| 70 6c 65 28 64 6f 75 62 | 6c 65 20 70 2c 6c 6f 6e |ple(doub|le p,lon| |000005c0| 67 20 6e 29 0d 2f 2a 0d | 52 65 74 75 72 6e 73 20 |g n)./*.|Returns | |000005d0| 61 20 61 20 73 61 6d 70 | 6c 65 20 66 72 6f 6d 20 |a a samp|le from | |000005e0| 61 20 62 69 6e 6f 6d 69 | 61 6c 20 64 69 73 74 72 |a binomi|al distr| |000005f0| 69 62 75 74 69 6f 6e 3a | 20 6e 75 6d 62 65 72 20 |ibution:| number | |00000600| 6f 66 20 68 65 61 64 73 | 20 69 6e 20 6e 20 66 6c |of heads| in n fl| |00000610| 69 70 73 20 6f 66 0d 63 | 6f 69 6e 20 77 69 74 68 |ips of.c|oin with| |00000620| 20 61 20 70 72 6f 62 61 | 62 69 6c 69 74 79 20 70 | a proba|bility p| |00000630| 20 6f 66 20 68 65 61 64 | 73 20 6f 6e 20 65 61 63 | of head|s on eac| |00000640| 68 20 66 6c 69 70 2e 20 | 54 68 69 73 20 69 73 20 |h flip. |This is | |00000650| 66 69 6e 65 20 69 66 20 | 6e 20 69 73 20 73 6d 61 |fine if |n is sma| |00000660| 6c 6c 2e 20 48 6f 77 65 | 76 65 72 2c 0d 69 66 20 |ll. Howe|ver,.if | |00000670| 6e 20 69 73 20 6c 61 72 | 67 65 2c 20 65 2e 67 2e |n is lar|ge, e.g.| |00000680| 20 6e 3e 31 30 30 2c 20 | 74 68 65 6e 20 74 68 69 | n>100, |then thi| |00000690| 73 20 72 6f 75 74 69 6e | 65 20 77 69 6c 6c 20 62 |s routin|e will b| |000006a0| 65 20 72 61 74 68 65 72 | 20 73 6c 6f 77 2c 20 61 |e rather| slow, a| |000006b0| 6e 64 20 79 6f 75 20 6d | 61 79 20 70 72 65 66 65 |nd you m|ay prefe| |000006c0| 72 0d 74 6f 20 75 73 65 | 20 74 68 65 20 6d 6f 72 |r.to use| the mor| |000006d0| 65 20 65 6c 61 62 6f 72 | 61 74 65 20 4e 75 6d 65 |e elabor|ate Nume| |000006e0| 72 69 63 61 6c 20 52 65 | 63 69 70 65 73 20 62 6e |rical Re|cipes bn| |000006f0| 6c 64 65 76 28 29 20 72 | 6f 75 74 69 6e 65 2e 0d |ldev() r|outine..| |00000700| 2a 2f 0d 7b 0d 09 6c 6f | 6e 67 20 6b 2c 69 3b 0d |*/.{..lo|ng k,i;.| |00000710| 0d 09 6b 3d 30 3b 0d 09 | 66 6f 72 28 69 3d 30 3b |..k=0;..|for(i=0;| |00000720| 69 3c 6e 3b 69 2b 2b 29 | 20 69 66 28 70 3e 55 6e |i<n;i++)| if(p>Un| |00000730| 69 66 6f 72 6d 53 61 6d | 70 6c 65 28 29 29 6b 2b |iformSam|ple())k+| |00000740| 2b 3b 0d 09 72 65 74 75 | 72 6e 20 6b 3b 0d 7d 0d |+;..retu|rn k;.}.| |00000750| 0d 69 6e 74 20 42 69 6e | 6f 6d 69 61 6c 53 61 6d |.int Bin|omialSam| |00000760| 70 6c 65 51 75 69 63 6b | 6c 79 28 69 6e 74 20 6e |pleQuick|ly(int n| |00000770| 29 0d 2f 2f 20 44 6f 65 | 73 20 6e 20 63 6f 69 6e |).// Doe|s n coin| |00000780| 20 66 6c 69 70 73 20 61 | 6e 64 20 72 65 74 75 72 | flips a|nd retur| |00000790| 6e 73 20 74 68 65 20 6e | 75 6d 62 65 72 20 6f 66 |ns the n|umber of| |000007a0| 20 68 65 61 64 73 2e 20 | 56 65 72 79 20 66 61 73 | heads. |Very fas| |000007b0| 74 2e 0d 7b 0d 09 72 65 | 67 69 73 74 65 72 20 69 |t..{..re|gister i| |000007c0| 6e 74 20 69 2c 6b 3d 30 | 3b 0d 09 72 65 67 69 73 |nt i,k=0|;..regis| |000007d0| 74 65 72 20 73 68 6f 72 | 74 20 72 3b 0d 09 0d 09 |ter shor|t r;....| |000007e0| 66 6f 72 28 69 3d 6e 3b | 69 3e 3d 38 3b 69 2d 3d |for(i=n;|i>=8;i-=| |000007f0| 38 29 7b 0d 09 09 72 3d | 72 61 6e 64 55 28 29 3b |8){...r=|randU();| |00000800| 09 2f 2f 20 55 73 65 20 | 6f 6e 6c 79 20 74 68 65 |.// Use |only the| |00000810| 20 75 70 70 65 72 20 38 | 20 62 69 74 73 2c 20 77 | upper 8| bits, w| |00000820| 68 69 63 68 20 61 72 65 | 20 72 65 70 75 74 65 64 |hich are| reputed| |00000830| 20 74 6f 20 62 65 20 6d | 6f 72 65 20 72 61 6e 64 | to be m|ore rand| |00000840| 6f 6d 2e 0d 09 09 69 66 | 28 72 3c 30 29 6b 2b 2b |om....if|(r<0)k++| |00000850| 3b 0d 09 09 72 3c 3c 3d | 31 3b 0d 09 09 69 66 28 |;...r<<=|1;...if(| |00000860| 72 3c 30 29 6b 2b 2b 3b | 0d 09 09 72 3c 3c 3d 31 |r<0)k++;|...r<<=1| |00000870| 3b 0d 09 09 69 66 28 72 | 3c 30 29 6b 2b 2b 3b 0d |;...if(r|<0)k++;.| |00000880| 09 09 72 3c 3c 3d 31 3b | 0d 09 09 69 66 28 72 3c |..r<<=1;|...if(r<| |00000890| 30 29 6b 2b 2b 3b 0d 09 | 09 72 3c 3c 3d 31 3b 0d |0)k++;..|.r<<=1;.| |000008a0| 09 09 69 66 28 72 3c 30 | 29 6b 2b 2b 3b 0d 09 09 |..if(r<0|)k++;...| |000008b0| 72 3c 3c 3d 31 3b 0d 09 | 09 69 66 28 72 3c 30 29 |r<<=1;..|.if(r<0)| |000008c0| 6b 2b 2b 3b 0d 09 09 72 | 3c 3c 3d 31 3b 0d 09 09 |k++;...r|<<=1;...| |000008d0| 69 66 28 72 3c 30 29 6b | 2b 2b 3b 0d 09 09 72 3c |if(r<0)k|++;...r<| |000008e0| 3c 3d 31 3b 0d 09 09 69 | 66 28 72 3c 30 29 6b 2b |<=1;...i|f(r<0)k+| |000008f0| 2b 3b 0d 09 09 72 3c 3c | 3d 31 3b 0d 09 7d 0d 09 |+;...r<<|=1;..}..| |00000900| 69 66 28 69 3e 30 29 7b | 0d 09 09 72 3d 72 61 6e |if(i>0){|...r=ran| |00000910| 64 55 28 29 3b 0d 09 09 | 66 6f 72 28 3b 69 3e 30 |dU();...|for(;i>0| |00000920| 3b 69 2d 2d 29 7b 0d 09 | 09 09 69 66 28 72 3c 30 |;i--){..|..if(r<0| |00000930| 29 6b 2b 2b 3b 0d 09 09 | 09 72 3c 3c 3d 31 3b 0d |)k++;...|.r<<=1;.| |00000940| 09 09 7d 0d 09 7d 0d 09 | 72 65 74 75 72 6e 20 6b |..}..}..|return k| |00000950| 3b 0d 7d 0d 0d 2f 2a 0d | 42 69 6e 6f 6d 69 61 6c |;.}../*.|Binomial| |00000960| 55 70 70 65 72 42 6f 75 | 6e 64 20 61 6e 64 20 42 |UpperBou|nd and B| |00000970| 69 6e 6f 6d 69 61 6c 4c | 6f 77 65 72 42 6f 75 6e |inomialL|owerBoun| |00000980| 64 20 72 65 74 75 72 6e | 20 61 20 50 20 63 6f 6e |d return| a P con| |00000990| 66 69 64 65 6e 63 65 20 | 69 6e 74 65 72 76 61 6c |fidence |interval| |000009a0| 20 66 6f 72 20 74 68 65 | 0d 75 6e 64 65 72 6c 79 | for the|.underly| |000009b0| 69 6e 67 20 62 69 6e 6f | 6d 69 61 6c 20 70 72 6f |ing bino|mial pro| |000009c0| 62 61 62 69 6c 69 74 79 | 20 61 73 73 75 6d 65 64 |bability| assumed| |000009d0| 20 74 6f 20 68 61 76 65 | 20 67 65 6e 65 72 61 74 | to have| generat| |000009e0| 65 64 20 74 68 65 20 64 | 61 74 61 2e 0d 0d 54 68 |ed the d|ata...Th| |000009f0| 65 20 66 6f 72 6d 75 6c | 61 20 69 73 20 62 61 73 |e formul|a is bas| |00000a00| 65 64 20 6f 6e 20 61 20 | 47 61 75 73 73 69 61 6e |ed on a |Gaussian| |00000a10| 20 61 70 70 72 6f 78 69 | 6d 61 74 69 6f 6e 2c 20 | approxi|mation, | |00000a20| 73 6f 6c 76 69 6e 67 20 | 66 6f 72 20 74 68 65 20 |solving |for the | |00000a30| 70 20 74 68 61 74 20 77 | 69 6c 6c 0d 70 75 74 20 |p that w|ill.put | |00000a40| 74 68 65 20 6f 62 73 65 | 72 76 65 64 20 72 65 73 |the obse|rved res| |00000a50| 75 6c 74 20 74 68 65 20 | 72 69 67 68 74 20 6e 75 |ult the |right nu| |00000a60| 6d 62 65 72 20 6f 66 20 | 73 74 61 6e 64 61 72 64 |mber of |standard| |00000a70| 20 64 65 76 69 61 74 69 | 6f 6e 73 20 61 77 61 79 | deviati|ons away| |00000a80| 20 66 72 6f 6d 20 74 68 | 65 0d 6d 65 61 6e 2c 20 | from th|e.mean, | |00000a90| 70 6c 75 73 20 6f 72 20 | 6d 69 6e 75 73 20 30 2e |plus or |minus 0.| |00000aa0| 35 2c 20 61 73 20 61 20 | 22 63 6f 6e 74 69 6e 75 |5, as a |"continu| |00000ab0| 69 74 79 20 63 6f 72 72 | 65 63 74 69 6f 6e 22 2e |ity corr|ection".| |00000ac0| 0d 0d 54 68 69 73 20 69 | 73 20 74 68 65 20 62 65 |..This i|s the be| |00000ad0| 73 74 20 66 6f 72 6d 75 | 6c 61 20 49 20 63 6f 75 |st formu|la I cou| |00000ae0| 6c 64 20 66 69 6e 64 20 | 69 6e 20 6c 6f 6f 6b 69 |ld find |in looki| |00000af0| 6e 67 20 74 68 72 6f 75 | 67 68 20 73 65 76 65 72 |ng throu|gh sever| |00000b00| 61 6c 20 73 74 61 74 69 | 73 74 69 63 73 0d 62 6f |al stati|stics.bo| |00000b10| 6f 6b 73 2e 20 48 6f 77 | 65 76 65 72 2c 20 74 68 |oks. How|ever, th| |00000b20| 65 20 72 65 73 75 6c 74 | 20 68 61 73 20 74 6f 20 |e result| has to | |00000b30| 62 65 20 74 61 6b 65 6e | 20 77 69 74 68 20 61 20 |be taken| with a | |00000b40| 67 72 61 69 6e 20 6f 66 | 20 73 61 6c 74 20 62 65 |grain of| salt be| |00000b50| 63 61 75 73 65 20 69 74 | 20 69 73 0d 6e 6f 74 20 |cause it| is.not | |00000b60| 70 6f 73 73 69 62 6c 65 | 20 74 6f 20 70 72 6f 64 |possible| to prod| |00000b70| 75 63 65 20 61 20 62 69 | 6e 6f 6d 69 61 6c 20 63 |uce a bi|nomial c| |00000b80| 6f 6e 66 69 64 65 6e 63 | 65 20 69 6e 74 65 72 76 |onfidenc|e interv| |00000b90| 61 6c 20 74 68 61 74 20 | 77 69 6c 6c 20 73 61 74 |al that |will sat| |00000ba0| 69 73 66 79 20 74 68 65 | 0d 73 74 72 69 63 74 20 |isfy the|.strict | |00000bb0| 64 65 66 69 6e 69 74 69 | 6f 6e 20 6f 66 20 61 20 |definiti|on of a | |00000bc0| 63 6f 6e 66 69 64 65 6e | 63 65 20 69 6e 74 65 72 |confiden|ce inter| |00000bd0| 76 61 6c 2c 20 6e 61 6d | 65 6c 79 20 6f 6e 65 20 |val, nam|ely one | |00000be0| 74 68 61 74 20 77 69 6c | 6c 20 68 61 76 65 20 74 |that wil|l have t| |00000bf0| 68 65 0d 73 70 65 63 69 | 66 69 65 64 20 70 72 6f |he.speci|fied pro| |00000c00| 62 61 62 69 6c 69 74 79 | 20 50 20 6f 66 20 63 6f |bability| P of co| |00000c10| 6e 74 61 69 6e 69 6e 67 | 20 74 68 65 20 75 6e 6b |ntaining| the unk| |00000c20| 6e 6f 77 6e 20 62 75 74 | 20 66 69 78 65 64 20 70 |nown but| fixed p| |00000c30| 61 72 61 6d 65 74 65 72 | 20 70 2e 20 54 68 61 74 |arameter| p. That| |00000c40| 27 73 0d 62 65 63 61 75 | 73 65 2c 20 75 6e 6c 69 |'s.becau|se, unli| |00000c50| 6b 65 20 74 68 65 20 4e | 6f 72 6d 61 6c 20 64 69 |ke the N|ormal di| |00000c60| 73 74 72 69 62 75 74 69 | 6f 6e 2c 20 74 68 65 20 |stributi|on, the | |00000c70| 42 69 6e 6f 6d 69 61 6c | 20 64 69 73 74 72 69 62 |Binomial| distrib| |00000c80| 75 74 69 6f 6e 20 69 73 | 20 6e 6f 74 0d 74 72 61 |ution is| not.tra| |00000c90| 6e 73 6c 61 74 69 6f 6e | 20 69 6e 76 61 72 69 61 |nslation| invaria| |00000ca0| 6e 74 2e 20 45 2e 67 2e | 20 69 66 20 70 20 69 73 |nt. E.g.| if p is| |00000cb0| 20 61 63 74 75 61 6c 6c | 79 20 7a 65 72 6f 20 74 | actuall|y zero t| |00000cc0| 68 65 6e 20 74 68 65 20 | 74 72 75 65 20 63 6f 6e |hen the |true con| |00000cd0| 66 69 64 65 6e 63 65 20 | 69 6e 74 65 72 76 61 6c |fidence |interval| |00000ce0| 0d 41 4c 57 41 59 53 20 | 63 6f 6e 74 61 69 6e 73 |.ALWAYS |contains| |00000cf0| 20 70 2e 0d 0d 49 20 67 | 61 76 65 20 73 6f 6d 65 | p...I g|ave some| |00000d00| 20 74 68 6f 75 67 68 74 | 20 74 6f 20 74 61 6b 69 | thought| to taki| |00000d10| 6e 67 20 61 20 42 61 79 | 65 73 69 61 6e 20 61 70 |ng a Bay|esian ap| |00000d20| 70 72 6f 61 63 68 2c 20 | 61 73 73 75 6d 69 6e 67 |proach, |assuming| |00000d30| 20 61 20 75 6e 69 66 6f | 72 6d 20 70 72 69 6f 72 | a unifo|rm prior| |00000d40| 20 70 64 66 0d 66 6f 72 | 20 70 20 61 6e 64 20 74 | pdf.for| p and t| |00000d50| 68 65 6e 20 63 6f 6d 70 | 75 74 69 6e 67 20 61 6e |hen comp|uting an| |00000d60| 20 61 20 70 6f 73 74 65 | 72 69 6f 72 69 20 63 6f | a poste|riori co| |00000d70| 6e 66 69 64 65 6e 63 65 | 20 69 6e 74 65 72 76 61 |nfidence| interva| |00000d80| 6c 2e 20 54 68 69 73 20 | 63 61 6e 20 62 65 20 64 |l. This |can be d| |00000d90| 6f 6e 65 2c 0d 62 75 74 | 20 69 74 27 73 20 68 61 |one,.but| it's ha| |00000da0| 72 64 2e 20 49 20 63 6f | 6e 63 6c 75 64 65 64 20 |rd. I co|ncluded | |00000db0| 74 68 61 74 20 69 74 27 | 73 20 70 6f 69 6e 74 6c |that it'|s pointl| |00000dc0| 65 73 73 20 62 65 63 61 | 75 73 65 20 74 68 65 20 |ess beca|use the | |00000dd0| 75 6e 69 66 6f 72 6d 20 | 70 64 66 0d 61 73 73 75 |uniform |pdf.assu| |00000de0| 6d 70 74 69 6f 6e 20 69 | 73 20 75 73 75 61 6c 6c |mption i|s usuall| |00000df0| 79 20 75 6e 77 61 72 72 | 61 6e 74 65 64 2e 0d 0d |y unwarr|anted...| |00000e00| 45 76 65 6e 20 74 68 6f | 75 67 68 20 74 68 69 73 |Even tho|ugh this| |00000e10| 20 63 6f 6e 66 69 64 65 | 6e 63 65 20 69 6e 74 65 | confide|nce inte| |00000e20| 72 76 61 6c 20 69 73 20 | 75 6e 73 61 74 69 73 66 |rval is |unsatisf| |00000e30| 61 63 74 6f 72 79 20 28 | 69 2e 65 2e 20 66 75 6e |actory (|i.e. fun| |00000e40| 64 61 6d 65 6e 74 61 6c | 6c 79 0d 66 61 6c 73 65 |damental|ly.false| |00000e50| 29 20 69 74 20 69 73 20 | 75 73 65 66 75 6c 20 69 |) it is |useful i| |00000e60| 6e 20 70 72 61 63 74 69 | 63 65 20 73 69 6e 63 65 |n practi|ce since| |00000e70| 20 69 74 20 74 79 70 69 | 63 61 6c 6c 79 20 62 65 | it typi|cally be| |00000e80| 68 61 76 65 73 20 73 69 | 6d 69 6c 61 72 6c 79 20 |haves si|milarly | |00000e90| 74 6f 20 74 68 65 0d 4e | 6f 72 6d 61 6c 20 63 61 |to the.N|ormal ca| |00000ea0| 73 65 2c 20 77 68 69 63 | 68 20 69 73 20 74 68 65 |se, whic|h is the| |00000eb0| 6f 72 65 74 69 63 61 6c | 6c 79 20 73 6f 75 6e 64 |oretical|ly sound| |00000ec0| 2e 0d 2a 2f 0d 64 6f 75 | 62 6c 65 20 42 69 6e 6f |..*/.dou|ble Bino| |00000ed0| 6d 69 61 6c 4c 6f 77 65 | 72 42 6f 75 6e 64 28 64 |mialLowe|rBound(d| |00000ee0| 6f 75 62 6c 65 20 50 2c | 6c 6f 6e 67 20 6b 2c 6c |ouble P,|long k,l| |00000ef0| 6f 6e 67 20 6e 29 0d 2f | 2a 0d 41 72 67 75 6d 65 |ong n)./|*.Argume| |00000f00| 6e 74 73 20 61 72 65 20 | 74 68 65 20 63 6f 6e 66 |nts are |the conf| |00000f10| 69 64 65 6e 63 65 20 50 | 2c 20 61 6e 64 20 74 68 |idence P|, and th| |00000f20| 65 20 6e 75 6d 62 65 72 | 20 6b 20 6f 66 20 68 65 |e number| k of he| |00000f30| 61 64 73 20 69 6e 20 6e | 20 66 6c 69 70 73 20 6f |ads in n| flips o| |00000f40| 66 20 61 20 63 6f 69 6e | 2e 0d 54 68 65 20 72 65 |f a coin|..The re| |00000f50| 74 75 72 6e 65 64 20 76 | 61 6c 75 65 20 70 20 69 |turned v|alue p i| |00000f60| 73 20 74 68 65 20 6c 6f | 77 65 72 20 65 6e 64 20 |s the lo|wer end | |00000f70| 6f 66 20 61 20 50 20 63 | 6f 6e 66 69 64 65 6e 63 |of a P c|onfidenc| |00000f80| 65 20 69 6e 74 65 72 76 | 61 6c 20 6f 6e 20 74 68 |e interv|al on th| |00000f90| 65 0d 75 6e 64 65 72 6c | 79 69 6e 67 20 70 72 6f |e.underl|ying pro| |00000fa0| 62 61 62 69 6c 69 74 79 | 20 6f 66 20 61 20 68 65 |bability| of a he| |00000fb0| 61 64 20 6f 6e 20 61 20 | 73 69 6e 67 6c 65 20 74 |ad on a |single t| |00000fc0| 72 69 61 6c 2e 0d 2a 2f | 0d 7b 0d 09 64 6f 75 62 |rial..*/|.{..doub| |00000fd0| 6c 65 20 72 69 67 68 74 | 2c 73 2c 73 73 2c 70 3b |le right|,s,ss,p;| |00000fe0| 0d 09 0d 09 69 66 28 6b | 3e 30 20 26 26 20 6b 3c |....if(k|>0 && k<| |00000ff0| 6e 29 7b 0d 09 09 72 69 | 67 68 74 3d 6b 2d 30 2e 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66 20 | 61 20 50 20 63 6f 6e 66 | end of |a P conf| |00001190| 69 64 65 6e 63 65 20 69 | 6e 74 65 72 76 61 6c 20 |idence i|nterval | |000011a0| 6f 6e 20 74 68 65 0d 75 | 6e 64 65 72 6c 79 69 6e |on the.u|nderlyin| |000011b0| 67 20 70 72 6f 62 61 62 | 69 6c 69 74 79 20 6f 66 |g probab|ility of| |000011c0| 20 61 20 68 65 61 64 20 | 6f 6e 20 61 20 73 69 6e | a head |on a sin| |000011d0| 67 6c 65 20 74 72 69 61 | 6c 2e 0d 2a 2f 0d 7b 0d |gle tria|l..*/.{.| |000011e0| 09 64 6f 75 62 6c 65 20 | 72 69 67 68 74 2c 73 2c |.double |right,s,| |000011f0| 73 73 2c 70 3b 0d 09 0d | 09 69 66 28 6b 3e 30 20 |ss,p;...|.if(k>0 | |00001200| 26 26 20 6b 3c 6e 29 7b | 0d 09 09 72 69 67 68 74 |&& k<n){|...right| |00001210| 3d 6b 2b 30 2e 35 3b 0d | 09 09 73 3d 49 6e 76 65 |=k+0.5;.|..s=Inve| |00001220| 72 73 65 4e 6f 72 6d 61 | 6c 28 73 71 72 74 28 50 |rseNorma|l(sqrt(P| |00001230| 29 29 3b 0d 09 09 73 73 | 3d 73 2a 73 3b 0d 09 09 |));...ss|=s*s;...| |00001240| 70 3d 28 72 69 67 68 74 | 2b 30 2e 35 2a 73 73 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|00001310| 6e 65 73 20 74 68 61 74 | 20 72 65 71 75 69 72 65 |nes that| require| |00001320| 20 4e 75 6d 65 72 69 63 | 61 6c 20 52 65 63 69 70 | Numeric|al Recip| |00001330| 65 73 20 69 6e 20 43 20 | 2a 2f 0d 09 23 69 6e 63 |es in C |*/..#inc| |00001340| 6c 75 64 65 20 3c 6e 72 | 2e 68 3e 09 09 09 09 2f |lude <nr|.h>..../| |00001350| 2a 20 70 72 6f 74 6f 74 | 79 70 65 20 66 6f 72 20 |* protot|ype for | |00001360| 62 65 74 61 69 28 29 20 | 2a 2f 0d 09 23 69 66 20 |betai() |*/..#if | |00001370| 30 0d 09 09 76 6f 69 64 | 20 6d 61 69 6e 28 76 6f |0...void| main(vo| |00001380| 69 64 29 0d 09 09 2f 2a | 20 61 20 71 75 69 63 6b |id).../*| a quick| |00001390| 20 61 6e 64 20 64 69 72 | 74 79 20 64 72 69 76 65 | and dir|ty drive| |000013a0| 72 20 74 6f 20 74 65 73 | 74 20 73 6f 6d 65 20 6f |r to tes|t some o| |000013b0| 66 20 74 68 65 73 65 20 | 72 6f 75 74 69 6e 65 73 |f these |routines| |000013c0| 20 2a 2f 0d 09 09 7b 0d | 09 09 09 64 6f 75 62 6c | */...{.|...doubl| |000013d0| 65 20 70 2c 70 55 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2c |d be 0.3|003.\n",| |000014a0| 70 4c 6f 77 65 72 29 3b | 0d 09 09 09 50 3d 30 2e |pLower);|....P=0.| |000014b0| 39 35 3b 0d 09 09 09 66 | 6f 72 28 69 3d 30 3b 69 |95;....f|or(i=0;i| |000014c0| 3c 31 30 3b 69 2b 2b 29 | 7b 0d 09 09 09 09 6b 3d |<10;i++)|{.....k=| |000014d0| 42 69 6e 6f 6d 69 61 6c | 53 61 6d 70 6c 65 28 20 |Binomial|Sample( | |000014e0| 70 2c 20 6e 29 3b 0d 09 | 09 09 09 70 4c 6f 77 65 |p, n);..|...pLowe| |000014f0| 72 3d 42 69 6e 6f 6d 69 | 61 6c 4c 6f 77 65 72 42 |r=Binomi|alLowerB| |00001500| 6f 75 6e 64 28 50 2c 6b | 2c 6e 29 3b 0d 09 09 09 |ound(P,k|,n);....| |00001510| 09 70 55 70 70 65 72 3d | 42 69 6e 6f 6d 69 61 6c |.pUpper=|Binomial| |00001520| 55 70 70 65 72 42 6f 75 | 6e 64 28 50 2c 6b 2c 6e |UpperBou|nd(P,k,n| |00001530| 29 3b 0d 09 09 09 09 50 | 4c 6f 77 65 72 3d 31 2e |);.....P|Lower=1.| |00001540| 30 2d 42 69 6e 6f 6d 69 | 61 6c 28 70 4c 6f 77 65 |0-Binomi|al(pLowe| |00001550| 72 2c 6b 2c 6e 29 3b 0d | 09 09 09 09 50 55 70 70 |r,k,n);.|....PUpp| |00001560| 65 72 3d 42 69 6e 6f 6d | 69 61 6c 28 70 55 70 70 |er=Binom|ial(pUpp| |00001570| 65 72 2c 6b 2b 31 2c 6e | 29 3b 0d 09 09 09 09 70 |er,k+1,n|);.....p| |00001580| 72 69 6e 74 66 28 22 6e | 20 25 34 64 20 70 20 25 |rintf("n| %4d p %| |00001590| 36 2e 34 66 20 25 36 2e | 34 66 b2 25 36 2e 34 66 |6.4f %6.|4f.%6.4f| |000015a0| b2 25 36 2e 34 66 20 20 | 25 36 2e 34 66 3c 20 3c |.%6.4f |%6.4f< <| |000015b0| 25 36 2e 34 66 5c 6e 22 | 0d 09 09 09 09 09 2c 6e |%6.4f\n"|......,n| |000015c0| 2c 70 2c 70 4c 6f 77 65 | 72 2c 6b 2f 28 64 6f 75 |,p,pLowe|r,k/(dou| |000015d0| 62 6c 65 29 6e 2c 70 55 | 70 70 65 72 2c 50 4c 6f |ble)n,pU|pper,PLo| |000015e0| 77 65 72 2c 50 55 70 70 | 65 72 29 3b 0d 09 09 09 |wer,PUpp|er);....| |000015f0| 09 70 4c 6f 77 65 72 3d | 49 6e 76 65 72 73 65 42 |.pLower=|InverseB| |00001600| 69 6e 6f 6d 69 61 6c 28 | 31 2e 30 2d 73 71 72 74 |inomial(|1.0-sqrt| |00001610| 28 50 29 2c 6b 2c 6e 29 | 3b 0d 09 09 09 09 70 55 |(P),k,n)|;.....pU| |00001620| 70 70 65 72 3d 49 6e 76 | 65 72 73 65 42 69 6e 6f |pper=Inv|erseBino| |00001630| 6d 69 61 6c 28 73 71 72 | 74 28 50 29 2c 6b 2b 31 |mial(sqr|t(P),k+1| |00001640| 2c 6e 29 3b 0d 09 09 09 | 09 50 4c 6f 77 65 72 3d |,n);....|.PLower=| |00001650| 31 2e 30 2d 42 69 6e 6f | 6d 69 61 6c 28 70 4c 6f |1.0-Bino|mial(pLo| |00001660| 77 65 72 2c 6b 2c 6e 29 | 3b 0d 09 09 09 09 50 55 |wer,k,n)|;.....PU| |00001670| 70 70 65 72 3d 42 69 6e | 6f 6d 69 61 6c 28 70 55 |pper=Bin|omial(pU| |00001680| 70 70 65 72 2c 6b 2b 31 | 2c 6e 29 3b 0d 09 09 09 |pper,k+1|,n);....| |00001690| 09 70 72 69 6e 74 66 28 | 22 20 20 25 34 64 20 70 |.printf(|" %4d p| |000016a0| 20 25 36 2e 34 66 20 25 | 36 2e 34 66 b2 25 36 2e | %6.4f %|6.4f.%6.| |000016b0| 34 66 b2 25 36 2e 34 66 | 20 20 25 36 2e 34 66 3c |4f.%6.4f| %6.4f<| |000016c0| 20 3c 25 36 2e 34 66 5c | 6e 22 0d 09 09 09 09 09 | <%6.4f\|n"......| |000016d0| 2c 6e 2c 70 2c 70 4c 6f | 77 65 72 2c 6b 2f 28 64 |,n,p,pLo|wer,k/(d| |000016e0| 6f 75 62 6c 65 29 6e 2c | 70 55 70 70 65 72 2c 50 |ouble)n,|pUpper,P| |000016f0| 4c 6f 77 65 72 2c 50 55 | 70 70 65 72 29 3b 0d 09 |Lower,PU|pper);..| |00001700| 09 09 7d 0d 09 09 7d 0d | 09 23 65 6e 64 69 66 0d |..}...}.|.#endif.| |00001710| 09 0d 09 64 6f 75 62 6c | 65 20 42 69 6e 6f 6d 69 |...doubl|e Binomi| |00001720| 61 6c 28 64 6f 75 62 6c | 65 20 70 2c 6c 6f 6e 67 |al(doubl|e p,long| |00001730| 20 6b 2c 6c 6f 6e 67 20 | 6e 29 0d 09 2f 2a 0d 09 | k,long |n)../*..| |00001740| 52 65 74 75 72 6e 73 20 | 74 68 65 20 70 72 6f 62 |Returns |the prob| |00001750| 61 62 69 6c 69 74 79 20 | 6f 66 20 6b 20 6f 72 20 |ability |of k or | |00001760| 6d 6f 72 65 20 68 65 61 | 64 73 20 69 6e 20 6e 20 |more hea|ds in n | |00001770| 66 6c 69 70 73 2c 20 77 | 68 65 72 65 20 70 72 6f |flips, w|here pro| |00001780| 62 61 62 69 6c 69 74 79 | 20 6f 66 0d 09 65 61 63 |bability| of..eac| |00001790| 68 20 68 65 61 64 73 20 | 69 73 20 70 2e 20 54 68 |h heads |is p. Th| |000017a0| 69 73 20 69 64 65 6e 74 | 69 74 79 20 61 70 70 65 |is ident|ity appe| |000017b0| 61 72 73 20 69 6e 20 4e | 75 6d 65 72 69 63 61 6c |ars in N|umerical| |000017c0| 20 52 65 63 69 70 65 73 | 20 69 6e 20 43 2c 20 70 | Recipes| in C, p| |000017d0| 61 67 65 20 31 38 32 2c | 20 61 6e 64 0d 09 69 6e |age 182,| and..in| |000017e0| 20 41 62 72 61 6d 6f 77 | 69 74 7a 20 61 6e 64 20 | Abramow|itz and | |000017f0| 53 74 65 67 75 6e 2c 20 | 70 2e 20 39 34 35 2e 20 |Stegun, |p. 945. | |00001800| 45 71 2e 32 36 2e 35 2e | 32 34 2e 0d 09 2a 2f 0d |Eq.26.5.|24...*/.| |00001810| 09 7b 0d 09 09 69 66 28 | 6b 3e 6e 29 72 65 74 75 |.{...if(|k>n)retu| |00001820| 72 6e 20 30 2e 30 3b 0d | 09 09 69 66 28 6b 3c 3d |rn 0.0;.|..if(k<=| |00001830| 30 4c 29 72 65 74 75 72 | 6e 20 31 2e 30 3b 0d 09 |0L)retur|n 1.0;..| |00001840| 09 69 66 28 70 3d 3d 30 | 2e 30 29 7b 0d 09 09 09 |.if(p==0|.0){....| |00001850| 69 66 28 6b 3d 3d 30 29 | 72 65 74 75 72 6e 20 31 |if(k==0)|return 1| |00001860| 2e 30 3b 0d 09 09 09 65 | 6c 73 65 20 72 65 74 75 |.0;....e|lse retu| |00001870| 72 6e 20 30 2e 30 3b 0d | 09 09 7d 0d 09 09 69 66 |rn 0.0;.|..}...if| |00001880| 28 70 3d 3d 31 2e 30 29 | 72 65 74 75 72 6e 20 31 |(p==1.0)|return 1| |00001890| 2e 30 3b 0d 09 09 72 65 | 74 75 72 6e 20 49 6e 63 |.0;...re|turn Inc| |000018a0| 6f 6d 70 6c 65 74 65 42 | 65 74 61 28 70 2c 6b 2c |ompleteB|eta(p,k,| |000018b0| 6e 2d 6b 2b 31 29 3b 0d | 09 7d 0d 09 0d 09 64 6f |n-k+1);.|.}....do| |000018c0| 75 62 6c 65 20 42 69 6e | 6f 6d 69 61 6c 50 64 66 |uble Bin|omialPdf| |000018d0| 28 64 6f 75 62 6c 65 20 | 70 2c 6c 6f 6e 67 20 6b |(double |p,long k| |000018e0| 2c 6c 6f 6e 67 20 6e 29 | 0d 09 2f 2a 0d 09 52 65 |,long n)|../*..Re| |000018f0| 74 75 72 6e 73 20 74 68 | 65 20 70 72 6f 62 61 62 |turns th|e probab| |00001900| 69 6c 69 74 79 20 6f 66 | 20 65 78 61 63 74 6c 79 |ility of| exactly| |00001910| 20 6b 20 68 65 61 64 73 | 20 69 6e 20 6e 20 66 6c | k heads| in n fl| |00001920| 69 70 73 2c 20 77 68 65 | 72 65 20 70 72 6f 62 61 |ips, whe|re proba| |00001930| 62 69 6c 69 74 79 20 6f | 66 0d 09 65 61 63 68 20 |bility o|f..each | |00001940| 68 65 61 64 73 20 69 73 | 20 70 2e 20 0d 09 49 27 |heads is| p. ..I'| |00001950| 6d 20 6e 6f 74 20 72 65 | 61 6c 6c 79 20 73 75 72 |m not re|ally sur| |00001960| 65 20 77 68 65 74 68 65 | 72 20 74 68 69 73 20 69 |e whethe|r this i| |00001970| 73 20 6d 6f 72 65 20 63 | 6f 6d 70 75 74 61 74 69 |s more c|omputati| |00001980| 6f 6e 61 6c 6c 79 20 65 | 66 66 69 63 69 65 6e 74 |onally e|fficient| |00001990| 20 74 68 61 6e 20 63 6f | 6d 70 75 74 69 6e 67 0d | than co|mputing.| |000019a0| 09 69 74 20 64 69 72 65 | 63 74 6c 79 2c 20 66 72 |.it dire|ctly, fr| |000019b0| 6f 6d 20 74 68 65 20 64 | 65 66 69 6e 69 74 69 6f |om the d|efinitio| |000019c0| 6e 20 6f 66 20 74 68 65 | 20 62 69 6e 6f 6d 69 61 |n of the| binomia| |000019d0| 6c 20 64 69 73 74 72 69 | 62 75 74 69 6f 6e 2c 20 |l distri|bution, | |000019e0| 62 75 74 20 69 74 20 69 | 73 20 76 65 72 79 20 65 |but it i|s very e| 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