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+--------+-------------------------+-------------------------+--------+--------+
|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 43 57 49 | 45 01 00 00 00 00 00 00 |.TEXTCWI|E.......|
|00000050| 00 00 00 00 00 1f e9 00 | 00 02 ca ac 33 4a bb ac |........|....3J..|
|00000060| 33 4a bb 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |3J......|........|
|00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 82 0d 00 00 |........|........|
|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 2c 31 39 39 34 20 |,1991,19|92,1994 |
|000000b0| 44 65 6e 69 73 20 47 2e | 20 50 65 6c 6c 69 0d 0d |Denis G.| Pelli..|
|000000c0| 56 61 72 69 6f 75 73 20 | 72 6f 75 74 69 6e 65 73 |Various |routines|
|000000d0| 20 74 68 61 74 20 64 65 | 61 6c 20 77 69 74 68 20 | that de|al with |
|000000e0| 42 69 6e 6f 6d 69 61 6c | 20 73 74 61 74 69 73 74 |Binomial| statist|
|000000f0| 69 63 73 2c 20 69 6e 63 | 6c 75 64 69 6e 67 20 67 |ics, inc|luding g|
|00000100| 65 6e 65 72 61 74 69 6e | 67 20 72 61 6e 64 6f 6d |eneratin|g random|
|00000110| 0d 73 61 6d 70 6c 65 73 | 20 61 6e 64 20 63 6f 6d |.samples| and com|
|00000120| 70 75 74 69 6e 67 20 63 | 6f 6e 66 69 64 65 6e 63 |puting c|onfidenc|
|00000130| 65 20 69 6e 74 65 72 76 | 61 6c 73 2e 0d 0d 54 68 |e interv|als...Th|
|00000140| 69 73 20 66 69 6c 65 20 | 63 6f 6e 74 61 69 6e 73 |is file |contains|
|00000150| 20 74 77 6f 20 67 65 6e | 65 72 61 74 69 6f 6e 73 | two gen|erations|
|00000160| 20 6f 66 20 72 6f 75 74 | 69 6e 65 73 20 74 68 61 | of rout|ines tha|
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|00000180| 64 65 6e 63 65 0d 69 6e | 74 65 72 76 61 6c 73 2e |dence.in|tervals.|
|00000190| 20 54 68 65 20 6f 72 69 | 67 69 6e 61 6c 20 72 6f | The ori|ginal ro|
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|00000200| 61 72 65 20 62 61 73 65 | 64 20 6f 6e 20 61 20 4e |are base|d on a N|
|00000210| 75 6d 65 72 69 63 61 6c | 20 52 65 63 69 70 65 73 |umerical| Recipes|
|00000220| 20 69 6e 20 43 0d 72 6f | 75 74 69 6e 65 20 74 6f | in C.ro|utine to|
|00000230| 20 63 6f 6d 70 75 74 65 | 20 74 68 65 20 69 6e 63 | compute| the inc|
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|00000300| 72 6d 2e 63 0d 0d 48 49 | 53 54 4f 52 59 3a 0d 31 |rm.c..HI|STORY:.1|
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|00000350| 65 6e 74 65 64 20 6f 75 | 74 20 62 79 20 73 65 74 |ented ou|t by set|
|00000360| 74 69 6e 67 20 4e 45 57 | 5f 42 49 4e 4f 4d 49 41 |ting NEW|_BINOMIA|
|00000370| 4c 20 74 6f 20 7a 65 72 | 6f 2e 0d 31 2f 37 2f 39 |L to zer|o..1/7/9|
|00000380| 31 20 52 65 77 72 6f 74 | 65 20 42 69 6e 6f 6d 69 |1 Rewrot|e Binomi|
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|000003d0| 09 09 64 65 73 69 72 65 | 64 20 63 6f 6e 66 69 64 |..desire|d confid|
|000003e0| 65 6e 63 65 20 6f 66 20 | 74 68 65 20 69 6e 74 65 |ence of |the inte|
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|00000400| 70 09 41 64 64 65 64 20 | 42 69 6e 6f 6d 69 61 6c |p.Added |Binomial|
|00000410| 53 61 6d 70 6c 65 51 75 | 69 63 6b 6c 79 28 29 2e |SampleQu|ickly().|
|00000420| 0d 34 2f 32 37 2f 39 32 | 09 64 67 70 09 4f 6f 70 |.4/27/92|.dgp.Oop|
|00000430| 73 2e 20 49 20 6a 75 73 | 74 20 6e 6f 74 69 63 65 |s. I jus|t notice|
|00000440| 64 20 74 68 61 74 20 74 | 68 69 73 20 66 69 6c 65 |d that t|his file|
|00000450| 20 77 61 73 20 61 6c 77 | 61 79 73 20 69 6e 63 6c | was alw|ays incl|
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|00000480| 68 61 76 65 20 69 66 20 | 79 6f 75 20 6f 77 6e 20 |have if |you own |
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|000004a0| 63 69 70 65 73 2e 20 49 | 20 6d 6f 76 65 64 20 74 |cipes. I| moved t|
|000004b0| 68 65 20 69 6e 63 6c 75 | 64 65 20 73 74 61 74 65 |he inclu|de state|
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|00000500| 0d 23 69 6e 63 6c 75 64 | 65 20 22 56 69 64 65 6f |.#includ|e "Video|
|00000510| 54 6f 6f 6c 62 6f 78 2e | 68 22 0d 0d 23 64 65 66 |Toolbox.|h"..#def|
|00000520| 69 6e 65 20 4e 45 57 5f | 42 49 4e 4f 4d 49 41 4c |ine NEW_|BINOMIAL|
|00000530| 20 30 0d 0d 6c 6f 6e 67 | 20 42 69 6e 6f 6d 69 61 | 0..long| Binomia|
|00000540| 6c 53 61 6d 70 6c 65 28 | 64 6f 75 62 6c 65 20 70 |lSample(|double p|
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|00000570| 72 6f 6d 20 61 20 62 69 | 6e 6f 6d 69 61 6c 20 64 |rom a bi|nomial d|
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|000005e0| 68 20 66 6c 69 70 2e 20 | 54 68 69 73 20 69 73 20 |h flip. |This is |
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|00000610| 6e 20 69 73 20 6c 61 72 | 67 65 2c 20 65 2e 67 2e |n is lar|ge, e.g.|
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|00000670| 65 20 65 6c 61 62 6f 72 | 61 74 65 20 4e 75 6d 65 |e elabor|ate Nume|
|00000680| 72 69 63 61 6c 20 52 65 | 63 69 70 65 73 20 62 6e |rical Re|cipes bn|
|00000690| 6c 64 65 76 28 29 20 72 | 6f 75 74 69 6e 65 2e 0d |ldev() r|outine..|
|000006a0| 2a 2f 0d 7b 0d 09 6c 6f | 6e 67 20 6b 2c 69 3b 0d |*/.{..lo|ng k,i;.|
|000006b0| 0d 09 6b 3d 30 3b 0d 09 | 66 6f 72 28 69 3d 30 3b |..k=0;..|for(i=0;|
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|00000da0| 72 72 61 6e 74 65 64 2e | 0d 0d 45 76 65 6e 20 74 |rranted.|..Even t|
|00000db0| 68 6f 75 67 68 20 74 68 | 69 73 20 63 6f 6e 66 69 |hough th|is confi|
|00000dc0| 64 65 6e 63 65 20 69 6e | 74 65 72 76 61 6c 20 69 |dence in|terval i|
|00000dd0| 73 20 6e 6f 74 20 77 68 | 6f 6c 6c 79 20 73 61 74 |s not wh|olly sat|
|00000de0| 69 73 66 61 63 74 6f 72 | 79 20 28 69 2e 65 2e 20 |isfactor|y (i.e. |
|00000df0| 66 75 6e 64 61 6d 65 6e | 74 61 6c 6c 79 0d 66 61 |fundamen|tally.fa|
|00000e00| 6c 73 65 29 20 69 74 20 | 69 73 20 75 73 65 66 75 |lse) it |is usefu|
|00000e10| 6c 20 69 6e 20 70 72 61 | 63 74 69 63 65 20 73 69 |l in pra|ctice si|
|00000e20| 6e 63 65 20 69 74 20 74 | 79 70 69 63 61 6c 6c 79 |nce it t|ypically|
|00000e30| 20 62 65 68 61 76 65 73 | 20 73 69 6d 69 6c 61 72 | behaves| similar|
|00000e40| 6c 79 20 74 6f 20 74 68 | 65 0d 4e 6f 72 6d 61 6c |ly to th|e.Normal|
|00000e50| 20 63 61 73 65 2c 20 77 | 68 69 63 68 20 69 73 20 | case, w|hich is |
|00000e60| 74 68 65 6f 72 65 74 69 | 63 61 6c 6c 79 20 73 6f |theoreti|cally so|
|00000e70| 75 6e 64 2e 0d 2a 2f 0d | 64 6f 75 62 6c 65 20 42 |und..*/.|double B|
|00000e80| 69 6e 6f 6d 69 61 6c 4c | 6f 77 65 72 42 6f 75 6e |inomialL|owerBoun|
|00000e90| 64 28 64 6f 75 62 6c 65 | 20 50 2c 6c 6f 6e 67 20 |d(double| P,long |
|00000ea0| 6b 2c 6c 6f 6e 67 20 6e | 29 0d 2f 2a 0d 41 72 67 |k,long n|)./*.Arg|
|00000eb0| 75 6d 65 6e 74 73 20 61 | 72 65 20 74 68 65 20 63 |uments a|re the c|
|00000ec0| 6f 6e 66 69 64 65 6e 63 | 65 20 50 2c 20 61 6e 64 |onfidenc|e P, and|
|00000ed0| 20 74 68 65 20 6e 75 6d | 62 65 72 20 6b 20 6f 66 | the num|ber k of|
|00000ee0| 20 68 65 61 64 73 20 69 | 6e 20 6e 20 66 6c 69 70 | heads i|n n flip|
|00000ef0| 73 20 6f 66 20 61 20 63 | 6f 69 6e 2e 0d 54 68 65 |s of a c|oin..The|
|00000f00| 20 72 65 74 75 72 6e 65 | 64 20 76 61 6c 75 65 20 | returne|d value |
|00000f10| 70 20 69 73 20 74 68 65 | 20 6c 6f 77 65 72 20 65 |p is the| lower e|
|00000f20| 6e 64 20 6f 66 20 61 20 | 32 3f 2d 73 69 64 65 64 |nd of a |2?-sided|
|00000f30| 20 50 20 63 6f 6e 66 69 | 64 65 6e 63 65 20 69 6e | P confi|dence in|
|00000f40| 74 65 72 76 61 6c 20 6f | 6e 20 74 68 65 0d 75 6e |terval o|n the.un|
|00000f50| 64 65 72 6c 79 69 6e 67 | 20 70 72 6f 62 61 62 69 |derlying| probabi|
|00000f60| 6c 69 74 79 20 6f 66 20 | 61 20 68 65 61 64 20 6f |lity of |a head o|
|00000f70| 6e 20 61 20 73 69 6e 67 | 6c 65 20 74 72 69 61 6c |n a sing|le trial|
|00000f80| 2e 0d 2a 2f 0d 7b 0d 09 | 64 6f 75 62 6c 65 20 72 |..*/.{..|double r|
|00000f90| 69 67 68 74 2c 73 2c 73 | 73 2c 70 3b 0d 09 0d 09 |ight,s,s|s,p;....|
|00000fa0| 69 66 28 6b 3e 30 20 26 | 26 20 6b 3c 6e 29 7b 0d |if(k>0 &|& k<n){.|
|00000fb0| 09 09 72 69 67 68 74 3d | 6b 2d 30 2e 35 3b 0d 09 |..right=|k-0.5;..|
|00000fc0| 09 73 3d 49 6e 76 65 72 | 73 65 4e 6f 72 6d 61 6c |.s=Inver|seNormal|
|00000fd0| 28 73 71 72 74 28 50 29 | 29 3b 0d 09 09 73 73 3d |(sqrt(P)|);...ss=|
|00000fe0| 73 2a 73 3b 0d 09 09 70 | 3d 28 72 69 67 68 74 2b |s*s;...p|=(right+|
|00000ff0| 30 2e 35 2a 73 73 2d 73 | 2a 73 71 72 74 28 72 69 |0.5*ss-s|*sqrt(ri|
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|00001010| 29 2b 30 2e 32 35 2a 73 | 73 29 29 2f 28 6e 2b 73 |)+0.25*s|s))/(n+s|
|00001020| 73 29 3b 0d 09 09 72 65 | 74 75 72 6e 20 70 3b 0d |s);...re|turn p;.|
|00001030| 09 7d 0d 09 69 66 28 6b | 3d 3d 30 29 20 72 65 74 |.}..if(k|==0) ret|
|00001040| 75 72 6e 20 30 2e 30 3b | 0d 09 69 66 28 6b 3d 3d |urn 0.0;|..if(k==|
|00001050| 6e 29 20 72 65 74 75 72 | 6e 20 70 6f 77 28 31 2e |n) retur|n pow(1.|
|00001060| 30 2d 50 2c 31 2e 30 2f | 6e 29 3b 0d 09 72 65 74 |0-P,1.0/|n);..ret|
|00001070| 75 72 6e 20 73 71 72 74 | 28 2d 31 2e 30 29 3b 09 |urn sqrt|(-1.0);.|
|00001080| 2f 2a 20 64 6f 6d 61 69 | 6e 20 65 72 72 6f 72 20 |/* domai|n error |
|00001090| 2a 2f 0d 7d 0d 0d 64 6f | 75 62 6c 65 20 42 69 6e |*/.}..do|uble Bin|
|000010a0| 6f 6d 69 61 6c 55 70 70 | 65 72 42 6f 75 6e 64 28 |omialUpp|erBound(|
|000010b0| 64 6f 75 62 6c 65 20 50 | 2c 6c 6f 6e 67 20 6b 2c |double P|,long k,|
|000010c0| 6c 6f 6e 67 20 6e 29 0d | 2f 2a 0d 41 72 67 75 6d |long n).|/*.Argum|
|000010d0| 65 6e 74 73 20 61 72 65 | 20 74 68 65 20 63 6f 6e |ents are| the con|
|000010e0| 66 69 64 65 6e 63 65 20 | 50 2c 20 61 6e 64 20 74 |fidence |P, and t|
|000010f0| 68 65 20 6e 75 6d 62 65 | 72 20 6b 20 6f 66 20 68 |he numbe|r k of h|
|00001100| 65 61 64 73 20 69 6e 20 | 6e 20 66 6c 69 70 73 20 |eads in |n flips |
|00001110| 6f 66 20 61 20 63 6f 69 | 6e 2e 0d 54 68 65 20 72 |of a coi|n..The r|
|00001120| 65 74 75 72 6e 65 64 20 | 76 61 6c 75 65 20 70 20 |eturned |value p |
|00001130| 69 73 20 74 68 65 20 75 | 70 70 65 72 20 65 6e 64 |is the u|pper end|
|00001140| 20 6f 66 20 61 20 32 3f | 2d 73 69 64 65 64 20 50 | of a 2?|-sided P|
|00001150| 20 63 6f 6e 66 69 64 65 | 6e 63 65 20 69 6e 74 65 | confide|nce inte|
|00001160| 72 76 61 6c 20 6f 6e 20 | 74 68 65 0d 75 6e 64 65 |rval on |the.unde|
|00001170| 72 6c 79 69 6e 67 20 70 | 72 6f 62 61 62 69 6c 69 |rlying p|robabili|
|00001180| 74 79 20 6f 66 20 61 20 | 68 65 61 64 20 6f 6e 20 |ty of a |head on |
|00001190| 61 20 73 69 6e 67 6c 65 | 20 74 72 69 61 6c 2e 0d |a single| trial..|
|000011a0| 2a 2f 0d 7b 0d 09 64 6f | 75 62 6c 65 20 72 69 67 |*/.{..do|uble rig|
|000011b0| 68 74 2c 73 2c 73 73 2c | 70 3b 0d 09 0d 09 69 66 |ht,s,ss,|p;....if|
|000011c0| 28 6b 3e 30 20 26 26 20 | 6b 3c 6e 29 7b 0d 09 09 |(k>0 && |k<n){...|
|000011d0| 72 69 67 68 74 3d 6b 2b | 30 2e 35 3b 0d 09 09 73 |right=k+|0.5;...s|
|000011e0| 3d 49 6e 76 65 72 73 65 | 4e 6f 72 6d 61 6c 28 73 |=Inverse|Normal(s|
|000011f0| 71 72 74 28 50 29 29 3b | 0d 09 09 73 73 3d 73 2a |qrt(P));|...ss=s*|
|00001200| 73 3b 0d 09 09 70 3d 28 | 72 69 67 68 74 2b 30 2e |s;...p=(|right+0.|
|00001210| 35 2a 73 73 2b 73 2a 73 | 71 72 74 28 72 69 67 68 |5*ss+s*s|qrt(righ|
|00001220| 74 2a 28 31 2e 30 2d 72 | 69 67 68 74 2f 6e 29 2b |t*(1.0-r|ight/n)+|
|00001230| 30 2e 32 35 2a 73 73 29 | 29 2f 28 6e 2b 73 73 29 |0.25*ss)|)/(n+ss)|
|00001240| 3b 0d 09 09 72 65 74 75 | 72 6e 20 70 3b 0d 09 7d |;...retu|rn p;..}|
|00001250| 0d 09 69 66 28 6b 3d 3d | 30 29 20 72 65 74 75 72 |..if(k==|0) retur|
|00001260| 6e 20 31 2e 30 2d 70 6f | 77 28 31 2e 30 2d 50 2c |n 1.0-po|w(1.0-P,|
|00001270| 31 2e 30 2f 6e 29 3b 0d | 09 69 66 28 6b 3d 3d 6e |1.0/n);.|.if(k==n|
|00001280| 29 20 72 65 74 75 72 6e | 20 31 2e 30 3b 0d 09 72 |) return| 1.0;..r|
|00001290| 65 74 75 72 6e 20 73 71 | 72 74 28 2d 31 2e 30 29 |eturn sq|rt(-1.0)|
|000012a0| 3b 09 2f 2a 20 64 6f 6d | 61 69 6e 20 65 72 72 6f |;./* dom|ain erro|
|000012b0| 72 20 2a 2f 0d 7d 0d 0d | 23 69 66 20 4e 45 57 5f |r */.}..|#if NEW_|
|000012c0| 42 49 4e 4f 4d 49 41 4c | 09 2f 2a 20 6e 65 77 20 |BINOMIAL|./* new |
|000012d0| 72 6f 75 74 69 6e 65 73 | 20 74 68 61 74 20 72 65 |routines| that re|
|000012e0| 71 75 69 72 65 20 4e 75 | 6d 65 72 69 63 61 6c 20 |quire Nu|merical |
|000012f0| 52 65 63 69 70 65 73 20 | 69 6e 20 43 20 2a 2f 0d |Recipes |in C */.|
|00001300| 09 23 69 6e 63 6c 75 64 | 65 20 22 6e 72 2e 68 22 |.#includ|e "nr.h"|
|00001310| 09 09 09 09 2f 2a 20 70 | 72 6f 74 6f 74 79 70 65 |..../* p|rototype|
|00001320| 20 66 6f 72 20 62 65 74 | 61 69 28 29 20 2a 2f 0d | for bet|ai() */.|
|00001330| 09 23 69 66 20 30 0d 09 | 09 76 6f 69 64 20 6d 61 |.#if 0..|.void ma|
|00001340| 69 6e 28 76 6f 69 64 29 | 0d 09 09 2f 2a 20 61 20 |in(void)|.../* a |
|00001350| 71 75 69 63 6b 20 61 6e | 64 20 64 69 72 74 79 20 |quick an|d dirty |
|00001360| 64 72 69 76 65 72 20 74 | 6f 20 74 65 73 74 20 73 |driver t|o test s|
|00001370| 6f 6d 65 20 6f 66 20 74 | 68 65 73 65 20 72 6f 75 |ome of t|hese rou|
|00001380| 74 69 6e 65 73 20 2a 2f | 0d 09 09 7b 0d 09 09 09 |tines */|...{....|
|00001390| 64 6f 75 62 6c 65 20 70 | 2c 70 55 70 70 65 72 2c |double p|,pUpper,|
|000013a0| 70 4c 6f 77 65 72 2c 50 | 2c 50 55 70 70 65 72 2c |pLower,P|,PUpper,|
|000013b0| 50 4c 6f 77 65 72 3b 0d | 09 09 09 69 6e 74 20 69 |PLower;.|...int i|
|000013c0| 2c 6e 2c 6b 3b 0d 09 09 | 09 0d 09 09 09 52 65 71 |,n,k;...|.....Req|
|000013d0| 75 69 72 65 28 30 29 3b | 0d 09 09 09 6e 3d 31 30 |uire(0);|....n=10|
|000013e0| 3b 0d 09 09 09 70 3d 2e | 32 3b 0d 09 09 09 70 4c |;....p=.|2;....pL|
|000013f0| 6f 77 65 72 3d 49 6e 76 | 65 72 73 65 42 69 6e 6f |ower=Inv|erseBino|
|00001400| 6d 69 61 6c 28 30 2e 30 | 35 2c 32 31 2c 35 30 29 |mial(0.0|5,21,50)|
|00001410| 3b 0d 09 09 09 70 72 69 | 6e 74 66 28 22 25 66 2c |;....pri|ntf("%f,|
|00001420| 20 41 62 72 61 6d 6f 77 | 69 74 7a 20 26 20 53 74 | Abramow|itz & St|
|00001430| 65 67 75 6e 20 70 2e 20 | 39 36 30 20 45 78 61 6d |egun p. |960 Exam|
|00001440| 70 6c 65 20 31 38 20 73 | 61 79 20 74 68 69 73 20 |ple 18 s|ay this |
|00001450| 73 68 6f 75 6c 64 20 62 | 65 20 30 2e 33 30 30 33 |should b|e 0.3003|
|00001460| 2e 5c 6e 22 2c 70 4c 6f | 77 65 72 29 3b 0d 09 09 |.\n",pLo|wer);...|
|00001470| 09 50 3d 30 2e 39 35 3b | 0d 09 09 09 66 6f 72 28 |.P=0.95;|....for(|
|00001480| 69 3d 30 3b 69 3c 31 30 | 3b 69 2b 2b 29 7b 0d 09 |i=0;i<10|;i++){..|
|00001490| 09 09 09 6b 3d 42 69 6e | 6f 6d 69 61 6c 53 61 6d |...k=Bin|omialSam|
|000014a0| 70 6c 65 28 20 70 2c 20 | 6e 29 3b 0d 09 09 09 09 |ple( p, |n);.....|
|000014b0| 70 4c 6f 77 65 72 3d 42 | 69 6e 6f 6d 69 61 6c 4c |pLower=B|inomialL|
|000014c0| 6f 77 65 72 42 6f 75 6e | 64 28 50 2c 6b 2c 6e 29 |owerBoun|d(P,k,n)|
|000014d0| 3b 0d 09 09 09 09 70 55 | 70 70 65 72 3d 42 69 6e |;.....pU|pper=Bin|
|000014e0| 6f 6d 69 61 6c 55 70 70 | 65 72 42 6f 75 6e 64 28 |omialUpp|erBound(|
|000014f0| 50 2c 6b 2c 6e 29 3b 0d | 09 09 09 09 50 4c 6f 77 |P,k,n);.|....PLow|
|00001500| 65 72 3d 31 2e 30 2d 42 | 69 6e 6f 6d 69 61 6c 28 |er=1.0-B|inomial(|
|00001510| 70 4c 6f 77 65 72 2c 6b | 2c 6e 29 3b 0d 09 09 09 |pLower,k|,n);....|
|00001520| 09 50 55 70 70 65 72 3d | 42 69 6e 6f 6d 69 61 6c |.PUpper=|Binomial|
|00001530| 28 70 55 70 70 65 72 2c | 6b 2b 31 2c 6e 29 3b 0d |(pUpper,|k+1,n);.|
|00001540| 09 09 09 09 70 72 69 6e | 74 66 28 22 6e 20 25 34 |....prin|tf("n %4|
|00001550| 64 20 70 20 25 36 2e 34 | 66 20 25 36 2e 34 66 b2 |d p %6.4|f %6.4f.|
|00001560| 25 36 2e 34 66 b2 25 36 | 2e 34 66 20 20 25 36 2e |%6.4f.%6|.4f  %6.|
|00001570| 34 66 3c 20 3c 25 36 2e | 34 66 5c 6e 22 0d 09 09 |4f< <%6.|4f\n"...|
|00001580| 09 09 09 2c 6e 2c 70 2c | 70 4c 6f 77 65 72 2c 6b |...,n,p,|pLower,k|
|00001590| 2f 28 64 6f 75 62 6c 65 | 29 6e 2c 70 55 70 70 65 |/(double|)n,pUppe|
|000015a0| 72 2c 50 4c 6f 77 65 72 | 2c 50 55 70 70 65 72 29 |r,PLower|,PUpper)|
|000015b0| 3b 0d 09 09 09 09 70 4c | 6f 77 65 72 3d 49 6e 76 |;.....pL|ower=Inv|
|000015c0| 65 72 73 65 42 69 6e 6f | 6d 69 61 6c 28 31 2e 30 |erseBino|mial(1.0|
|000015d0| 2d 73 71 72 74 28 50 29 | 2c 6b 2c 6e 29 3b 0d 09 |-sqrt(P)|,k,n);..|
|000015e0| 09 09 09 70 55 70 70 65 | 72 3d 49 6e 76 65 72 73 |...pUppe|r=Invers|
|000015f0| 65 42 69 6e 6f 6d 69 61 | 6c 28 73 71 72 74 28 50 |eBinomia|l(sqrt(P|
|00001600| 29 2c 6b 2b 31 2c 6e 29 | 3b 0d 09 09 09 09 50 4c |),k+1,n)|;.....PL|
|00001610| 6f 77 65 72 3d 31 2e 30 | 2d 42 69 6e 6f 6d 69 61 |ower=1.0|-Binomia|
|00001620| 6c 28 70 4c 6f 77 65 72 | 2c 6b 2c 6e 29 3b 0d 09 |l(pLower|,k,n);..|
|00001630| 09 09 09 50 55 70 70 65 | 72 3d 42 69 6e 6f 6d 69 |...PUppe|r=Binomi|
|00001640| 61 6c 28 70 55 70 70 65 | 72 2c 6b 2b 31 2c 6e 29 |al(pUppe|r,k+1,n)|
|00001650| 3b 0d 09 09 09 09 70 72 | 69 6e 74 66 28 22 20 20 |;.....pr|intf("  |
|00001660| 25 34 64 20 70 20 25 36 | 2e 34 66 20 25 36 2e 34 |%4d p %6|.4f %6.4|
|00001670| 66 b2 25 36 2e 34 66 b2 | 25 36 2e 34 66 20 20 25 |f.%6.4f.|%6.4f  %|
|00001680| 36 2e 34 66 3c 20 3c 25 | 36 2e 34 66 5c 6e 22 0d |6.4f< <%|6.4f\n".|
|00001690| 09 09 09 09 09 2c 6e 2c | 70 2c 70 4c 6f 77 65 72 |.....,n,|p,pLower|
|000016a0| 2c 6b 2f 28 64 6f 75 62 | 6c 65 29 6e 2c 70 55 70 |,k/(doub|le)n,pUp|
|000016b0| 70 65 72 2c 50 4c 6f 77 | 65 72 2c 50 55 70 70 65 |per,PLow|er,PUppe|
|000016c0| 72 29 3b 0d 09 09 09 7d | 0d 09 09 7d 0d 09 23 65 |r);....}|...}..#e|
|000016d0| 6e 64 69 66 0d 09 0d 09 | 64 6f 75 62 6c 65 20 42 |ndif....|double B|
|000016e0| 69 6e 6f 6d 69 61 6c 28 | 64 6f 75 62 6c 65 20 70 |inomial(|double p|
|000016f0| 2c 6c 6f 6e 67 20 6b 2c | 6c 6f 6e 67 20 6e 29 0d |,long k,|long n).|
|00001700| 09 2f 2a 0d 09 52 65 74 | 75 72 6e 73 20 74 68 65 |./*..Ret|urns the|
|00001710| 20 70 72 6f 62 61 62 69 | 6c 69 74 79 20 6f 66 20 | probabi|lity of |
|00001720| 6b 20 6f 72 20 6d 6f 72 | 65 20 68 65 61 64 73 20 |k or mor|e heads |
|00001730| 69 6e 20 6e 20 66 6c 69 | 70 73 2c 20 77 68 65 72 |in n fli|ps, wher|
|00001740| 65 20 70 72 6f 62 61 62 | 69 6c 69 74 79 20 6f 66 |e probab|ility of|
|00001750| 0d 09 65 61 63 68 20 68 | 65 61 64 20 69 73 20 70 |..each h|ead is p|
|00001760| 2e 20 54 68 69 73 20 69 | 64 65 6e 74 69 74 79 20 |. This i|dentity |
|00001770| 61 70 70 65 61 72 73 20 | 69 6e 20 4e 75 6d 65 72 |appears |in Numer|
|00001780| 69 63 61 6c 20 52 65 63 | 69 70 65 73 20 69 6e 20 |ical Rec|ipes in |
|00001790| 43 2c 20 70 61 67 65 20 | 31 38 32 2c 20 61 6e 64 |C, page |182, and|
|000017a0| 0d 09 69 6e 20 41 62 72 | 61 6d 6f 77 69 74 7a 20 |..in Abr|amowitz |
|000017b0| 61 6e 64 20 53 74 65 67 | 75 6e 2c 20 70 2e 20 39 |and Steg|un, p. 9|
|000017c0| 34 35 2e 20 45 71 2e 32 | 36 2e 35 2e 32 34 2e 0d |45. Eq.2|6.5.24..|
|000017d0| 09 2a 2f 0d 09 7b 0d 09 | 09 69 66 28 6b 3e 6e 29 |.*/..{..|.if(k>n)|
|000017e0| 72 65 74 75 72 6e 20 30 | 2e 30 3b 0d 09 09 69 66 |return 0|.0;...if|
|000017f0| 28 6b 3c 3d 30 4c 29 72 | 65 74 75 72 6e 20 31 2e |(k<=0L)r|eturn 1.|
|00001800| 30 3b 0d 09 09 69 66 28 | 70 3d 3d 30 2e 30 29 7b |0;...if(|p==0.0){|
|00001810| 0d 09 09 09 69 66 28 6b | 3d 3d 30 29 72 65 74 75 |....if(k|==0)retu|
|00001820| 72 6e 20 31 2e 30 3b 0d | 09 09 09 65 6c 73 65 20 |rn 1.0;.|...else |
|00001830| 72 65 74 75 72 6e 20 30 | 2e 30 3b 0d 09 09 7d 0d |return 0|.0;...}.|
|00001840| 09 09 69 66 28 70 3d 3d | 31 2e 30 29 72 65 74 75 |..if(p==|1.0)retu|
|00001850| 72 6e 20 31 2e 30 3b 0d | 09 09 72 65 74 75 72 6e |rn 1.0;.|..return|
|00001860| 20 49 6e 63 6f 6d 70 6c | 65 74 65 42 65 74 61 28 | Incompl|eteBeta(|
|00001870| 70 2c 6b 2c 6e 2d 6b 2b | 31 29 3b 0d 09 7d 0d 09 |p,k,n-k+|1);..}..|
|00001880| 0d 09 64 6f 75 62 6c 65 | 20 42 69 6e 6f 6d 69 61 |..double| Binomia|
|00001890| 6c 50 64 66 28 64 6f 75 | 62 6c 65 20 70 2c 6c 6f |lPdf(dou|ble p,lo|
|000018a0| 6e 67 20 6b 2c 6c 6f 6e | 67 20 6e 29 0d 09 2f 2a |ng k,lon|g n)../*|
|000018b0| 0d 09 52 65 74 75 72 6e | 73 20 74 68 65 20 70 72 |..Return|s the pr|
|000018c0| 6f 62 61 62 69 6c 69 74 | 79 20 6f 66 20 65 78 61 |obabilit|y of exa|
|000018d0| 63 74 6c 79 20 6b 20 68 | 65 61 64 73 20 69 6e 20 |ctly k h|eads in |
|000018e0| 6e 20 66 6c 69 70 73 2c | 20 77 68 65 72 65 20 70 |n flips,| where p|
|000018f0| 72 6f 62 61 62 69 6c 69 | 74 79 20 6f 66 0d 09 65 |robabili|ty of..e|
|00001900| 61 63 68 20 68 65 61 64 | 20 69 73 20 70 2e 20 0d |ach head| is p. .|
|00001910| 09 49 27 6d 20 6e 6f 74 | 20 72 65 61 6c 6c 79 20 |.I'm not| really |
|00001920| 73 75 72 65 20 77 68 65 | 74 68 65 72 20 74 68 69 |sure whe|ther thi|
|00001930| 73 20 69 73 20 6d 6f 72 | 65 20 63 6f 6d 70 75 74 |s is mor|e comput|
|00001940| 61 74 69 6f 6e 61 6c 6c | 79 20 65 66 66 69 63 69 |ationall|y effici|
|00001950| 65 6e 74 20 74 68 61 6e | 20 63 6f 6d 70 75 74 69 |ent than| computi|
|00001960| 6e 67 0d 09 69 74 20 64 | 69 72 65 63 74 6c 79 2c |ng..it d|irectly,|
|00001970| 20 66 72 6f 6d 20 74 68 | 65 20 64 65 66 69 6e 69 | from th|e defini|
|00001980| 74 69 6f 6e 20 6f 66 20 | 74 68 65 20 62 69 6e 6f |tion of |the bino|
|00001990| 6d 69 61 6c 20 64 69 73 | 74 72 69 62 75 74 69 6f |mial dis|tributio|
|000019a0| 6e 2c 20 62 75 74 20 69 | 74 20 69 73 20 76 65 72 |n, but i|t is ver|
|000019b0| 79 20 65 61 73 79 0d 09 | 74 6f 20 77 72 69 74 65 |y easy..|to write|
|000019c0| 2c 20 61 6e 64 20 61 76 | 6f 69 64 73 20 74 68 65 |, and av|oids the|
|000019d0| 20 64 69 66 66 69 63 75 | 6c 74 79 20 6f 66 20 63 | difficu|lty of c|
|000019e0| 6f 6d 70 75 74 69 6e 67 | 20 74 68 65 20 62 69 6e |omputing| the bin|
|000019f0| 6f 6d 69 61 6c 20 63 6f | 65 66 66 69 63 69 65 6e |omial co|efficien|
|00001a00| 74 0d 09 77 69 74 68 6f | 75 74 20 6f 76 65 72 66 |t..witho|ut overf|
|00001a10| 6c 6f 77 2e 0d 09 2a 2f | 0d 09 7b 0d 09 09 69 66 |low...*/|..{...if|
|00001a20| 28 6b 3c 30 4c 20 7c 7c | 20 6b 3e 6e 29 72 65 74 |(k<0L ||| k>n)ret|
|00001a30| 75 72 6e 20 30 2e 30 3b | 0d 09 09 69 66 28 6b 3d |urn 0.0;|...if(k=|
|00001a40| 3d 6e 29 20 72 65 74 75 | 72 6e 20 70 6f 77 28 70 |=n) retu|rn pow(p|
|00001a50| 2c 6e 29 3b 0d 09 09 72 | 65 74 75 72 6e 20 42 69 |,n);...r|eturn Bi|
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