19世纪,美国天文学家西蒙·纽康在翻阅对数表时,偶然发现表中以1开头的数出现的频率更高.约半个世纪后,物理学家本福特又重新发现这个现象,从实际生活得出的大量数据中,以1开头的数出现的频率约为总数的三成,接近期望值
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的3倍,并提出本福特定律,即在大量b进制随机数据中,以n开头的数出现的概率为
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmsub%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmsub%3E%3Cmrow%3E%3Cmi%3Elog%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmsub%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmrow%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmi%3En%3C%2Fmi%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,如斐波那契数、阶乘数、素数等都比较符合该定律.后来常有数学爱好者用此定律来检验某些经济数据、选举数据等大数据的真实性.根据本福特定律,在某项大量经济数据(十进制)中,以6开头的数出现的概率为
;若
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmstyle+displaystyle%3D%22true%22%3E%3Cmunderover%3E%3Cmo%3E%E2%88%91%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Ek%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmunderover%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmn%3E10%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmstyle%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmsub%3E%3Cmi%3EP%3C%2Fmi%3E%3Cmrow%3E%3Cmn%3E10%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmrow%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,
![](https://math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmo%3E%E2%88%88%3C%2Fmo%3E%3Cmtext%3EN%3C%2Fmtext%3E%3Cmo%3E%2A%3C%2Fmo%3E%3Cmtext%3E++%EF%BC%8C+%3C%2Fmtext%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmo%3E%E2%89%A4%3C%2Fmo%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,则k的值为
.