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Ley de Benford

Ley de Benford. Primer dígito significativo. 0,0174. 299.959. 1,74 · 10 -2 [1,74] = 1. 2,99959 · 10 5 [2,99959] = 2. Las barras negras representan las frecuencias de aparición como primer dígito significativo (d = 1,2,3,...,9) en una lista de N = 201 constantes físicas.

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Ley de Benford

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  1. Ley de Benford

  2. Primer dígito significativo 0,0174 299.959 1,74 ·10-2 [1,74] = 1 2,99959 ·105 [2,99959] = 2

  3. Las barras negras representan las frecuencias de aparición como primer dígito significativo (d = 1,2,3,...,9) en una lista de N = 201 constantes físicas.

  4. En barras blancas aparecen las frecuencias de aparición como primer dígito de los números 1 a 9 en el tamaño en bytes de N = 1.295.777 ficheros.

  5. Simon Newcomb (1835-1909). Note on the frequency of use of the different digits in natural numbers.Amer. J. Math. 4 (1881) 39-40.

  6. Frank Benford 1 Title 2 3 4 5 6 7 8 9 Sampls Rivers, Area 31.0 16.4 10.7 11.3 7.2 8.6 5.5 4.2 5.1 335 Population 33.9 20.4 14.2 8.1 7.2 6.2 4.1 3.7 2.2 3259 Constants 41.3 14.4 4.8 8.6 10.6 5.8 1.0 2.9 10.6 104 Newspapers 30.0 18.0 12.0 10.0 8.0 6.0 6.0 5.0 5.0 100 Specific Heat 24.0 18.4 16.2 14.6 10.6 4.1 3.2 4.8 4.1 1389 Pressure 29.6 18.3 12.8 9.8 8.3 6.4 5.7 4.4 4.7 703 H.P. Lost 30.0 18.4 11.9 10.8 8.1 7.0 5.1 5.1 3.6 690 Mol. Wgt. 26.7 25.2 15.4 10.8 6.7 5.1 4.1 2.8 3.2 1800 Drainage 27.1 23.9 13.8 12.6 8.2 5.0 5.0 2.5 1.9 159 Atomic Wgt. 47.2 18.7 5.5 4.4 6.6 4.4 3.3 4.4 5.5 91      , 25.7 20.3 9.7 6.8 6.6 6.8 7.2 8.0 8.9 5000 Design 26.8 14.8 14.3 7.5 8.3 8.4 7.0 7.3 5.6 560 Reader's Digest 33.4 18.5 12.4 7.5 7.1 6.5 5.5 4.9 4.2 308 Cost Data 32.4 18.8 10.1 10.1 9.8 5.5 4.7 5.5 3.1 741 X-Ray Volts 27.9 17.5 14.4 9.0 8.1 7.4 5.1 5.8 4.8 707 Am. League 32.7 17.6 12.6 9.8 7.4 6.4 4.9 5.6 3.0 1458 Blackbody 31.0 17.3 14.1 8.7 6.6 7.0 5.2 4.7 5.4 1165 Addresses 28.9 19.2 12.6 8.8 8.5 6.4 5.6 5.0 5.0 342 The law of anomalous numbers. Proc. Am. Philos. Soc. 78 (1938) 551-538.    , 25.3 16.0 12.0 10.0 8.5 8.8 6.8 7.1 5.5 900 Death Rate 27.0 18.6 15.7 9.4 6.7 6.5 7.2 4.8 4.1 418 Average 30.6 18.5 12.4 9.4 8.0 6.4 5.1 4.9 4.7 1011 Probable Error

  7. Las barras representan las frecuencias de aparición como primer dígito de los números 10 a 99 en los N = 1.295.777 ficheros medidos. La línea continua representa la ley de Benford generalizada para dos dígitos.

  8. Invarianza de base y de escala en la densidad de probabilidad Theodore Hill Invarianza de escala Invarianza de base No toda lista de números que cumple la Ley de Benford proviene de una distribución invariante de escala. Pero seguro que es invariante de base.

  9. Procesos multiplicativos

  10.  = -1 5 décadas 5 décadas

  11. Para una lista de números que siga una distribución de probabilidad en forma de ley de potencias N-1, tendremos que la probabilidad del primer dígito significativo es independiente de la década y sigue la ley de Benford: Normalizando:

  12. The demonstration of Benford’s Law (and also for the distribution of the second digit) was done in 1996 by Professor Theodore Hill (School of Mathematics, Center for Applied Probability, Georgia Institute of Technology) in his article: “A Statistical Derivation of the Significant‐Digit law”. Hill later showed there was a kind of central limit theorem that applied to a wide variety of distributions--that combinations of distributions tend towards the distribution predicted by Benford’s law even when the original distributions do not [Hill1996].

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