Power laws for monkeys typing randomly: the case of unequal probabilities
An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N>1) and a space bar, where a space separates words. A space is hit with probability p; all other letters are hit with equal probability (1-p)/N....
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| Published in: | IEEE transactions on information theory Vol. 50; no. 7; pp. 1403 - 1414 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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New York
IEEE
01.07.2004
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N>1) and a space bar, where a space separates words. A space is hit with probability p; all other letters are hit with equal probability (1-p)/N. Miller proved that in this experiment, the rank-frequency distribution of words follows a power law. The case where letters are hit with unequal probability has been the subject of recent confusion, with some suggesting that in this case the rank-frequency distribution follows a lognormal distribution. We prove that the rank-frequency distribution follows a power law for assignments of probabilities that have rational log-ratios for any pair of keys, and we present an argument of Montgomery that settles the remaining cases, also yielding a power law. The key to both arguments is the use of complex analysis. The method of proof produces simple explicit formulas for the coefficient in the power law in cases with rational log-ratios for the assigned probabilities of keys. Our formula in these cases suggests an exact asymptotic formula in the cases with an irrational log-ratio, and this formula is exactly what was proved by Montgomery. |
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| AbstractList | An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N>1) and a space bar, where a space separates words. A space is hit with probability p; all other letters are hit with equal probability (1-p)/N. Miller proved that in this experiment, the rank-frequency distribution of words follows a power law. The case where letters are hit with unequal probability has been the subject of recent confusion, with some suggesting that in this case the rank-frequency distribution follows a lognormal distribution. We prove that the rank-frequency distribution follows a power law for assignments of probabilities that have rational log-ratios for any pair of keys, and we present an argument of Montgomery that settles the remaining cases, also yielding a power law. The key to both arguments is the use of complex analysis. The method of proof produces simple explicit formulas for the coefficient in the power law in cases with rational log-ratios for the assigned probabilities of keys. Our formula in these cases suggests an exact asymptotic formula in the cases with an irrational log-ratio, and this formula is exactly what was proved by Montgomery. [PUBLICATION ABSTRACT] An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N>1) and a space bar, where a space separates words. A space is hit with probability p; all other letters are hit with equal probability (1-p)/N. Miller proved that in this experiment, the rank-frequency distribution of words follows a power law. The case where letters are hit with unequal probability has been the subject of recent confusion, with some suggesting that in this case the rank-frequency distribution follows a lognormal distribution. We prove that the rank-frequency distribution follows a power law for assignments of probabilities that have rational log-ratios for any pair of keys, and we present an argument of Montgomery that settles the remaining cases, also yielding a power law. The key to both arguments is the use of complex analysis. The method of proof produces simple explicit formulas for the coefficient in the power law in cases with rational log-ratios for the assigned probabilities of keys. Our formula in these cases suggests an exact asymptotic formula in the cases with an irrational log-ratio, and this formula is exactly what was proved by Montgomery. |
| Author | Conrad, B. Mitzenmacher, M. |
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| Cites_doi | 10.1103/PhysRevE.57.1347 10.1103/PhysRevE.60.1412 10.2307/2333389 10.1103/PhysRevE.54.220 10.1007/3-540-45465-9_11 10.1002/9781118032770 10.2307/1419346 10.1080/15427951.2004.10129088 10.1016/S0378-4371(02)01507-8 |
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| References | ref12 Pitt (ref13) 1958 ref11 Widder (ref14) 1941 ref10 Mahmoud (ref9) 1992 ref2 ref16 ref7 Flajolet (ref8) ref4 ref3 Huberman (ref15) 1999 ref5 Gong (ref6) Mandelbrot (ref1) 1953 |
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| Snippet | An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N>1) and... An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N > 1)... |
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| SubjectTerms | Analysis Asymptotic properties Coefficients Computer aided software engineering Confusion Frequency History Information Information analysis Information theory Internet Keyboards Keys Mathematics Monkeys Monkeys & apes Natural languages Number theory Power law Psychology Typing |
| Title | Power laws for monkeys typing randomly: the case of unequal probabilities |
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