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Some Results on the Distribution of the Most Significant Digit

A. K. Adhikari
Sankhyā: The Indian Journal of Statistics, Series B (1960-2002)
Vol. 31, No. 3/4 (Dec., 1969), pp. 413-420
Published by: Springer on behalf of the Indian Statistical Institute
Stable URL: http://www.jstor.org/stable/25051694
Page Count: 8
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Some Results on the Distribution of the Most Significant Digit
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Abstract

This paper finds the distribution of the most significant digit of some functions of random variables $X_{1},X_{2},\ldots ,X_{n}$, where these variables are independent and distributed uniformly in (0, 1). The probability that the most significant digit of $Y_{n}$ is A (A=1,..., 9) has been found, where $Y_{n}$ is defined as the product of reciprocals of n such random variables. It has been shown that this probability distribution tends to ${\rm log}_{10}$ (A+1)/A as n tends to infinity. Similarly if $Z_{n}$ is defined as $Z_{n}=X_{1}/X_{2}/\ldots /X_{n+1}$, it has been proved that the probability distribution of the most significant digit of $Z_{n}$ also tends to the same limit as n tends to infinity. More generally it is found that if $V_{1},V_{2},\ldots ,V_{n}$ are defined as $V_{1}=B/X,\ldots,\ V_{n}=V_{n-1}/X_{n}$ where B is any random variable defined on the positive axis of the real line, the probability distribution of the most significant digit tends to ${\rm log}_{10}$ (A+1)/A as n tends to infinity.

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