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Introduction to Probability phần 5

Hình 5.4: Lãnh đạo các chữ số trong tờ khai thuế của Tổng thống Clinton. Theodore Hill2 đưa ra một mô tả chung về sự phân bố Benford, khi xem xét các chữ số d đầu tiên của số nguyên trong một tập hợp dữ liệu. Chúng tôi sẽ hạn chế sự chú ý của chúng tôi để các chữ số đầu tiên. | 196 CHAPTER 5. DISTRIBUTIONS AND DENSITIES Figure 5.4 Leading digits in President Clinton s tax returns. Theodore Hill2 gives a general description of the Benford distribution when one considers the first d digits of integers in a data set. We will restrict our attention to the first digit. In this case the Benford distribution has distribution function f k logio k 1 - logi0 k for 1 k 9. Mark Nigrini3 has advocated the use of the Benford distribution as a means of testing suspicious financial records such as bookkeeping entries checks and tax returns. His idea is that if someone were to make up numbers in these cases the person would probably produce numbers that are fairly uniformly distributed while if one were to use the actual numbers the leading digits would roughly follow the Benford distribution. As an example Nigrini analyzed President Clinton s tax returns for a 13-year period. In Figure 5.4 the Benford distribution values are shown as squares and the President s tax return data are shown as circles. One sees that in this example the Benford distribution fits the data very well. This distribution was discovered by the astronomer Simon Newcomb who stated the following in his paper on the subject That the ten digits do not occur with equal frequency must be evident to anyone making use of logarithm tables and noticing how much faster the first pages wear out than the last ones. The first significant figure is oftener 1 than any other digit and the frequency diminishes up to 9. 4 2T. P. Hill The Significant Digit Phenomenon American Mathematical Monthly vol. 102 no. 4 April 1995 pgs. 322-327. 3 M. Nigrini Detecting Biases and Irregularities in Tabulated Data working paper 4 S. Newcomb Note on the frequency of use of the different digits in natural numbers American Journal of Mathematics vol. 4 1881 pgs. 39-40. 5.1. IMPORTANT DISTRIB UTIONS 197 Exercises 1 For which of the following random variables would it be appropriate to assign a uniform distribution a .