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# A statistical study of randomness among the first 5,00,000 digits of Pi (π)

Author Affiliations

• 1Maharashtra University of Health Sciences, Nashik, MS, India

Res. J. Mathematical & Statistical Sci., Volume 6, Issue (3), Pages 1-5, March,12 (2018)

## Abstract

A large amount of work has been done on the randomness of the digits of Pi (π) with various statistical tests of randomness which are used to distinguish good from not-so-good random number generators when applie d to the digits of Pi (π). Sampling and simulation are the vital are as in statistics. In both of these areas random sample i s the basic requirement to arrive at correct decision. To draw random sample, lots of methods ranging from lottery method to computer based random number generation are readily available. The digits of Pi (π) have to pass the tests as well as from the good random number generator (RNG) can be easily and rapid ly generate in the computer. I have made an interesting study in the statement in which first 5,00,000 digits of Pi (π) were divided into various consecutive blocks and each block was tested for randomness by using Chi-square test goodness of fit. A statistical analysis of the first 5,00,000 digits of Pi (π) was carried out with a view to examine, in close detail, the degree of randomness in the frequency and in the order of appear ance of the various digits therein. Frequency counts were done fo r single digits within blocks of 5000; 10,000 and 20 ,000 of digits of Pi (π). Calculation of the various statistical quan tities shows that the sets of digits under analysis confirm closely to the hypothesis of perfect randomness.

## References

1. Sourabh S.K., Chakraborty S. and Das B.K. (2009)., Are Subsequences of Decimal Digits of PI Random?. Annals., Computer Science Series, 7(2).
2. L, Testing random number generators., In Winter Simulation Conference: Proceedings of the 24 th conference on Winter simulation, 13(16), 305-313.
3. Marsaglia G. (1985)., A current view of random number generators., Computer Science and Statistics, 16th Symposium on the Interface. Elsevier Science Publishers, North-Holland, Amsterdam, 3-10.
4. Maurer U.M. (1992)., A universal statistical test for random bit generators., Journal of cryptology, 5(2), 89-105. IJCSNS International Journal of Computer Science and Network Security, 90, 9(9), September 2009.
5. Peizari B. and Dakhilalian M. (2003)., Improvement of runs test and its application on sub blocks., 2nd Iranian Society of Cryptography Conference, Sharif University, September 2003.
6. Golomb S.W., Sequences S.R. and Holden-Day S.F. (1982)., CA, 1967., Aegean Park.
7. Beker H. and Piper F. (1982)., Cipher systems: the protection of communications., Northwood Books., London.
8. Rukhin A., Soto J., Nechvatal J., Smid M. and Barker E. (2001)., A statistical test suite for random and pseudorandom number generators for cryptographic applications., Booz-Allen and Hamilton Inc Mclean Va. , NIST Special Publication 800-22, 15 May 2001. See http://csrc.nist.gov/rng/.
9. Anderson T.W. and Darling D.A. (1954)., A test of goodness of fit., Journal of the American statistical association, 49(268), 765-769.
10. Gustafson H., Dawson E., Nielsen L. and Caelli W. (1994)., A computer package for measuring the strength of encryption algorithms., Computers & Security, 13(8), 687-697. See http://www.isi.qut.edu.au/resources/cryptx.
11. Knuth D.E. (1998)., The Art of Computer Programming., Seminumerical Algorithms, 3rd ed., Addison-Wesley, Reading, Mass, 2.
12. L'Ecuyer P., Simard R. and Test U01 (2001)., A Software Library in ANSI C for Empirical Testing of Random Number Generators., Software user's guide. See http://www.iro.umontreal.ca/ simardr/testu01/tu01.html.
13. Marsaglia G. (2015)., DIEHARD: a battery of tests of randomness (1996)., See http://stat.fsu.edu/geo/diehard.html.
14. Marsaglia G. and Tsang W.W. (2002)., Some difficult-to-pass tests of randomness., Journal of Statistical Software, 7(3), 1-9. See http://www.jstatsoft.org/v07/i03/tuftests.pdf.
15. Wegenkittl S. (1995)., Empirical Testing of Pseudorandom Number Generators., Master of Science Thesis, Salzbufg University.
16. Wegenkittl S. (1998)., Generalized-divergence and Frequency Analysis in Markov Chains., Ph.D. thesis, University of Salzburg. See http://random.mat.sbg.ac.at/team/.
17. Pathria R.K. (1962)., A statistical study of randomness among the first 10,000 digits of π., Mathematics of Computation, 16(78), 188-197.
18. https://www.angio.net/pi/digits.html (http://www.angio.net/pi/digits/pi1000000.txt), undefined, undefined
19. Kendall M.G. and Smith B.B. (1939)., Random Sampling Numbers, Tracts for Computers., No. 24, R. Statitical Soc., 101, 147-66, Cambridge Univ. Press
20. Kasture Madhukar, Pandharikar Nanda and Mathankar Mayura (2012)., Index of First Order Independence within the set up of Markov Dependence., International Journal of Management Studies, Statistics and Applied Economics, 2(1), 15-20.
21. Patharia R.K. (1963)., Study of Randomness Among The First 60,000 Digits of e, Department of Physics, University of Delhi, Delhi 6, (Received September 2, 1963) Communicated by P. V. Krishna Iyer, F.N.I.
22. Pathria R.K. (1961)., A Statistical Analysis of the First 2,500 Decimal Places of e and 1/e., Department of Physics, University of Delhi, Delhi 6, (Received February 2, 1963) Communicated by P. V. Krishna Iyer, F.N.I., 27, 270-282.
23. Ruhkin A.L. (2001)., Testing randomness: A suite of statistical procedures., Theory of Probability & Its Applications, 45(1), 111-132.