By Daniel J. Velleman
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Extra info for American Mathematical Monthly, volume 116, number 1, january 2009
C. P. Jargodzki and F. Potter, Mad About Physics: Braintwisters, Paradoxes, and Curiosities, John Wiley, New York, 2001. 12. P. B. Johnson, Leaning tower of lire, Amer. J. Phys. 23 (1955) 240. 13. G. M. , Clarendon, Oxford, 1907. 14. M. Paterson and U. Zwick, Overhang, in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), Society for Industrial and Applied Mathematics, Philadelphia, 2006, 231–240. 15. M. Paterson, Y. Peres, M. Thorup, P. Winkler, and U. 0093. 16.
19. 20. R. Sutton, A problem of balancing, Amer. J. Phys. 23 (1955) 547. 21. W. , Deighton, Bell, Cambridge, 1855. 22. U. Zwick, Jenga, in Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’02), Association for Computing Machinery, New York, 2002, 243–246. , FRS) took degrees in mathematics at Cambridge. A glimpse of his early interests can be found in the 1964 paper referenced above, before he rose to fame as the co-inventor with John Conway of Sprouts. He evolved from president of the Trinity Mathematical Society to president of the European Association for Theoretical Computer Science, and migrated from MIT to the University of Warwick, where he has been in the Computer Science department for 37 years.
Then, computing the right-hand side in Stein’s identity (9), we find E[X f (X )] = E[(B − p) f (B − p)] = p(1 − p) f (1 − p) − (1 − p) p f (− p) = σ 2 [ f (1 − p) − f (− p)] 1− p = σ2 −p f (u)du = σ 2 E f (U ), for U having uniform density over [− p, 1 − p]. Hence, a version of Stein’s identity holds for this nonnormal variable upon replacing the X appearing on the left-hand side of (9) by a variable with a distribution different from that of the given X . This calculation gives one instance of the zero bias transformation: the centered Bernoulli B − p is transformed to the uniform U over [− p, 1 − p].
American Mathematical Monthly, volume 116, number 1, january 2009 by Daniel J. Velleman