Some Estimates of the Average Number of Guesses to Determine a Random Variable

REFERENCES
E. Arikan, “An Inequality On Guessing and its Application to Sequential Decoding,” IEEE Transactions on Information Theory, Vol. 42, pp. 99-105, January 1996.
S. Boztag, “Comments on “An Inequality On Guessing and its
11. THE NEW BOUNDS
Using an identity (see [3], p. 242) due to A. Korkine, we obtain: T h e o r e m 1 For any guessing sequence, we hawe the bound
Chebyshev’s Inequality for Sequences of Real Numbers,” 1996, manuscript under review for publication.
0-7803-3956-8/97/$10.00 0 1 9 9 7 IEEE
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S. S. Dragomir and B. Mond, “Some Mappings Associated with Another approach to upper bounding E[G] is by utilizing the Cauchy-Schwan inequality. This yields the ffollowing estimate in terms of the Euclidean norm of p = ( P I , . . , P M ) : Theorem 2 For any guessing sequence,
E[G]2
M+1 ( -- ( M - l ) 6 M + l ) 2
lj:<~<M
IPi - P j l ,
E,”=,
111. CONCLUSION
New upper and lower bounds were presented on the expected number of guesses required to determine a random variable.
ISIT 1997, Ulm, Germany, June 29
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Some Estimates of the Average Number of Guesses to Determine a Random Variable
Sever S. Dragomir and Serdar Boztq Department of Applied Mathematics University of Transkei Transkei, South Africa and Department of Mathematics Royal Melbourne Institute of Technology GPO Box 2476V, Melbourne 3001 Australia Email: serdar@rmit.edu.au
below on the distance of the average number of guesses from the uniform case:
Application to Sequential Decoding”,” to appear in the IEEE Transactions on Information Theory. D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
Abstract - N e w u p p e r a n d lower b o u n d s are given for bounding t h e average n u m b e r of guesses to determ i n e a r a n d o m variable. holds.
is tighter than that in [ I ] by roughly a factor of 2, and applies to a class of nonminimal guessing sequences. Here, we obtain further upper bounds on the first moment of the number of guesses. These bounds apply to any guessing sequence. Consider a random variable X with finite range X = {zI,z~, Z M } and distribution P x ( E k ) = P k , for k = ..., 1 , 2 , . .. ,M. A one-to-one function G : X -+ ( 1 , . . . , M} is a guessing function for X. Thus, E[G”] = kmph is the mth moment of this function, provided we renumber the zI such that X k is always the k t h guess. Note that if we assume that pk 2 Pk+l for k 2 1 this corresponds to the optimal guessingof the random variable X (in the sense of minimizing E[G]) the guessing sequence 2 1 , ZZ, . . . ,Z M . with
Applying the methods of [4] we can also obtain another estimate of the average number of guesses: Theorem 3 For any guessing sequence,
I. INTRODUCTION
In [ l ] , Arikan obtained an asymptotically tight upper bound to the positive moments of the minimal number of guesses required to determine the value of a random variable. In [2], Boztq obtained an upper bound to the mth moment of the number of guesses (where m 2 1 is an integer). When m = 1, this bound becomes
The results obtained in the above Theorems can be used in a straightforward manner to obtain sufficient conditions on IE[G]being bounded above by a constant E > 0. It should also be noted that for the case of optimal guessing, it is a priori known that E[G]< ( M + 1)/2, so that the bounds in Theorems 1-3 become lower b o u n d s on E[G].As an example, Theorem 1 becomes,