Abstract
Consider infinite random words over a finite alphabet where the letters occur as an i.i.d. sequence according to some arbitrary distribution on the alphabet. The expectation and the variance of the waiting time for the first completed $h$-run of any letter (i.e., first occurrence of $h$ subsequential equal letters) is computed. The expected waiting time for the completion of $h$-runs of $j$ arbitrary distinct letters is also given.
Highlights
In [7], Szekely presented the following paradox: In measuring the regularity of a die one may use waiting times for sequences of the same side of certain lengths
Consider infinite random words over a finite alphabet where the letters occur as an i.i.d. sequence according to some arbitrary distribution on the alphabet
One would expect that a smaller number of throws is needed to get such sequences with a biased die. This leads to the definition to call one die more regular than another die if more throws are needed to get sequences of one side of a certain length
Summary
In [7], Szekely presented the following paradox: In measuring the regularity of a die one may use waiting times for sequences of the same side of certain lengths. The consequence of this paradox is that one cannot use the mean waiting times for such runs as a (sufficient) criterion for the definition of regularity of a die (or whatever random sequence of digits from a finite alphabet). This paradox gave motivation to compute first and second moments of such waiting times for so called h-runs in this article.
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