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A study on the negative binomial distribution motivated by Chv\'atal's theorem
Let denote a binomial random variable with parameters and .
Chv\'{a}tal's theorem says that for any fixed , as ranges over
, the probability is the smallest when
is closest to . Motivated by this theorem, in this note we
consider the infimum value of the probability , where is a
negative binomial random variable. As a consequence, we give an affirmative
answer to the conjecture posed in [Statistics and Probability Letters, 200
(2023) 109871].Comment: 10 page
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