Let $q_m=P(X\le m)$, where $m$ is a positive integer and $X$ a binomial random variable with parameters $n$ and $m/n$. Va\v{s}ek Chv\a'atal conjectured that, for fixed $n\ge 2$, $q_m$ attains its minimum when $m$ is the integer closest to $2n/3$. As shown by Svante Janson, this conjecture is true for large $n$. Here, we prove that the conjecture is actually true for every $n\ge 2$.
Barabesi Lucio, Pratelli Luca, Rigo Pietro (2023). On the Chvatal-Janson conjecture. STATISTICS & PROBABILITY LETTERS, 194(March), 1-6 [10.1016/j.spl.2022.109744].
On the Chvatal-Janson conjecture
Rigo Pietro
2023
Abstract
Let $q_m=P(X\le m)$, where $m$ is a positive integer and $X$ a binomial random variable with parameters $n$ and $m/n$. Va\v{s}ek Chv\a'atal conjectured that, for fixed $n\ge 2$, $q_m$ attains its minimum when $m$ is the integer closest to $2n/3$. As shown by Svante Janson, this conjecture is true for large $n$. Here, we prove that the conjecture is actually true for every $n\ge 2$.File in questo prodotto:
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11585_903684.pdf
Open Access dal 01/04/2024
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