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

#### 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\$.
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2023
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].
Barabesi Lucio; Pratelli Luca; Rigo Pietro
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/903684`