On the probability that a binomial variable is at most its expectation
Svante Janson · Statistics & Probability Letters · 2020
Consider the probability that a binomial random variable Bi ( n , m ∕ n ) with integer expectation m is at most its expectation. Chvátal conjectured that for any given n , this probability is smallest when m is the integer closest to 2 n ∕ 3 . We show that this holds when n is large.