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On the probability that a discrete IFR random variable exceeds its mean
Abstract
Let X be an integer-valued random variable with 𝔼(|X|)<∞ whose survival probabilities Sj=ℙ(X≥ j), j∈ℤ, form a log-concave sequence. Suppose that for some integer m and some γ∈ (0, 1/2] it holds 𝔼(X)≥ m and ℙ(X=m+1)≥γ·ℙ(X=m)> 0. It is shown that ℙ(X ≥ m+1) ≥ γ/(1+2γ) , with the bound being best possible. As a consequence, it is shown that if H ∼ Hyp(n, i, k) is a hypergeometric random variable for which both ik/n and $(n-i)(n-k)/n$ are at least log(4/3), then it holds ℙ(H >𝔼(H)) ≥ 1/4 when 𝔼(H) is not an integer, and ℙ(H >𝔼(H)) ≥ 1/6 when 𝔼(H) is an integer, both bounds being best possible.
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BibTeX
@misc{ai2026probability,
title = {{On the probability that a discrete IFR random variable exceeds its mean}},
author = {Jianhang Ai and Vassilis Karagiannis and Ondřej Kuželka and Christos Pelekis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24726},
eprint = {2610.24726},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24726},
}
RIS
TY - UNPB
TI - On the probability that a discrete IFR random variable exceeds its mean
AU - Jianhang Ai
AU - Vassilis Karagiannis
AU - Ondřej Kuželka
AU - Christos Pelekis
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24726
UR - https://arxiv.org/abs/2610.24726
AB - Let X be an integer-valued random variable with 𝔼(|X|)<∞ whose survival probabilities S_j=ℙ(X≥ j), j∈ℤ, form a log-concave sequence. Suppose that for some integer m and some γ∈ (0, 1/2] it holds 𝔼(X)≥ m and ℙ(X=m+1)≥γ·ℙ(X=m)> 0. It is shown that ℙ(X ≥ m+1) ≥ γ/(1+2γ) , with the bound being best possible. As a consequence, it is shown that if H ∼ Hyp(n, i, k) is a hypergeometric random variable for which both ik/n and $(n-i)(n-k)/n$ are at least log(4/3), then it holds ℙ(H >𝔼(H)) ≥ 1/4 when 𝔼(H) is not an integer, and ℙ(H >𝔼(H)) ≥ 1/6 when 𝔼(H) is an integer, both bounds being best possible.
ER -