2021/01/10 by Mark Huber, Huber, Mark · 1 citation
Mathematics · #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2101.03452
Let X be an integrable discrete random variable over \0, 1, 2, …\ with ℙ(X = i + 1) ≤ ℙ(X = i) for all i. Then for any integer a ≥ 1, ℙ(X ≤ a) ≤ 𝔼[X] / (2a - 1). Let W be an discrete random variable over \…, -2, -1, 0, 1, 2, …\ with finite second moment where the ℙ(W = i) values are unimodal. Then ℙ(|W - 𝔼[W]| ≥ a) ≤ (\mathbbV(W) + 1 / 12) / (2(a - 1 / 2)2).