On the anti-concentration functions of some familiar families of distributions

Let $\{X_{\alpha}\}$ be a family of random variables following a certain type of distributions with finite expectation $\mathbf{E}[X_{\alpha}]$ and finite variance ${\rm Var}(X_{\alpha})$, where $\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function $(0, \infty)\ni y\to \inf_{\alpha}\mathbf{P}\left(|X_{\alpha}-\mathbf{E}[X_{\alpha}]|\geq y \sqrt{{\rm Var}(X_{\alpha})}\right)$ and find its explicit expression. We show that, for certain familiar families of distributions, including uniform distributions, exponential distributions, non-degenerate Gaussian distributions and student's $t$-distribution, the anti-concentration function is not identically zero, while for some other familiar families of distributions, including binomial, Poisson, negative binomial, hypergeometric, Gamma, Pareto, Weibull, log-normal and Beta distributions, the anti-concentration function is identically zero.
Comment: 15 pages