{"ID":2841883,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.11566","arxiv_id":"2511.11566","title":"The Maximal Variance of Unilaterally Truncated Gaussian and Chi Distributions","abstract":"This work explores the bounds of the variance of unilaterally truncated Gaussian distributions (UTGDs) and scaled chi distributions (UTSCDs) with fixed means. For any arbitrary Gaussian distribution function, $f(x;μ,σ)$, with a fixed, finite mean $M$ on the truncated domain $x \\ge a$, where $a \\in \\mathbb{R}$, it is proven that the variance is bounded: specifically, $\\sup \\mathrm{Var}(x)_{|x \\ge a}= \\sup \\mathrm{Var}(x)_{|x \\le a} =(M-a)^2$. For a fixed cutoff, $a$, the variance can be considered a function of only $M$, $a$, and the location parameter $μ$. Examples of such approximating functions, which can be used for model calibration, are developed in addition to other, related calibration methods. For UTSCDs, numerical evidence is presented indicating that for $n \\in \\mathbb{Z+}$ degrees of freedom, or dimensions, and a fixed, finite mean, the variance, $\\mathrm{Var}(R)$, over $R \\in [a,\\infty)$ reaches its maximum value $M^2(π-2)/2$ at $a=0$, $n=1$. For a fixed cutoff value, there is a local maximum in the variance as a function of $n$, and the number of dimensions resulting in the maximal variance, $n_{\\mathrm{vmx}}$, increases with cutoff value. However, for $n \\in \\mathbb{R}$, as the cutoff approaches $0$, $n_{\\mathrm{vmx}}$ approaches $-1$, while $\\mathrm{Var}(R)$ appears to grow without bound.","short_abstract":"This work explores the bounds of the variance of unilaterally truncated Gaussian distributions (UTGDs) and scaled chi distributions (UTSCDs) with fixed means. For any arbitrary Gaussian distribution function, $f(x;μ,σ)$, with a fixed, finite mean $M$ on the truncated domain $x \\ge a$, where $a \\in \\mathbb{R}$, it is pr...","url_abs":"https://arxiv.org/abs/2511.11566","url_pdf":"https://arxiv.org/pdf/2511.11566v1","authors":"[\"Robert J. Petrella\"]","published":"2025-11-14T18:56:57Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}