Properties of Doubly-Truncated Gamma Variables

Area of Science:

• Statistics
• Probability Theory
• Reliability Engineering

Background:

• The gamma distribution is extensively used in reliability and life-testing.
• Existing research often assumes only an upper bound (0, u) for the variable's support.
• Doubly-truncated distributions are less explored, particularly with known truncation points.

Purpose of the Study:

• To derive and present simplified mathematical forms for a doubly-truncated gamma distribution.
• To extend these findings to the doubly-truncated noncentral chi-square distribution.
• To provide a comprehensive characterization of these distributions.

Main Methods:

• Derivation of density and cumulative distribution function (CDF) for the doubly-truncated gamma distribution.
• Formulation of moment generating function, cumulant generating function, and characteristic function.
• Extension of methods to the doubly-truncated noncentral chi-square distribution.

Main Results:

• Provided simple, explicit forms for the density, CDF, and moments of the doubly-truncated gamma distribution.
• Derived the density, CDF, and moments for the doubly-truncated noncentral chi-square distribution.
• Established a clear mathematical framework for these truncated distributions.

Conclusions:

• The study successfully characterized the doubly-truncated gamma distribution with known bounds.
• The derived mathematical forms offer practical tools for statistical analysis in relevant fields.
• The extension to the noncentral chi-square distribution broadens the applicability of these findings.