New Statistical Distribution SMART1

Authors

Saad Naji Al-Azzawi, Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, IraqFollow
Mahmood A. Shamran, Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, IraqFollow
Aseel H. Ali, Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, IraqFollow

Abstract

This paper introduces a new nonclassical statistical distribution, SMART1, derived from a nonpolynomial function. This function is used in ecology to model population growth rates, in medicine to classify the relationship between activity and tumor volume in cancer, and in economics to analyze the relationship between supply and demand. The proposed distribution differs from the Gompertz distribution, which is fundamentally based on the exponential function. By contrast, the SMART1 distribution is constructed on principles of mathematical analysis, specifically through the identification of local maximum endpoints of the Gompertz growth function and the subsequent verification that the resulting function satisfies the criteria of a probability density function. This distribution (SMART1) is flexible and can model various real-world phenomena on a bounded interval (0, β ), especially in reliability analysis or survival modeling, where an upper lifetime limit exists. All statistical concepts are expressed in terms of the distribution's scale parameter β defines the upper bound (or support limit) of the distribution. The random variable X can only take values in the interval (0, β ). In other words, β is a ``lifetime limit'' or ``maximum capacity''; and the shape parameter α determines how the risk or likelihood is distributed over time: Low α : high initial risk that decreases (e.g., early failures). High α : low initial risk that increases with time (e.g., aging or wear-out). These include the probability density function, the cumulative distribution function, the reliability function, the hazard function, order statistics, moments, and key measures such as the mode and the median.

Keywords

Cumulative distribution function (C.d.f), Gompertz growth function, Hazard function H(x), Median, Mode, Moments function M(x), Probability density function (pdf), Reliability function R(x)

Subject Area

Mathematics

Article Type

Article

First Page

1027

Last Page

1039

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite this Article

Al-Azzawi, Saad Naji; Shamran, Mahmood A.; and Ali, Aseel H. (2026) "New Statistical Distribution SMART1," Baghdad Science Journal: Vol. 23: Iss. 3, Article 25.
DOI: https://doi.org/10.21123/2411-7986.5250