A Review of Recent Generalized Probability Distribution Families: Advances and Applications

Probability distributions are essential tools for modeling, prediction, and statistical inference. In recent years, several generalized families of distributions have been proposed to extend classical models and increase their flexibility in capturing complex data behaviors. This paper reviews selected generalized families published between 2023 and 2025, focusing on their construction mechanisms, statistical properties, estimation methods, and real-world applications. The families discussed include trigonometric-based, inverse, Lomax-generated, Topp–Leone, and hybrid forms. To illustrate their performance, five families were combined with the exponential distribution and fitted to a real dataset. The comparison shows that all extended models provide an adequate fit, while the standard exponential model performs poorly. The findings confirm the practical value of generalized families in improving data modeling.Let η ≥ 3 be an integer with primitive root π. For a simple connected graph G of order n, a bijective function f : V (G) → {1, 2, …, n} is called a logarithmic cordial labeling to the base π modulo η if the induced function f ∗ π,η : E(G) → {0, 1}, defined by f ∗ π,η(ab) = 0 if indπ,η(f(a) + f(b)) ≡ 0(mod 2) or gcd(f(a) + f(b), η) ̸= 1, and f ∗ π,η(ab) = 1 if indπ,η(f(a) + f(b)) ≡ 1(mod 2), satisfies the condition |ef∗π,η (0) − ef∗π,η (1)| ≤ 1 where ef∗π,η (i) is the number of edges with label i(i = 0, 1). In this paper, we study the logarithmic cordial labeling of various classes of graphs, including path graphs, cycle graphs, star graphs, and complete graphs.