Résumé: This study investigates the goodness-of-fit test, fuzzy reliability analysis, and Bayesian estimation for a novel one-parameter probability distribution. Specifically, we introduce and analyze the exponential-Lindley and exponential-X-Lindley distributions as extensions of the proposed model. Using comprehensive analytical techniques, several key statistical properties of the distribution are derived and thoroughly examined. To assess the model's behavior under uncertainty, fuzzy reliability measures are developed, demonstrating its robustness and practical applicability in scenarios involving imprecise or vague data. Furthermore, a variety of parameter estimation methods—including classical and Bayesian approaches—are explored to assess the flexibility and precision of the proposed model. A simulation study is conducted using randomly generated datasets to evaluate the performance of the estimation techniques and to gain deeper insights into the model’s adaptability across different conditions. Finally, the model’s adequacy is validated using a goodness-of-fit test, confirming its potential usefulness in reliability and lifetime data analysis.

Résumé: The power XLindley (PXL) distribution is introduced in this study. It is a two-parameter distribution that extends the XLindley distribution established in this paper. Numerous statistical characteristics of the suggested model were determined analytically. The proposed model’s fuzzy dependability was statistically assessed. Numerous estimation techniques have been devised for the purpose of estimating the proposed model parameters. The behaviour of these factors was examined using randomly generated data and developed estimation approaches. The suggested model seems to be superior to its base model and other well-known and related models when applied to the COVID-19 data set.

Résumé: The main purpose of this paper is to introduce and investigate stochastic orders of scalar products of random vectors. We study the problem of finding maximal expected utility for some functional on insurance portfolios involving some additional (independent) randomization. Furthermore, applications in policy limits and deductible are obtained, we consider the scalar product of two random vectors which separates the severity effect and the frequency effect in the study of the optimal allocation of policy limits and deductibles. In that respect, we obtain the ordering of the optimal allocation of policy limits and deductibles when the dependence structure of the losses is unknown. Our application is a further study of [1 − 6].

Résumé: The paper deals with several types of stochastic order affecting random variables and linear combinations of random variables. We study the problem of finding maximal expected utility for some functionals on insurance portfolios involving some additional (independent) randomization. Applications in policy limits and deductible are obtained, and some relationships with other actuarial main topics (comparison of copulas, individual and collective risk models, reinsurance contracts, etc.) are studied too

Résumé: This paper focus on stochastic order and comparing risks. To this end, we deal some relationship of different stochastic order. Some examples and applications in actuarial science are given.

Résumé: This paper focus on several types of stochastic order affecting random variables and linear combinations of random variables which we obtain the ordering of the optimal allocation of policy limits. In addition, many applications and some examples of convex order are given: Individual and collective risk model, Reinsurance contracts, dependent portfolios increase risk.