iosr-Jm   Volume-19 ~ Issue-6 ~ Nov. – Dec. 2023

Abstract : This article addresses two estimation methods for the unknown parameters of beta Kumar-aswamy-exponential (BKw-E) distribution under complete samples. Also, we discussed the estimation of the reliability and hazard functions. A comparison between maximum likelihood method and Bayes method for two unknown parameters of BKw-E distribution is provided. Further, the Bayes estimators are studied under three types of loss functions; squared error, linear-exponential and general entropy, using importance sampling technique. Finally, a simulation study is presented to study the performance of the estimated parameters.

Keywords: Beta Kumaraswamy exponential distribution; maximum likelihood estimator; Bayes estimators. 2010 Mathematics Subject Classification: 62E10, 62Q05

[1]. Abo-Kasem, O. E., El Saeed, A. R., And El Sayed, A. I. (2023). Optimal Sampling And Statistical Inferences For Kumaraswamy Distribution Under Progressive Type-Ii Censoring Schemes. Scientific Reports, 13(1):12063.
[2]. Adepoju, K. And Chukwu, O. (2015). Maximum Likelihood Estimation Of The Kumaraswamy Expo- Nential Distribution With Applications. Journal Of Modern Applied Statistical Methods, 14(1):208– 214.
[3]. Al-Saiary, Z. A., Bakoban, R. A., And Al-Zahrani, A. A. (2020). Characterizations Of The Beta Kumaraswamy Exponential Distribution. Mathematics, 8(1):23.
[4]. Alduais, F. S., Yassen, M. F., Almazah, M. M., And Khan, Z. (2022). Estimation Of The Ku- Maraswamy Distribution Parameters Using The E-Bayesian Method. Alexandria Engineering Jour- Nal, 61(12):11099–11110.
[5]. Awodutire, P., Nduka, E., And Ijomah, M. (2020). The Beta Type I Generalized Half Logistic Distribution: Properties And Application. Asian Journal Of Probability And Statistics, 6(2):27–41.

Abstract : The article gives a new proof of Fermat's last theorem. The proof is based on the study of the properties of natural numbers, an analysis of the constraints on the proposed solutions, and uses some general theorems on the roots of algebraic equations. The connection between Fermat's theorem and Beal conjecture is discussed. Beal conjecture is proved by induction using the same reasoning as in the proof of Fermat's theorem.

Keywords: number theory, natural numbers, Fermat's theorem, Beal conjecture

[1]. Wiles A. Modular Elliptic Curves And Fermat's Last Theorem // Annals Of Mathematics, 1995, Vol. 142, Pp. 443-551.
[2]. Waerdan Van Der B.L. Algebra. Moscow, Publishing House "Nauka", 1976 (In Russian)..