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Advances in Differential Equations and Control Processes

Advances in Differential Equations and Control Processes
Volume 29, , Pages 47 - 63 (November 2022)
http://dx.doi.org/10.17654/0974324322033

CLOSED FORM EXPRESSIONS OBTAINED FROM THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS OF THE PROBABILITY DENSITY FUNCTION OF THE BETA DISTRIBUTION

Olasunmbo O. Agboola, Hilary I. Okagbue, Adedayo F. Adedotun and Paulinus O. Ugwoke

Abstract:

In this paper, some closed form expressions for selected parameters for the probability density function (PDF) of the beta distribution are obtained. The closed form expressions are recovered from the solution of the ordinary differential equations (ODEs), obtained from the differentiation of the PDF of the distribution. The paper shows that the shape of the distributions also determines the nature of the resulting ODE which has shown how distributions related to the beta distribution can be traced via the solutions of the ODEs. Numerical methods are unnecessary because the closed form expressions are the same with the values obtained from the standard statistical software.

Keywords and phrases:

Beta distribution, closed form expression, ordinary differential equation, probability density function, shape parameter statistics.

Received: August 12, 2022; Revised: August 30, 2022; Accepted: September 5, 2022; Published: October 7, 2022

How to cite this article: Olasunmbo O. Agboola, Hilary I. Okagbue, Adedayo F. Adedotun and Paulinus O. Ugwoke, Closed form expressions obtained from the solution of ordinary differential equations of the probability density function of the Beta distribution, Advances in Differential Equations and Control Processes 29 (2022), 47-63. http://dx.doi.org/10.17654/0974324322033

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] H. I. Okagbue, M. O. Adamu, A. A. Opanuga and P. E. Oguntunde, Boundary properties of bounded interval support probability distributions, Far East Journal of Mathematical Sciences (FJMS) 99(9) (2016), 1309-1323.
[2] H. I. Okagbue, T. A. Anake, P. E. Oguntunde and A. A. Opanuga, Near exact quantile estimates of the beta distribution, Advances and Applications in Statistics 70(1) (2021), 109-128.
[3] H. J. Mohammed and M. A. Mohammed, Solving the created equations from power function distribution, Iraqi Journal of Science (2022), 3073-3087.
[4] H. I. Okagbue, M. O. Adamu, E. A. Owoloko and E. A. Suleiman, Classes of ordinary differential equations obtained for the probability functions of Harris extended exponential distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 883-888.
[5] H. I. Okagbue, O. A. Odetunmibi, A. A. Opanuga and P. E. Oguntunde, Classes of ordinary differential equations obtained for the probability functions of half-normal distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 876-882.
[6] H. I. Okagbue, O. O. Agboola, A. A. Opanuga and J. G. Oghonyon, Classes of ordinary differential equations obtained for the probability functions of Gumbel distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 871-875.
[7] H. I. Okagbue, O. O. Agboola, P. O. Ugwoke and A. A. Opanuga, Classes of ordinary differential equations obtained for the probability functions of exponentiated Pareto distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 865-870.
[8] H. I. Okagbue, P. E. Oguntunde, A. A. Opanuga and E. A. Owoloko, Classes of ordinary differential equations obtained for the probability functions of exponential and truncated exponential distributions, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 858 864.
[9] H. I. Okagbue, M. O. Adamu, E. A. Owoloko and S. A. Bishop, Classes of ordinary differential equations obtained for the probability functions of half-Cauchy and power Cauchy distributions, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 552 558.
[10] H. I. Okagbue, A. A. Opanuga, E. A. Owoloko and M. O. Adamu, Classes of ordinary differential equations obtained for the probability functions of exponentiated Fréchet distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 546-551.
[11] H. I. Okagbue, M. O. Adamu, A. A. Opanuga and J. G. Oghonyon, Classes of ordinary differential equations obtained for the probability functions of 3 parameter Weibull distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 539-545.
[12] H. I. Okagbue, M. O. Adamu, E. A. Owoloko and A. A. Opanuga, Classes of ordinary differential equations obtained for the probability functions of Gompertz and gamma Gompertz distributions, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 405-411.
[13] H. I. Okagbue, S. A. Bishop, A. A. Opanuga and M. O. Adamu, Classes of ordinary differential equations obtained for the probability functions of Burr XII and Pareto distributions, Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 399-404.
[14] H. I. Okagbue, A. A. Opanuga, E. A. Owoloko and M. O. Adamu, Classes of ordinary differential equations obtained for the probability functions of Cauchy, standard Cauchy and log-Cauchy distributions, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 198 204.
[15] H. I. Okagbue, P. E. Oguntunde, P. O. Ugwoke and A. A. Opanuga, Classes of ordinary differential equations obtained for the probability functions of exponentiated generalized exponential distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 192 197.
[16] H. I. Okagbue, P. E. Oguntunde, A. A. Opanuga and E. A. Owoloko, Classes of ordinary differential equations obtained for the probability functions of Fréchet distribution, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2017, 25-27 October, 2017, San Francisco, U.S.A., pp. 186-191.
[17] R. Torishnyi and V. Sobol, Smooth approximation of probability and quantile functions: vector generalization and its applications, Journal of Physics: Conference Series 1925 (2021) 012034.
[18] H. Okagbue, M. O. Adamu and T. A. Anake, Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication, Heliyon 6(11) (2020) e05523.
[19] H. I. Okagbue, M. O. Adamu and T. A. Anake, Closed form expression of the quantile function of Maxwell-Boltzmann distribution, Adv. Appl. Stat. 54(2) (2019), 179-197.
[20] H. I. Okagbue, M. O. Adamu and T. A. Anake, Closed form expressions for the quantile function of the Erlang distribution used in engineering models, Wireless Personal Communications 104(4) (2019), 1393-1408.
[21] H. Okagbue, M. O. Adamu and T. A. Anake, Closed form expression for the inverse cumulative distribution function of Nakagami distribution, Wireless Networks 26(7) (2020), 5063-5084.
[22] H. I. Okagbue, M. O. Adamu and T. A. Anake, Closed-form expressions for the quantile function of the Chi square distribution using the hybrid of quantile mechanics and spline interpolation, Wireless Personal Communications 115(3) (2020), 2093-2112.
[23] H. I. Okagbue, M. O. Adamu and T. A. Anake, Ordinary differential equations of probability functions of convoluted distributions, International Journal of Advanced and Applied Sciences 5(10) (2018), 46-52.
[24] G. M. Cordeiro, M. Alizadeh, T. G. Ramires and E. M. Ortega, The generalized odd half-Cauchy family of distributions: properties and applications, Communications in Statistics-Theory and Methods 46(11) (2017), 5685-5705.
[25] A. Saboor, M. N. Khan, G. M. Cordeiro, M. A. Pascoa, P. L. Ramos and M. Kamal, Some new results for the transmuted generalized gamma distribution, Journal of Computational and Applied Mathematics 352 (2019), 165-180.