# On Reliability and Ratio in the Beta Case

• Woo, Jung-Soo (Department of Statistics, Yeungnam University)
• Published : 2009.05.31
• 67 17

#### Abstract

We consider distribution, reliability and moment of ratio in two independent beta random variables X and Y, and reliability and $K^{th}$ moment of ratio are represented by a mathematical generalized hypergeometric function. We introduce an approximate maximum likelihood estimate(AML) of reliability and right-tail probability in the beta distribution.

#### Keywords

Beta distribution;generalized hypergeometric function;psi-function;reliability;right tail probability

#### References

1. Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions, Dover Publications, New York
2. Ali, M. M. and Woo, J. (2005). Inferences on reliability P(Y < X) in a power function distribution, Journal of Statistics & Management Systems, 8, 681-686 https://doi.org/10.1080/09720510.2005.10701186
3. Ali, M. M., Woo, J. and Nadarajah, S. (2005). On the ratio X/(X + Y) for the power function distribu-tion, Pakistan Journal of Statistics, 21, 131-138
4. Balakrishnan, N. and Cohen, A. C. (1991). Order Statistics and Inference: Estimation Methods, Academic Press, New York
5. Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products, Academic Press, New York
6. Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distribution-1, Houghton Muffin, Boston
7. Oberhettinger, F. (1974). Tables of Mellin Transforms, Springer-Verlag, New York
8. Rohatgi, V. K. (1976). An Introduction to Probability Theory and Mathematical Statistics, John Wiley & Sons, New York
9. Saunders, S. C. (2007). Reliability, Life Testing, and Prediction of Service Lives, Springer, New York
10. Son, H. and Woo, J. (2007a). The approximate MLE in a skew-symmetric Laplace distribution, Journal of the Korean Data & Information Science Society, 18, 573-584
11. Son, H. and Woo, J. (2007b). Estimating a skewed parameter and reliability in a skew-symmetric double Rayleigh distribution, Journal of the Korean Data & Information Science Society, 18, 1205-1214
12. Woo, J. (2006). Reliability P(Y
13. Woo, J. (2007). Reliability in a half-triangle distribution and a skew-symmetric distribution, Journal of the Korean Data & information Sciences Science Socieiy, 18, 543-552