Two-variate Exponential Distribution and Its Numerical Table for Engineering Application

Abstract

This study aims to develop the fundamental theory of a two-variate gamma distribution, especially of a two-variate exponential distribution for engineering application. In outline, the study is as follows: (1) Methods of estimating the parameters included in the probability density fuction of the distribution, the shape parameter in the marginal distribution of which is the same in each, are developed by using the techniques of maximum likelihood and moments. The results show that the estimator for the correlation parameter by the latter is coincident with the ordinary Pearsonian definition of correlation coefficient, but that by the former is not. (2) The characteristics of the two-variate exponential distribution, which is a special type of gamma distribution, especially the characteristics of a correlation surface and locus of the mode of the conditional probability density function are clarified theoretically and numerically in relation to the correlation parameters. (3) For convenience of engineering application of two-variate exponential distribution, numerical values of the conditional probability function are provided in a table. That is, for the fixed values of one variate, the computational values of the other variate are prepared under various conditional probabilities and correlation parameters.