The Research Of The Distributions Of The Linear Combination Of The Dependent Random Variables

The distribution ofαX+βY has been studied by several authors especially when X and Y are independent random variables and come from the same family. However, there is relatively little work of this kind when X and Y are correlated random variables.1 haven't found related results in domestic so far.Based on this consideration, in this paper,we takes bivariate exponential distribution which is widely applied in reliability as an example, derive the exact distributions ofαX+βY, and we provide the plot of the probability density function of Z with the changing of the parameter. In addition, we give the research of the distribution of the specialist linear combination of X and Y - X+Y, on the condition that X and Y obey the standard bivariate normal distribution, at the same time the accompanying application is also provided in the paper. Further more, the distribution of X+Y is derived for the general bivariate normal distribution at last.