Exact Distributions of the Sum of Two Standard Bivariate Normal Dependent Random Variables

The distribution of αX + 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. Based on this consideration, in this paper, we takes bivariate standard normal distribution which is widely applied in our life as an example, derive the exact distributions of X +Y . The expressions for the pdf of the sum of two dependent random variables are given. The plots for the pdf, and statistical application of the distribution have been provided. The work is motivated by real-life examples in quality and reliability engineering.