General form of the alternative expectation formula when the variables are defined on the whole real axis

Hong (2015) showed that the expectation of the product of two nonnegative random variables can be expressed as a double integral involving their joint survival function. In this paper, I extend this result to random variables defined on the entire real line. The obtained identity decomposes the expectation into four terms corresponding to the four quadrants of the Cartesian plane and combines probabilities associated with both joint distribution and joint survival functions in a symmetric way. Three different proofs are presented, all based on a decomposition of the integration domain into four quadrants. The first is based on indicator functions. The second follows the first approach introduced by Hong (2015) for nonnegative random variables and extends it to the whole real axis. The third, extending the second approach of Hong (2015), relies on integration by parts and provides an alternative derivation of the same result. The proposed identity generalizes the alternative expectation formula for a large class real-valued random variables. Extensions to multivariate settings and higher-order moments are discussed, and a numerical verification based on the bivariate normal distribution is reported. (abstract generated with the support of AI Copilot)