This article proposes a particular member of the weighted bivariate distribution, namely, bivariate weighted generalized exponential distribution. This distribution is obtained via conditioning, starting from three independent generalized exponential distributions with different shape but equal scale parameters. Several structural properties of the proposed bivariate weighted generalized exponential distribution including total positivity of order two, marginal moments, reliability parameter and estimation of the model parameters are studied. A multivariate extension of the proposed model is discussed with some properties. Small simulation experiments have been performed to see the behavior of the maximum likelihood estimators, and one data analysis has been presented for illustrative purposes.