We introduce two uncertainty measures: weighted past varentropy (WPVE) and weighted paired dynamic varentropy (WPDVE). Several properties of these proposed measures are studied. In addition, the non-parametric kernel estimates of the WPVE and WPDVE are proposed. Furthermore, the maximum likelihood estimation technique is employed to estimate WPVE and WPDVE for an exponential population. The performance of the non-parametric and parametric estimators of the WPVE and WPDVE is compared in terms of the absolute bias and mean squared error. Finally, we have reported an application of WPVE related to reliability engineering.