Expected Mordell-Weil rank heuristics through Sato-Tate, Birch and Swinnerton-Dyer conjectures

We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjecture. We do calculations in CM and non-CM cases and raise questions about their relations, if any, with the predictions of various average rank models that have been considered. We also point out incompatibility of a conjecture in the literature with the conjecture of Birch and Swinnerton-Dyer.