This work presents a control design method for linear sampled-data systems whose random sampling intervals form a Poisson process. Unlike a previous result in the literature, the proposed stabilization conditions, based on linear feedbacks of both the state and the past input values, are necessary and sufficient for the mean exponential stability of the system. Moreover, such non-conservative conditions correspond to linear matrix inequalities, implying then that the stabilization problem can be efficiently addressed through semidefinite programming. As a second contribution, the characterization and optimization of the mean exponential convergence rate of the closed-loop system is given in form of a generalized eigenvalue problem. A numerical example illustrates the theoretical results.