Abstract
In this typescript, we present nonlinear implicit random fractional integro-differential equation in mean square sense and its corresponding coupled system. For the existence, uniqueness and at least one solution of the said nonlinear equation, we use Banach contraction and Schauder fixed point theorems, respectively. Uniqueness and at least one solution of corresponding coupled form of the proposed nonlinear system will be prove through Banach contraction theorem and Arzela`-Ascoli theorem, respectively. Under some hypothesis, we scrutinize Hyers-Ulam stability of the mentioned nonlinear equation and its corresponding coupled nonlinear system. For the support of our main results, we present examples.