By using Poisson distribution series we introduce and derive different subclasses of regular and bi-univalent functions in the open unit disk$. $We then present special estimates for the Taylor coefficient inequalities $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ of functions belonging to this new subclass. In addition, several consequences of our results are pointed out which are new and have not yet been discussed in association with bounded boundary rotation.