In the present paper, we introduce two new subclasses of the function class of bi-univalent functions defined in the open unit disc U . We find the bounds on the initial coefficients c2 and c3 and upper bounds for the Fekete-Szego functional for the functions in this class.Motivated by the work of H. M. Srivastava et al. construct a new subclass of biunivalent functions governed by the Pascal distribution series. Then, we investigate the optimal bounds for the Taylor - Maclaurin coefficients c2 and c3 in our new subclass. In Communications and Engineering elds the Pascal distribution has been widely used (see [11]). Recently, in geometric function theory, there has been a growing interest in studying the geometric properties of analytic functions associated with the Pascal
distribution. This distribution is based on the binomial theorem with a negative exponent and it
describes the probability of m success and n failure in (n + m-1) trials, and success on
(n + m)th trials where (1- q) is the probability of success.