Expectation formulas for $q$-probability distributions: a new extension via Andrews-Askey integral

In this paper, we utilize the $q$-Chu-Vandermonde formula to derive a novel expectation formula for the $q$-probability distribution $W(x,y;q)$, extending previously known results. Several applications are presented, including a broader generalization of the Andrews-Askey integral. Although fractional $q$-calculus is not directly employed in this work, its potential for future extensions is discussed, as non-integer order derivatives and integrals could offer deeper insights into $q$-series and probability distributions.