  Using matrix model, Mironov and Morozov recently gave a formula which
represents Kontsevich-Witten tau-function as a linear expansion of Schur
Q-polynomials. In this paper, we will show directly that the Q-polynomial
expansion in this formula satisfies the Virasoro constraints, and consequently
obtain a proof of this formula without using matrix model. We also give a proof
for Alexandrov's conjecture that Kontsevich-Witten tau-function is a
hypergeometric tau-function of the BKP hierarchy after re-scaling.
