On the basis of a second-order perturbation theory, some analytical formulas are derived for describing spontaneous and stimulated Raman scatterings of a molecular system in the presence of weak pulse-mode chirped laser field. We are concerned, in particular, with the effect of chirping of the incident laser pulse on the Raman spectroscopic profile and the molecular vibrational distribution. From the derived formulas in the small chirp rate limit, it was found that the molecular vibrational distribution by the spontaneous emission is enhanced in the chirped pulse case relative to the transform-limited pulse and that the difference caused by the sign of the chirp rate essentially does not exist. On the other hand, the overtone molecular vibrational distribution by the stimulated emission was found to change considerably, depending on the sign of the chirp rate.