First of all, am I correct in assuming that given a normally distributed random variable A, and an independent log-normally distributed random variable B, the random variable A·B is normally distributed?
Assuming I am correct in that, I guess the resultant distribution has $\mu=\mu_a \mu_b$... what would the variance be? It doesn't quite seem like it should be $\sigma=\sigma_a \sigma_b$.
And if I'm incorrect in that, is there some other normal-esque distribution I can scale a normal RV by to keep the result normal?