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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?

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Answer to first question: No. Hint on second question: $A$ and $B$ are independent and for any random variable $X$ with finite second moment, $\mathbb{V}\mathrm{ar}(X) = \mathbb{E} X^2 - (\mathbb{E} X)^2$. Pseudo-answer to third question: Well, if you're scaling random variable has a variance of zero, then the resulting product is still normally distributed. (But, this last comment is admittedly a little tongue-in-cheek.) Additional comment: Using the notation $A$ and $B$ for random variables is sufficiently nonstandard that it may confuse the casual reader. – cardinal May 10 '11 at 23:15

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