Product of Log Normal Distributions

If i have two independent log normal distributions, where;

Y1~LN(Mu1,Sigma1^2) and Y2~LN(Mu2, Sigma2^2), and Y=Y1*Y2

Am i right in thinking;

Y~LN(Mu1+Mu2, Sigma1^2+Sigma2^2) ?

If this is correct, how do i go about proving this? I have tried to substitute Y=exp(y) for y1 and y2 accordingly, and multiply the pdf's together due to independence, but do not seem to get anywhere near my desired answer.

If this is incorrect, is there an identity i can use?

Thanks

Hey Mcoolta.

The best way to prove these kinds of results is to look at the moment generating functions:

Moment-generating function - Wikipedia, the free encyclopedia

After looking at the following:

Log-normal distribution - Wikipedia, the free encyclopedia

It appears that you need to use the characteristic function, however the same deal applies: if the product of the two functions has the same form then you can use this to show that the addition of two distributions has the same distribution form.