Product of two normal distributions

Am I right in thinking that if there are two random wariables with normal distributions X~N(mu_{x}, sigma^{2}_{x}) and Y~N(mu_{y},C), then the product of these two random variables, say Z=XY has the distribution Z~N(mu_{z}, sigma^{2}_{z})

where mu_{z} = (mu_{x}*sigma^{2}_{y} + mu_{y}*sigma^{2}_{x})/(sigma^{2}_{x}+sigma^{2}_{y) }and sigma^{2}_{z} = (sigma^{2}_{x}*sigma^{2}_{y})/(sigma^{2}_{x}+sigma^{2}_{y})

Re: Product of two normal distributions

Hey Mullineux.

I thought the answer would be no but it turns out that the answer is that the product of two normals is again normal and this is done by using properties of logarithms of random variables where the sum of two log normals is lognormal and then exponentiating this gives back a normal.

Obtained from the discussion here:

Product distribution - Wikipedia, the free encyclopedia