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})