# Adding two distributions with proportional means and standard deviations [duplicate]

I need to know the mean $\mu_z$ and standard deviation $\sigma_z$ of a log normal distribution $Z(\mu_z,\sigma_z)$ which is the sum of two other log normal distributions--$X(\mu_x,\sigma_x)$ and $Y(k\mu_x,k\sigma_x)$--each of which has the same mean and standard deviation up to a factor $k$.

$$Z(\mu_z,\sigma_z)=X(\mu_x,\sigma_x)+Y(k\mu_x,k\sigma_x)$$

$Z(\mu_z,\sigma_z)$ might be, for example, a distribution of net revenues, where $X(\mu_x,\sigma_x)$ and $Y(k\mu_x,k\sigma_x)$ are revenue and cost distributions, respectively.

I would think that $\mu_z$ and $\sigma_z$ are proportional to $\mu_x$ and $\sigma_x$, but not sure how to derive the constant of proportionality.

• What makes you think the sum of two log normal random variables is log normal? You also need to say something about the joint distribution of $X$ and $Y$ to be able to talk about the distribution of the sum. In any case $\mu_z = (k+1) \mu_x$ without any assumption since expectation is linear. – dsaxton Aug 16 '15 at 17:47
• Finding the distribution of a sum of (independent) lognormal random variables is aknown hard problem, search this site for "lognormal" you will find some information but not a real solution. – kjetil b halvorsen Aug 16 '15 at 18:37