I have two lognormally distributed random variables $Y\_i=e^{X\_i}$ where $X\_i \sim \mathcal{N}\big(\mu\_i, \: \sigma\_i^2 \big)$ for $i=1,2$, and $X\_1$ and $X\_2$ are correlated by $\rho\_{12}$. Now, Let $Z=\alpha Y\_1 - \beta Y\_2$.


***Question 1***


Is $Z$ lognormally distributed?


***Question2***


When $\mu\_i$, $\sigma\_i$, and $\rho\_{12}$, as well as the mean and variance of $Z$: $\mu\_Z$, $\sigma\_Z^2$ are known, how can I construct a confidence interval for $\mu\_Z$ using these known parameters?


I also know how to calculate the mu and sigma for $Y\_i$ using these relationship.


$\mu\_{Y\_i} =\mathbb{E}\big[Y\_i\big] = e^{\mu\_i+\frac{1}{2}\sigma\_i^2}\\
 \sigma\_{Y\_i}^2 =\mathbb{V} \big[Y\_i\big] = \Big[e^{\sigma\_i^2}-1 \Big]\: e^{(2\mu\_i+\sigma\_i^2)}.$


I was looking at [Wikipedia](https://en.wikipedia.org/wiki/Log-normal_distribution#Statistics) and [this reference](http://jse.amstat.org/v13n1/olsson.html#:%7E:text=For%20example%2C%20in%20a%20sample,of%20the%20median%20of%20X.) but struggling to understand how to adopt it for $Z$. I am not from mathematics background. I would appreciate if you can point me out to the right directions...



Confidence interval for the difference of lognormally distributed random variables
Question 1: $Z$ will not be lognormally distributed in general. E.g., if $\beta>0$ or $\alpha<0$, then $P(Z<0)>0$, and hence $Z$ is not lognormally distributed.


Question 2: If $\mu\_Z$ is known, then it does not make sense to use a confidence interval for $\mu\_Z$.


If $\mu\_Z$ and $\sigma\_Z$ are both unknown, then, in view of the central limit theorem, an approximate $(1-p)$-confidence interval for $\mu\_Z$ based on a large iid sample $Z\_1,\dots,Z\_n$ from the distribution of $Z$ is
$$[\bar Z-z\_{p/2}S\_Z/\sqrt n,\bar Z+z\_{p/2}S\_Z/\sqrt n], \tag{1}$$
where
$$\bar Z:=\frac1n\,\sum\_1^n Z\_i,\quad S\_Z:=\sqrt{\frac1{n-1}\,\sum\_1^n(Z\_i-\bar Z)^2},$$
$z\_{p/2}:=\Phi^{-1}(1-p/2)$, and $\Phi$ is the standard normal cdf. Here we have also used the law of large numbers, implying that $S\_Z^2\to\sigma\_Z^2$ in probability.


If $\mu\_Z$ is unknown but $\sigma\_Z$ is known, then one can replace $S\_Z$ in (1) by $\sigma\_Z$.


