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 and this reference 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...