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?


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...

  • $\begingroup$ For question 1, perhaps try the properties of the exponential functional/logartihm? For question 2, what formula for the confidence interval is given? $\endgroup$ Mar 5 at 18:17
  • $\begingroup$ @Sam Sanders thank you for your comments. I think the accepted answer addressed my/your question about Q.2 :) $\endgroup$
    – yufiP
    Mar 10 at 9:44

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 $\sigma_Z$.

  • $\begingroup$ Thankyou @losif Pinelis. Every point makes sense. Yes I was not correct to say "known" should have said "estimated" so those parameters I mentioned are sample mean, sd etc. $\endgroup$
    – yufiP
    Mar 10 at 9:42

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