Variance of $Y = a \cdot e^X$ if $X$ is Gussian random variable

Question

I have the following function

$$Y = a e^X$$

where $X$ is a normally distributed random variable. I would like to compute the variance of $Y$. So far I did the following

\begin{eqnarray} \text{Var}(Y) &= \text{Var}(ae^X)\\ &= a^2 \text{Var}(e^X)\\ &\approx a^2 e^{2E(X)} \text{Var}(X) \end{eqnarray}

Where I used information I have found on this site on variance propagation. I would like to know if this calculation is correct and if there are better and more correct ways to compute $\text{Var}(Y)$? So far, this is only an approximation, if what I did is right.

Answer 1

If $X\sim N(\mu;\sigma^2)$ then $Y=e^X\sim\text{LogNormal}$ thus you can calculate $\mathbb{V}[Y]$ exactly

Lognormal distribution

$$\mathbb{V}[Y]=a^2\cdot e^{2\mu+\sigma^2}(e^{\sigma^2}-1)$$