variance of lognormal distribution when given median and mode

Question

Given is a lognormal distribution with median $e$ and mode $\sqrt{e}$. What is the variance of the lognormal distribution?

I'm not sure how to solve this. A variable Y has a lognormal distribution if $\log(Y)$ has a normal distribution. So I'm thinking you can solve the question by finding the mean and standard deviation of the associated normal distribution by using the given median and mode. But I don't know how to. For a normal distribution, the median and mode equal the mean, but for a lognormal distribution they evidently do not. How to use these values to find the variance?

Answer 1

you know that

1. mode=$e^{\mu-\sigma^2}$

2. Median=$e^{\mu}$

3. Variance=$e^{2\mu+\sigma^2}(e^{\sigma^2}-1)$

using your data you get

$$\begin{cases} e^{\mu-\sigma^2}=e^{\frac{1}{2}} \\ e^{\mu}=e \end{cases}$$

That means

$$\begin{cases} \mu-\sigma^2=\frac{1}{2} \\ \mu=1 \end{cases}$$

$$\begin{cases} \sigma^2=\frac{1}{2} \\ \mu=1 \end{cases}$$

now you can calculate your variance substituting $\mu$ and $\sigma$ in your variance expression finding

$$\mathbb{V}[Y]=\sqrt{e^3}(\sqrt{e}-1)$$