# variance of lognormal distribution when given median and mode

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?

• Wikipedia gives expressions for the median, mode and variance of a log-normal distribution in terms of the mean and variance of the underlying normal distribution. In effect you need to solve simultaneous equations Sep 17, 2020 at 14:35

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

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