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?

  • $\begingroup$ 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 $\endgroup$
    – Henry
    Sep 17, 2020 at 14:35

1 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



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