 # Given is a lognormal distribution with median e and mode <msqrt> e </msqrt> . What is t rose2904ks 2022-06-21 Answered
Given is a lognormal distribution with median $e$ and mode $\sqrt{e}$. What is the variance of the lognormal distribution?
Not sure how to solve this. A variable $Y$ has a lognormal distribution if $\mathrm{log}\left(Y\right)$ 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?
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We know that
1. Mode = ${e}^{\mu -{\sigma }^{2}}$
2. Median = ${e}^{\mu }$
3. Variance = ${e}^{2\mu +{\sigma }^{2}}\left({e}^{{\sigma }^{2}}-1\right)$
Using your data you get
$\left\{\begin{array}{l}{e}^{\mu -{\sigma }^{2}}={e}^{\frac{1}{2}}\\ {e}^{\mu }=e\end{array}$
That means
$\left\{\begin{array}{l}\mu -{\sigma }^{2}=\frac{1}{2}\\ \mu =1\end{array}$
$\left\{\begin{array}{l}{\sigma }^{2}=\frac{1}{2}\\ \mu =1\end{array}$
Now you can calculate your variance substituting $\mu$ and $\sigma$ in your variance expression finding
$\mathbb{V}\left[Y\right]=\sqrt{{e}^{3}}\left(\sqrt{e}-1\right)$

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