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 e . What is the variance of the lognormal distribution?
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?
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Answers (1)

Aaron Everett
Answered 2022-06-22 Author has 18 answers
We know that
1. Mode = e μ σ 2
2. Median = e μ
3. Variance = e 2 μ + σ 2 ( e σ 2 1 )
Using your data you get
{ e μ σ 2 = e 1 2 e μ = e
That means
{ μ σ 2 = 1 2 μ = 1
{ σ 2 = 1 2 μ = 1
Now you can calculate your variance substituting μ and σ in your variance expression finding
V [ Y ] = e 3 ( e 1 )

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