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Wikipedia says that the skewness of the log-normal distribution only depends on the variance of the underlying normal distribution.

Skewness: enter image description here

However, from my point of view the skewness increases as the mean of the underlying normal distribution increases (described here).

Can anyone show which point is valid?

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I believe the ambiguity comes from how you define skewness. Wikipedia uses $$ \mathrm{E}\left(\frac{X - \mu}{\sigma}\right)^3, \text{ with } \mu=\mathrm{E}X \text{ and } \sigma^2=\mathrm{E}(X-\mu)^2. $$ If you compute this for $X\sim \text{log-normal}(\mu,\sigma^2)$, you arrive at the given formula. During the calculation, the dependency on $\mu$ is normalized out when you divide by $\sigma^3$. To check this, write $X = \exp(\mu+\sigma Z)$ for $Z\sim\text{Normal}(0,1)$ and try calculating the moments.

Alternatively, the third moment, $\mathrm{E}X^3$, and the third central moment, $\mathrm{E}(X-\mathrm{E}X)^3$ certainly do depend on $\mu$. Your cited web page doesn't define skewness, but I assume they are referring to the third central (unnormalized) moment.

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