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I have gotten into the habit of notating a lognormally distributed random variable $X$ as:

$$X \sim \ln\mathcal{N}(\mu,\sigma^2)$$

I am now starting to question where I picked this habit up and whether this is non-standard potentially incorrect notation. It is possibly misleading since it implies that the logarithm of normal random variables is being taken rather than the exponential. A safer choice of notation would be:

$$\ln(X) \sim \mathcal{N}(\mu,\sigma^2)$$

However what is the best approach in the case that I want to write:

$$X \sim \,\,?$$

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There is no "best" approach, you can use either of the below:

$$ \begin{align} \ln(X) &\sim \mathcal{N}(\mu,\sigma^2) \\ X &\sim \mathcal{LN}(\mu,\sigma^2) \\ X &\sim \mathrm{Lognormal}(\mu,\sigma^2) \\ \end{align} $$

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