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Suppose I am trying to estimate a future population that I believe to be log-normally distributed from a current value. But, every $n$ periods, I remove a fixed amount in the future.

For example, I may have a fish research pond where I have 1,000 fish. If I believe the population will grow 10%, plus or minus per period, for future years I will forecast the following years population (in each year) by mutiplying the the current year's log-normally-distributed estimate/projection by a log-normal where the $\mu$ of the underlying distribution would have the value $ln(1.10)$ (and some appropriate $\sigma$).

But, when one of those $n$ periods occurs, I would remove a fixed number of fish (say 100) for examination. After doing that, I seem to be left with what some people call a three-parameter log-normal, where the third parameter ('a') would convert every occurrence of $x$ in the pdf to $(x-a$) - in other words, the pdf would be shifted $a$ units to the left.

My question is this. If that is done, will multiplying by the annual growth log-normal multiplier continue to generate log-normal distributions?

I recognize that one issue would need to be some care and possibly re-scaling of parameters. But aside from this, does 'subtracting' a fixed amount from a log-normal distribution destroy the underlying property that the product of log-normals is, itself, log-normal?

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Your thought is correct: if $X,Y$ were independent lognormals, $X(Y-c)$ would not actually be lognormal.

[If instead of removing a constant number you removed a constant fraction $\delta Y$, that would leave you with a lognormal.]

However, you can't actually remove $c$ for every possible value for $Y$ (this won't be an issue for you in practice, because if the population dropped very low the approach would change, but it's potentially relevant to your model for the situation if a change in what you assume at the low end affects the conclusions you draw. It probably won't make a difference, but it's sometimes worth checking to see if a policy that bounds the number removed by a fixed constant $c$ but also keeps it below some small fraction of the population changes results.)

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