How to transform one Poisson distributed random variable to another with a different mean? Since the simple affine transformation does not preserve Poisson distribution, I'm wondering if there is any trick to apply a (deterministic) transformation to a Poisson random variable with mean $\lambda_1$ such that it remains Poisson but with mean $\lambda_2$?
One idea I had is to do the Anscombe transformation to get an approximate normally distributed random variable, and then apply a linear transformation to get the desired mean, followed by the inverse Anscombe. Of course, this is only approximate and I'm not sure if it's even valid.
 A: If you want it to be reversible and the result to have a Poisson distribution, it's not really possible.
Poisson distributions assign probability to the non-negative integers.

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*What monotonic transformations take non-negative integers to non-negative integers?
Unless you leave gaps in the middle of the sequence, I think all you have is shifts to the right, either of which doesn't leave you with another Poisson.
So that rules out readily interpretable (monotonic) transformations. Assuming you want a 1-to-1 transformation (otherwise you cannot transform back), you're at best going to be left with "shuffling" the values in some way.
But now note that the Poisson is unimodal (with an "edge case" of two adjacent modes when the parameter is an integer); the probabilities decrease either side of that. That severely limits your options (without even getting to the specific functional form)


*Now consider a specific Poisson -- say a Poisson(1). It has a specific set of probabilities. Any deterministic 1-to-1 transformation will simply move those probabilities somewhere else. In general no other Poisson can share more than a fraction of those probabilities; where do the others go?  e.g. with a Poisson(1) you have two probabilities of 1/e -- you might perhaps be able to find another Poisson that has a probability of 1/e, but can you find one with two of them, both the largest possible probabilities? It turns out you can't.


*If you don't require a 1-to-1 transformation, so you can steal any number of probabilities from the far tail as needed, you might in some cases be able to have a transformation from a large parameter to a good approximation of a small one, but it might be hard to give a finite-time/space construction of one; I think this would not be a practical exercise in general.
A: Not in general.
It's not possible to do it exactly if $\lambda_2>\lambda_1$, since a Poisson variable with mean $\lambda_2$ has higher entropy than one with mean $\lambda_1$, so it takes more information to specify it, even if you are willing to have a crazy non-monotone transformation.
For $\lambda_2<\lambda_1$, it is at least not always possible.  Suppose $\lambda$ is small, so that the variable basically has only values 0 and 1, and the probability of 0 is $\exp(-\lambda)$. You can't transform between two distributions like this.
I can't see any easy way to rule out that it's possible in some cases with $1 \ll \lambda_2 \ll \lambda_1$.
