# 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.

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 $$1 \ll \lambda_2 \ll \lambda_1$$.