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Given a random variable $Y$ from $\operatorname{Poisson}(λ)$ I am trying to understand what the distribution of $Y-λ$ is. I tried writing out the PMF of $Y-λ$ and obtained: ${λ^{k+λ}e^{-λ}}/{(k+λ)!}$. So I am wondering if the obtained PMF is a known distribution?

Michael Hardy
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Dick
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    Why did you try writing out the pdf if this doesn't help you? What are you expecting to help you? Could you explain this in your question. – Sextus Empiricus Jun 17 '22 at 12:29
  • Unfortunately, I am not sure what can help me. I hoped to find a clue from the density function but in vein. – Dick Jun 17 '22 at 12:30
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    Then, what is your question? It's a bit difficult for us to know what your question means if you don't even know it yourself. – Sextus Empiricus Jun 17 '22 at 12:31
  • My problem is to find the distribution of $Y-λ$. – Dick Jun 17 '22 at 12:32
  • Where do you get stuck? What step is your problem in finding it? To me it seems like you already found it with ${λ^{k+λ}e^{-λ}}/{(k+λ)!}$. It is unclear what this question is further about. – Sextus Empiricus Jun 17 '22 at 12:34
  • I am trying to find out if the distribution of the $Y-λ$ is still a Poisson? Or does it follow another known distribution? – Dick Jun 17 '22 at 12:35
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    @Dick if you subtract the mean, the values can be negative, while Poisson is a distribution with non-negative support, so no. – Tim Jun 17 '22 at 12:39
  • @tim does the distribution have a special name? – Erik Hambardzumyan Jun 17 '22 at 12:40
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    No, because it is derived in such a simple, transparent manner from a Poisson distribution. A good description would be "The distribution of $Y-\lambda$ is Poisson of parameter $\lambda$ shifted left by $\lambda.$" – whuber Jun 17 '22 at 13:05
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    Note that if $\lambda$ is not an integer then the support looks a bit odd: e.g. if $\lambda=1.4$ then the support is $-1.4, -0.4,0.6,1.6,2.6,\ldots$. The only helpful property is that the expectation is $0$, while a curious feature is that the variance is the negative of the minimum possible value. – Henry Jun 17 '22 at 13:14
  • "I am trying to find out if the distribution of the −λ is still a Poisson? Or does it follow another known distribution?" I do not get this specific question from your original post. Could you edit/update your question if this is what you really want to ask. – Sextus Empiricus Jun 17 '22 at 13:50

1 Answers1

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I am wondering if the obtained PMF is a known distribution?

Any distribution can been generalized by converting the distribution to a family with a shift or location parameter (and also a scaling is often added).

For several well known distributions these converted distributions with a location (and shift) parameter have some uses, such that they get a specific name. An example is the generalized student's t-distribution (not to be confused with the non-central t-distribution).

When a distribution family that has no location parameter is converted to a family with location parameter (or other additional parameters) then often we speak about the distribution with the addition of the word 'generalized'. So in your case we could speak about a generalized Poisson distribution ${λ^{k+λ-\mu}e^{-λ}}/{(k+λ-\mu)!}$ with $\mu = 0$ (with zero mean).

It is difficult to find an example of a previous use of this. When we look for it then we also come across a problem with these names using 'general' and that is that they are very general. There are more ways how distributions can be generalized. In the case of 'generalized Poisson distribution' we see a lot of examples that use a generalization that relates to overdispersion.

Sidenote 1: I am not sure why you would like to define $Y-\lambda$ as a seperate distribution. But when I think of it then the Skellam distribution pops up in my head. You might want to read about that.

Sidenote 2: Asside from a shift, another common transformation occuring with variables is truncation. For that case of transformation, the resulting distribution often has no special name, and is simply called the 'truncated ... distribution'. For instance the truncated normal distribution. It is for these types of simple transformations, truncation, scaling, shifting, that often no special new name exists for the distribution of the resulting new variable.

Sextus Empiricus
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