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What is Poisson density multiplied by integer-Guassian?

$$p(X=x)\propto\frac{\theta^x}{x!}\exp(-x^2),\quad x\in\mathbb{N}$$

Is it a distribution being studied?

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    $\begingroup$ It's also unclear what you seek when you say "what is...". Assuming you can resolve Artem's question, what is it you want to know about the resulting pmf? Are you seeking a name for it, or some property of it? $\endgroup$
    – Glen_b
    Apr 1 '19 at 22:19
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    $\begingroup$ I don't see how "Poisson multiplied by Gaussian" describes this pdf. It's identical to the Poisson pdf except that it uses exp(-x^2) instead of exp(-x). I've never heard of any use for that distribution. $\endgroup$ Apr 2 '19 at 3:52
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    $\begingroup$ @HeleneHoegsbroThygesen Poisson is $\exp\left(-\theta\right)$ $\endgroup$
    – ZHU
    Apr 2 '19 at 4:27
  • $\begingroup$ oh yes of course, thanks! $\endgroup$ Apr 2 '19 at 21:53

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