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Is it possible to define a geometric distributed variable with parameter p on an interval $[a,b]$, such that

$$ c \cdot \sum_{j=a}^b (1-p)^j p=1$$ with a scale factor $c$. Is that reasonable?

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It is possible to define it, although the name "geometric distribution" may be confusing, since it commonly refers to a distribution on a different domain. "Geometric distribution on a bounded domain" seems correct – maybe it's used somewhere. You might also call it a "geometrically varying" distribution if you like, since subsequent probabilities are in a fixed ratio, just like for the ordinary geometric distribution.

Such a distribution can in fact be rewritten in the form $\mathrm{p}(i \vert \lambda) = \frac{1}{Z(\lambda)} \exp(\lambda\ i)$ with $$Z(\lambda):=\frac{\exp[(b+1/2)\ \lambda]-\exp[(a-1/2)\ \lambda]}{\exp(\lambda/2)-\exp(-\lambda/2)}\ ,$$ obtainable using the expression of the partial sum of a geometric series. Note that the parameter $\lambda \equiv \ln(1-p)$ is determined by the mean of the distribution, and vice versa. I don't think there's an analytic expression giving $\lambda$ as a function of the mean, but I'm not sure to be honest.

Also, the mean (ie, first moment) is a sufficient statistics because of the special exponential form (see ref. below). So this is also a maximum-entropy distribution with the mean (ie, first moment) as constraint.

I think that the parameter $p$ in your expression ceases to have the specific meaning it has for the ordinary geometric distribution.

Whether this distribution is reasonable or not depends completely on the specific problem you're applying it to – for example, can you consider the mean to be a sufficient statistics, in your problem?

See eg Bernardo, Smith: Bayesian Theory (Wiley 2000), § 4.5.

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    $\begingroup$ "geometric distribution" is an established name for a different distribution, so you may cause confusion. This can simply be called a maximum-entropy distribution, or a geometrically varying distribution if you like, since the subsequent probabilities are in a fixed ratio, just like for the ordinary geometric distribution. $\endgroup$
    – pglpm
    Nov 30, 2020 at 22:34
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    $\begingroup$ I don't consider it as "scaled", because the change of scale is enforced by the normalization, it isn't a free choice. But this is my subjective opinion. I suppose people will have different opinions on whether it's a reasonable name or not. $\endgroup$
    – pglpm
    Nov 30, 2020 at 22:41
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    $\begingroup$ ...I mean, "geometric distribution on a bounded domain" for me is a fine name. But consider that there are always hurried readers that may skip over the "on a bounded domain" part and thus misunderstand you. You're the best judge for your specific circumstances. $\endgroup$
    – pglpm
    Nov 30, 2020 at 22:45
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    $\begingroup$ Yeah you are right. Thank you very much for your explanation:) $\endgroup$
    – Alif
    Nov 30, 2020 at 22:48
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    $\begingroup$ I think if I describe my probem in the right way, I can somehow refer to the geometrix distribution since it has this specific form. $\endgroup$
    – Alif
    Nov 30, 2020 at 22:51

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