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I'm working on a project and I needed to calculate the distribution of the difference between two iid discrete uniform variables (sorry for the long title).
That is, let $I, J$ be two iid discrete uniform variables with support $\{1, \ldots, k\}$. I calculated the distribution of $Z = |I - J|$:
$P(Z = 0) = \frac{1}{k}$, $P(Z = y) = \frac{2(k-y)}{k^2}\ \forall y \in \{1, \ldots, k-1\}$.
I was wondering if such a distribution has a name, so that I can get more info and maybe some results about it.

Thanks a lot.

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    $\begingroup$ this is called a triangular distribution $\endgroup$
    – Carlo Beenakker
    Commented Sep 28, 2020 at 18:27
  • $\begingroup$ Thanks @CarloBeenakker. I see how that distribution is similar, but I don't exactly understand how you would discretize it to yield the distribution I calculated... $\endgroup$
    – Simon
    Commented Sep 28, 2020 at 19:51
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    $\begingroup$ This is not a discrete triangular distribution, since the probability that the difference is 0 is roughly half the probability that it is 1. $\endgroup$
    – user44143
    Commented Sep 28, 2020 at 20:28
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    $\begingroup$ The random variable $I-J$ could be called "discrete triangular"; it is centered at zero. But the poster asks for the absolute value of that, namely $Z=|I-J|$. $\endgroup$
    – Jeppe Stig Nielsen
    Commented Sep 29, 2020 at 9:15
  • $\begingroup$ Perhaps it doesn't have a (very known) name and "absolute value of a discrete triangular random variable" is good enough. Thank you all anyway. :) $\endgroup$
    – Simon
    Commented Sep 30, 2020 at 15:52

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