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Let X be a discrete random variable (positive integers in my specific case). I am considering a function $f$ with $Y=f(X)$ such as:

$P(Y=n) = \sum_{m \geq n}P(X=m)/ \Bbb{E}[X]$

I am wondering if $f$ has a name in the literature (as the expression is simple, I would be very surprised to be the first to stumble on this), and if there are more "canonical" ways to write the same function, maybe through convolutions or something along similar lines. Also, I am seeking insights on the mathematical properties of this function. Thanks!

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  • $\begingroup$ Lets say $X$ is measured in seconds, the units of $Y$ are $\frac{1}{sec}$, why would that make sense ? $\endgroup$ – Uri Goren Sep 26 '15 at 12:24
  • $\begingroup$ It does make sense, there is an interesting interpretation in supply chain. When I get the time, I will update the post with a link toward the interpretation. $\endgroup$ – Joannes Vermorel Sep 27 '15 at 8:24
  • $\begingroup$ Have you checked whether this sums to 1? $\endgroup$ – tchakravarty Sep 27 '15 at 14:51
  • $\begingroup$ To Fg Nu, in the case where X is positive and discrete, it seemed to be that Y was correctly summing to 1. $\endgroup$ – Joannes Vermorel Sep 27 '15 at 17:01
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The PMF written for $Y$ is not (in general) normalized. If I assume that instead of "=" it should be "$\propto$", the division by $\mathbb{E}[x]$ becomes irrelevant (it's a constant). So then we're left with:

$$ P(Y=n) \propto \sum_{m \geq n} P(X=m) = 1 - G(n-1) $$ where $G$ is the CDF of $X$.

So the density of $Y$ is proportional to the (shifted) tail probability of $X$, and looks a lot like a normalized survival function.

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I really like this problem and will continue to think about it. I'll share what I've been thinking. If $X$ has a mean, define the random variable $W$ by the PDF

$$ P(W=w) = \frac{w\cdot P(X=w)}{\mathbb{E}[X]}. $$

Then consider the following process:

$$ W \sim P(W) \\ Y \sim \mathcal{U}(\{1\dots W \}) $$

Then $$ P(Y=y) = \sum_{w \geq y} \frac{P(W=w)}{w} = \sum_{w \geq y} \frac{P(X=w)}{\mathbb{E}[X]}. $$

So if I were to try and extend this idea to thinking about a density on positive reals, as opposed to a PMF on positive integers, I would imagine sampling uniformly from the area under the graph of $X \cdot P(X)$ and taking the $y$ coordinate. Does that last bit make any sense?

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