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I'm looking for an exponential function which returns values in $[0,1]$ while $X$ can assume any positive integer value.

Specifically, in my case the $X$ represents the number of failed operations over to total number $Y$ of operation, with $X \leq Y$. Consequently, I would like to associate a weight to $X$ which defines the slope of the curve, so that the larger $X$ becomes, the faster the curve goes to $0$ (or $1$).

Is there any specific function that can meet my requirements?

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    $1-e^{-x}$ or such? – 2017-02-13
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    Yes I was already thinking to that function. However, I need to associate a weight to x, so that the curve goes to 0 ( or 1) more rapidly when x increase. It is like to say, if I'm sending at time t, y=100 requests to a system and x=90 requests fail, then the curve must be close to 1 (or 0). Viceversa, if x=5 the curve must be close to 0 ( or 1). – 2017-02-13

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