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I've done some research and gathered data for plotting on a histogram. I believe it is a bell-shaped curve, but given what I know am not sure. I know it is centered around 0, with half of it negative and the other half is positive. The distribution has this ratio:

for b > 0, a < 0, $$ 1/((b/|a|)+1) = f(b) / (f(b) + f(a)) $$ I hope the formula is clear. To say it in words: if b = a, then the ratio is 50%; if b is twice as big as a, then it's 33%; if b is half as big as a, then it's 66%. And the count of values at a and b, given as f(a) and f(b), follow these ratios as well; it's like what percent of the total are at b.

I am fairly sure the max value at 0 is near 1 (ex, if b is almost 0, and a is -infinity), and that the tails flatten out to almost 0 as a goes to -infinity and b goes to infinity.

Given this info, how do I find the function f? I suspect it is a bell curve, but I am not sure around 0 if its smooth, or if it is a point.

Thank you for your help, I hope my question is clear, feel free to ask if not because I study math as a hobby not in a formal class, I might be using incorrect terms or descriptions.

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up vote 1 down vote accepted

You have a problem at $0$. Consider that if $b=0$ you have $1/((0/|a|)+1)=1$ and if $f(0)$ is non-zero then $\frac{f(0) }{f(0) + f(a)}=1$ implies $f(a)=0$, while if $f(0)=0$ then $\frac{f(0) }{f(0) + f(a)}=0$ not $1$.

One set of curves which satisfy your ratio formula is $f(x)=k/|x|$ for any nonzero $k$ since $$\frac{f(b) }{f(b) + f(a)} = \dfrac{\frac{k}{|b|}}{\frac{k}{|b|}+\frac{k}{|a|}}=\dfrac{1}{1+\frac{|b|}{|a|}}.$$

This is not a probability distribution for any $k$ as it cannot have a finite integral over the real line, though for parts of the distribution it may be a reasonable approximation. In particular it is infinite or undefined at $0$.

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Thank you for the answer, this helps me learn a lot. I had a feeling there was something suspicious about it but did not know how to prove it; I'll keep in mind the "implies" logic. I suppose k would be the count for a score. If it's not a probability distribution, should I change the tag? –  d l Feb 3 '12 at 21:44
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