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.