Please don't get too cross if this is not phrased well.
I am doing some work on the following problem.... Take $n$ identically shaped distributions with different means $m_1,m_2...m_n$ Ensure that the if $n_{low}$ has the lowest value of $m$ and $n_{high}$ has the highest value of $m$ that there is an overlap between these two distributions.
Take a random value from each distribution $r_1, r_2 ... r_n$.
Let $p_1,p_2...p_n$ be the probability that the random number from that particular distribution be the maximum of $r_1, r_2 ... r_n$
The overlap condition previously ensures that all values of $p$ are non zero.
So as far as I can determine by montecarlo methods if the distributions are continuous then the relationship between $p$ and $m$ can be well approximated by
$p(x)= \exp(a\cdot m(x)^2+b \cdot m(x)+c)$
My problems are
I don't know what type of problem this is so can't search out papers for it. Blatantly it is statistics of some sort anyone any ideas what I should be looking for.
Is it already well known (see above) and my approximation is an approximation to a known formula? (I have tried to find others but the above is the best fit so far)
Finally is there a proper mathematical proof somewhere for the above?
I've been playing with this for at least 5 years on and off and I'm coming to a wall where my mathematical skills can't see over.
Thanks in advance.