max of independant uniform random numbers

[edtost]

a friend posed this problem to me as one that came up in his work on a computer program; i thought i had found a theoretical solution, but it didn't match the results of his simulation.

stripping the problem down to its probabilistic elements, we have:

x1, x2, ..., xn independent and identically distributed, approximately uniform integers from 0 to some large number L (i think it was 2^32, but i'm not sure).

a1, a2, ..., an desired probabilities of picking index 1, 2, ..., n.

q1, q2, ..., qn coefficients to multiply x1...xn by so that the probability of (qm*xm) being the maximum over {q1x1, q2x2, ..., qnxn} is am.

the problem is: how can one derive the vector q?

[pzhon]

For simplicity, assume the xi are uniformly distributed on [0,1].

Given the coefficients {qi}, can one determine {ai} simply? If so, determining {qi} to match a particular choice of {ai} should be relatively easy, at worst an application of Newton's method in n dimensions.

Suppose the {qi} are increasing.

For j=1,...n, consider the event Ej that the maximum of {qixi} is less than qj. This imposes no restriction on xi for i=1,...,j. It restricts x(j+1) through xn.

Compute the probability of Ej: Product qj/qk where k ranges from j+1 to n.

Compute the probability of Ej\E(j-1), where E0 is empty: P(Ej)-P(E(j-1)).

The conditional probability that qixi is the greatest, given Ej\E(j-1), is 0 if i<j, and 1/(n-j+1) if i>=j.

To find the probability qixi is the maximum, sum the terms P(Ej\E(j-1)) P(qixi is the maximum | Ej\E(j-1)).

This looks like it might be simple to invert.