Hi,

I have the following minimization problem:

where Y is Uniform(0,1) and k from 1 to M. a is also given and ranges from 1/2 to 1.

Of course I can just plug in the different values of k and find the minimum but I need an approximation or a closed form solution.

There are 3 expressions for the Probability which all don't help we finding the minimum

where B is the Beta function (incomplete/complete). By admitting k to be a non-integer (Works in the 3rd formula) I can also treat the probability as differentiable which could help me find the minimum but I cannot find a derivative wrt k.

Any ideas?

An approximation idea or something would also be OK since the approach is already an approximation.

Thanks for reading.