Hi,

I have the following minimization problem:

$\displaystyle

k = argmin_k k + (M-2k) Pr( Y_{(k)} < a )

$

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

$\displaystyle

Pr( Y_{(k)} < a ) = \sum_{i=k}^M {M \choose i} a^i (1-a)^{M-i}

$

$\displaystyle

Pr( Y_{(k)} < a ) = \int_0^a \frac{M!}{(k-1)!(M-k)!} t^{k-1}(1-t)^{M-k} dt

$

$\displaystyle

Pr( Y_{(k)} < a ) = B_a(k,N-k+1)/B(k,N-k+1)

$

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.