Order Statistics Problem

#1
Hi there

I have random variables X1,X2,X3,...,XN which are I.I.D. Each of these random variables are Erlang Distributed. X=min{X1,...,XN}. Now the sample size N changes with each iteration according to the following rule

If Yn<=X<Yn+1 Then
N=2^n
where n is an element of {1,2,3,4}.............................(1)
Initially, N is set to 2, N=2. Yn and Yn+1 are constants that partition the domain of X. X is defined between 0 and Infinity. If X falls into a certain range say Y2 and Y3, then the sample size for the next iteration is changed from N=2 to N=2^2=4. So if X=min{X1,X2} initially, and we find that this X value lies between Y3 and Y4, the next sample size for the next iteration becomes N=2^3=8 according to Rule (1). The problem is to find the CDF and hence the PDF of Random variable X.

Regards