We have G groups. Each group has $M$ (i.i.d) variables $X_{g,1}$,...,$X_{g,M}$ following the exponential distribution exp(-$\mu$) ($\mu$ is a constant).
Focus on each group g (g=1,...,G), we take the $k$th ($k\leq M$ and fixed) order statistic $X_{g,sta}$, i.e., $X_{g,sta}$ is the $k$th-smallest value of each group. By taking the maximum of the G order statistics, we get a new varaible $X_{max}$=max{$X_{1,sta}, X_{2,sta},..., X_{G,sta}$}.
The question is: how to get the expectation E[$X_{max}$] of $X_{max}$?