Note : The equation is taken from a paper titled "Performance Analysis of Monostatic Multi-Tag Backscatter Systems With General Order Tag Selection" and given reference is of "Order Statistics" by H.A.David.
I am trying to derive the following equation (1) using order statistics theory.
Let $Z$ =r-th max ($\gamma_1,\gamma_2...\gamma_m$) where $m=1,2,...,M$ and the PDF, CDF of $\gamma_m$ is: $f_{\gamma_m}(z)$=$\frac{\zeta\cdot e^{-\zeta\sqrt{z}}}{2\sqrt{z}}u(z)$ and $F_{\gamma_m}(z)=1-e^{-\zeta\sqrt{z}}u(z)$. Then using these above PDF and CDF of $\gamma_m$ and using order statistics theory, the PDF of $Z$ is $f_Z(z)= r\binom{M}{r}\frac{\zeta\cdot e^{-\zeta r \sqrt{z}}(1-e^{-\zeta\sqrt{z}})^{M-r}}{2\sqrt{z}}u(z)$ ---(1).
My doubt is how the equation (1) is coming. Any help in this regard is highly appreciated.