I am trying to find the expected value of $X_{(k)}$ Given cdf $$ F(x) = \begin{cases} 1-\left(\frac{\sigma}{x}\right)^\alpha, & x > \sigma\\ 0, & \text{else.} \end{cases}$$

My attempt: $$f_{X_{(k)}}(x)=\frac{n!}{(k-1)!(n-k)!}f_X(x)[F_{X}(x)]^{k-1}[1-F_X(x)]^{n-k}$$

$$E(X_{(k)})=\frac{n!}{(k-1)!(n-k)!}\alpha \sigma^{\alpha(n-k+1)}\int_\sigma^\infty [x^{\alpha}-\sigma^{\alpha}]^{k-1}[x]^{-n\alpha}dx$$

I cannot simplify the integral further.


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