I am trying to solve for the expected value of fourth order statistic, $$E[X_{(4)}]. $$
Suppose I have the following random variables:
$$ X_1, X_2, ... X_5 $$
With pdf: $$ f(x) = e^{-x} $$ $$ x>0 $$
And their corresponding order statistics:
$$ X_{(1)} < X_{(2)} ... <X_{(5)}$$
First, I have solved for the CDF: $$ F(x) = \int_0^x{f(t)}dt = (1-e^{-x})$$
Using the following equation for the PDF of the kth order statistic, I have found the PDF of the 4th order statistic.
$$f_{x_{(k)}} = n f(x)*\begin{pmatrix} n-1\\k-1\end{pmatrix} *F(x)^{k-1}*(1-F(x))^{n-k}$$
$$f_{x_{(4)}} = 5*e^{-x}*\begin{pmatrix} 5-1\\4-1\end{pmatrix} *(1-e^{-x})^{4-1}*(1-F(x))^{5-4}$$
$$f_{x_{(4)}} = 20e^{-2x}(1-e^{-x})^3 $$
Now to solve for the expected value, $E[X_{(4)}]. $
$$ E[X_{(4)}] = \int_0^{\infty}x*f_{x_{(4)}}dx$$
$$ E[X_{(4)}] = \int_0^{\infty}x*20e^{-2x}(1-e^{-x})^3 dx$$
$$ E[X_{(4)}] = ? $$
First, I am not sure that I can possibly evaluate this integral analytically, and I would be fine to evaluate numerically. However, when I try to evaluate numerically using a tool like Wolfram, it says that the integral does not converge, and it will not solve.
Can somebody please verify that I have taken the correct approach and applied all formulas correctly.
Thank you very much in advance!