Let $X_i \sim^{iid} exp(1)$ be a sequence of $n$ iid random variables where $n$ is even. Consider the sample median:
$$\hat\theta_n = X_{(n/2)}/2 + X_{(n/2 + 1)}/2$$
Compute the expected value, $\mathbb E \hat \theta_n$.
what I've tried
Applying standard results on order statistics, we obtain the density of $X_{(n/2)}$:
$$f(x) = e^{-x}e^{-xn/2}(1-e^{-x})^{n/2-1}$$
To compute the expected value:
$$\mathbb E X_{(n/2)} = \int_0^\infty xf(x) dx$$
But this integral is difficult to compute (Mathematica gives a result involving harmonic numbers). Is there a more straightforward way to compute this expectation? Or is there a trick to computing this integral that I am missing?
