I am trying to compute the expected absolute difference between $n$ iid uniformly between $[- \epsilon, \epsilon]$ sampled points to a given point $- \epsilon< c < \epsilon$:
$$ \mathrm{E}(\mathrm{min}(|x_1-c|,|x_2-c|,...,|x_n-c|)) $$
I know, that the expected minimum of n iid uniformly between $[0, \epsilon]$ sampled points is $\frac{\epsilon}{n+1}$.
How would I go about this problem though?
Since $\min|X_i - c|$ is non-negative and you want to compute its expectation, we can compute its survival function first. Consider the survival function of individual $|X_i - c|$:
$$ \begin{align} S(x)&=\Pr\{|X_i - c| > x\} \\ &= \Pr\{X_i > c + x\} + \Pr\{X_i < c - x\} \\ &= \begin{cases} 1 & \text {if} & x \leq 0 \\ \displaystyle 1 - \frac {x} {\epsilon} & \text{if} & 0 < x \leq \epsilon - |c|\\ \displaystyle \frac {\epsilon + |c| - x} {2\epsilon} & \text {if} & \epsilon - |c| < x < \epsilon + |c| \\ 0 & \text{if} & x \geq \epsilon + |c| \end{cases} \end{align}$$
Then the survival function of $\min |X_i - c|$ will be $S(x)^n$ and thus $$ \begin{align} E[\min|X_i - c|] &= \int_0^{+\infty}S(x)^ndx \\ &= \int_0^{\epsilon - |c|} \left(1 - \frac {x} {\epsilon} \right)^n dx + \int_{\epsilon - |c|}^{\epsilon + |c|} \left(\frac {\epsilon + |c| - x} {2\epsilon} \right)^n dx \\ &= \left. \frac {-\epsilon} {n+1}\left(1 - \frac {x} {\epsilon} \right)^{n+1} \right|_0^{\epsilon - |c|} + \left.\frac {-2\epsilon} {n+1} \left(\frac {\epsilon + |c| - x} {2\epsilon} \right)^{n+1} \right|_{\epsilon - |c|}^{\epsilon + |c|} \\ &= \frac {\epsilon} {n+1}\left(1 - \frac {|c|^{n+1}} {\epsilon^{n+1}} + 2\frac {|c|^{n+1}} {\epsilon^{n+1}}\right) \\ &= \frac {\epsilon} {n+1}\left(1 + \frac {|c|^{n+1}} {\epsilon^{n+1}} \right) \end{align}$$
As you see, the answer is minimized when $c = 0$ and it reduced back to the previous solution.