An array $A[1\ldots n]$ contains $n$ distinct numbers that are randomly ordered, with each permutation of the $n$ numbers equally likely. What is the expectation of the index of the maximum element in the array? What is the expectation of the minimum element of the array?

The expectation of the max element having an index $i$ is $\Pr\{X = i\} = \frac 1 n$.

$$ \E[X] = \sum_{i=1}^n i \cdot \Pr\{X = i\} = \sum_{i=1}^n i \cdot \frac 1 n = \frac 1 n \sum_{i=1}^n i = \frac 1 n \frac{n(n+1)}{2} = \frac{n+1}{2} $$

Not a surprising result.

Curious: the same logic applies for the minimum element. Thus, the expectation for the index of the minimum element is the same.