I have a $n$ sided die. You pick any number $1$ to $n$. Call it $x$. You then roll my $n$ sided die $x$ times. Sum all roles greater than or equal to $x$. What is the optimal $x$ you should choose to maximize the expectation of the sum?
Tower time:
Let $S_x$ be the sum when choosing $x$ let $N_x$ be the random number of rolls you get greater than or equal to $x$.
$\mathbb{E}[S_x] = \mathbb{E}[\mathbb{E}[S_x | N_x]] = \mathbb{E}[\frac{x+n}{2} \cdot N_x] = \frac{x+n}{2} \cdot \mathbb{E}[N_x] = \frac{x+n}{2} \cdot x \frac{n+1-x}{n} = \frac{1}{2n} \cdot x(x+n)(n+1-x) $
Maxima of this cubic is at $\sqrt{\frac{n^2+4}{3}}$ then fiddle around with floors and ceilings at the turning point.
Does this all seem okay? Anyone got a nicer method?