What's the expected value of the expression: $x^{random(y, y+1)}$ where $random(a,b)$ is a uniformly random real number between a and b.
Simple testing confirms my suspicion that this is not simply $x^{E[random(y, y+1)]}$, i.e. $x^{y+0.5}$.
(I'm coming up with some exponential backoff retry logic for a system that involves this randomization to prevent "collisions", and I'm trying to analyze its expected cumulative retry time.)