Let $F$ be the cdf of a discrete (for example, binary) distribution. I'm looking at $g = \int_0^\alpha x(i)^\beta di$, where each $x(i)$ is an independent draw from $F$, and the integral is to be understood as a Riemann integral.
How can I represent $\mathbf E[g]$? My intuition:
- A LLN applies, since there are infinitely many draws.
- I don't care about the individual draw of each $x(i)$, all that matters is that their distribution is given by $F$.
Yet, I can't conceptually simplify this -- how should I approach it?