For example (but I'm most interested in a general procedure), consider $f(x) = \sin(x)$. For a random, continuous range of values of x of width $\Delta x$, what is the average maximum value of $f(x)$. In other words, for a uniform random variable $X$, what is the expectation value for: $$\max\left[f(x)\right] \,\,\,\mathrm{st} \,\,\, x \in [X, X+\Delta x]$$
I don't really know how to approach the range of $x$ aspect of this. For single value of $x$, I think the probability, $P(y < y') \propto \int_0^{y'} \sin^{-1}(y) \, dy $. You could use this to calculate the distribution of expected $y$ values for independent, random $x$, but now for a continuous range (i.e. dependent, random) $x$.
I was thinking about the problem in a bad way. For a given $\Delta x$, the maximum value of $f(x)$ over that range is:
$$ \begin{cases} \sin(x + \frac{\Delta x}{2}) & x < \frac{1}{2}(\pi - \Delta x) \\ 1 & \frac{1}{2}(\pi - \Delta x) < x < \frac{1}{2}(\pi + \Delta x) \\ \sin(x - \frac{\Delta x}{2}) & \frac{1}{2}(\pi - \Delta x) < x \end{cases} $$
Integrating this function, and dividing by $\pi$ then gives the average value for all $x$.
The answer (confirmed numerically), is $$\langle \max\left[\sin(x|\Delta x)\right]\rangle = \frac{1}{\pi}(2\cos(\Delta x/2) + \Delta x)$$