Understanding an expectation formula

Question

Consider a puddle of water which evaporates at random points over time. Let $f(x,t)$ be the fraction of evaporated water at site $x$ on the puddle at a time $t$ after the start of evaporation. I want to calculate the expected time of evaporation $t_E(x)$ at a site $x$. Is that simply given by $$ t_E(x)=\int_0^\infty t \frac{\partial f(x,t)}{\partial t}\,dt $$ since $f$ can be seen as the CDF of the process? Or am I missing something?

Answer 1

The fraction of water evaporated is not the CDF of the distribution of water fractions of some random process. For example, consider a linearly increasing $f(x,t) = t$ on $[0, 1]$, and constant $1$ thereafter. The expected time of evaporation is exactly $1$, while your formula would give you $1/2$.