A "simple" expected value inequality confuse me.
If $f(x)$ is a monotone increasing function defined on the interval $[a,b]$, and $c \in [a,b]$, is it possible to prove the following inequality?
$$\mathrm E\left(\sum_{t=a}^{c}f(t)\right)\leq \mathrm E \left(\sum_{t=a}^{b}f(t)\right)$$