Given a continuous function $r(t) \geq 0$, $t \in \mathbb{R}$, what is $\mathbb{E}[r(t-X)]$ where $X \geq 0$ is a continuous random variable with PDF $f(x)$?
I am thinking that since $r(t)$ is continuous, $\mathbb{E}[r(t-X)] = r(d)$ for some $d \in (-\infty,t]$. Intuitively, I might expect $d = \mathbb{E}[X]$ such that $\mathbb{E}[r(t-X)] = r(t-\mathbb{E}[X])$. Is my reasoning completely wrong? If not, how could I prove this?