Let $U$ be random variable with uniform distribution $\mathcal{U}(0,T)$
Let $X(t) =P_T(t)h(t-A)$ the stochastic processes where $$P_T(t) = \begin{cases}8 & t \in [0,T] \\ 0 & \text{otherwise}\end{cases}$$ and $$h(t-a) = \begin{cases}1 & a < t \\ 0 & \text{otherwise}\end{cases}$$
Calculate the mean of the stochastic process X(t).
I know that the mean value of X(t) is E(X(t)).
My problem is that the function $P_T(t)$, with the function $h$ is not related to the fact that A is an uniform random variable. In other words, I'm not sure what integral to calculate.
The teacher has given us only a typical example with a trigonometric function and has sent us a series of exercises, I have been able to do the others, but here I cannot see which is the integral that I must perform.
I need some help, please.
Hint: $P_T(t)$ is deterministic, so \begin{align*}\mathbb{E}[X(t)] &= \mathbb{E}[P_T(t)h(t-A)] \\ &= P_T(t)\mathbb{E}[h(t-A)]. \end{align*}
Do you see which integral you need to compute to evaluate $\mathbb{E}[h(t-A)]$?