My friend asked me to help with the problem on the random processes, but I am stuck as well, because I don't understand the notation
$X_t = _{[U,1]}(t), t \in [0,1]$
Could anyone explain this one to me?
Here U is a random variable uniformly distributed in [0, 1].
$1_B$ denotes the indicator function of a set $B$ and, by definition, $$1_B(t) = \begin{cases} 0, & t \notin B, \\ 1, & t \in B. \end{cases}$$
Therefore,
$$X_t(\omega) = \begin{cases} 0, & t \in [0,U(\omega)), \\ 1, & t \in [U(\omega),1]. \end{cases}$$