$U$ exp. distributed, $E[U]=1$. $X(t)=\chi_{\{U \le t\}}$. Looking for one-dimensional marginal and finite dimensional distribution of $X$

Let $ U$ be a random variable with exponential distribution and $ \mathbb E[U]=1$ and

$ $ X(t)=\chi_{\{U \le t\}},\ Y(t)=\chi_{\{U < t\}},\ t \in \mathbb{R_+}$ $

($ \chi$ denotes the indicator function).

Then $ (X(t), t \ge 0)$ and $ (Y(t),t \ge 0)$ are stochastic processes in continuous time with values in $ \{0,1\}$

I need to describe the one-dimensional marginal distributions of $ X$ and $ Y$ , that is the distribution of $ X(t)$ and $ Y(t)$ for a $ t \ge 0$ . Furthermore I need to describe the finite-dimensional distributions of $ X$ and $ Y$ . According to the task it is sufficient to state the probabilites for a single point.

We have never done anything with stochastic processes before so I don’t really know what I need to do so any help is appreciated!