# $U$ 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!