A random process $X(t)$ is defined as $X(t)=1, 0\leq t \leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$
I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of $X(t)$ is $$Pr(\{ X(t, \xi)=1 \})=Pr(\{ (t, \xi) | 0\leq t \leq \xi \})=\int_0^{\infty}\int_0^\xi f(t,\xi)dtd\xi$$
However, I am confused how I can calculate $f(t, \xi)$. I know from the question that $f(\xi) = \lambda e^{-\lambda \xi}$ where $\lambda$ is the parameter of the geometric distribution. But I don't see how to get $f(t,\xi)$