Let $\Omega=[0,\infty)$, $A=B([0,\infty))$ and $P$ be a probability measure on $A$ which has a density. Define two stochastic processes $(X(t):t\ge 0)$ and $(Y(t):t\ge 0)$ by \begin{align} X(t)(\omega)&=\begin{cases}1 & \text{if }t=\omega \\ 0 & \text{otherwise}\end{cases}\\ Y(t)(ω)&=0 \end{align}
How can I prove that $X$ and $Y$ have the same distribution?