I understand the space of such parametric curves is no longer a linear space. But I think it could be a valid question to ask whether the density of the random curve are exists or not.

The context of my question is: I have a set of parametric curves $\beta(t)=(x(t),y(t))', t\in [0,1]$ (for example $x(t)=sin t$ and $y(t)=cos t$). I consider these curves are random sample from population of parametric curves and I try to understand whether I can define probability density function of this random curves.

@TheBridge: Yes, $P_X$ is push forward measure.