# Is Monte Carlo uncertainty estimation equivalent to analytical error propagation?

If I have a deterministic, analytic model, $y=f(x)$, I can analytically calculate the uncertainty in $y$ from a known uncertainty in $x$, $\sigma$. Or I can do a Monte Carlo integration: sample from the distribution $x$, and run those samples through the model, and get an estimate of the distribution of $y$ from the model's output.

Is there any guarantee that the results of the two methods will be (asymptotically) the same? If so, is it equivalent for a larger class of models than just those with analytical solutions?