My textbook says, for a multivariate case, say $f(x_1, x_2, x_3) = f(x_3 |x_2, x_1)f(x_2|x_1)f(x_1)$, we would simulate a value from the density $f(x_1)$, then using this value simulate from $f(x_2|x_1)$ and then $f(x_3 |x_2, x_1)$..
My question is, how is this simulation from each distribution done in practice? Do we use the inverse transform for each distribution? (assuming these are from standard distributions)
This is supposed to be setting the foundations for Markov chain's and then moving onto MCMC where sampling from more complicated distributions is required.