Does the accepted sample based on the acceptance-rejection algorithm has the same distribution as X~$f(x)$?

Question

In Bayes Statistics, does the accepted sample based on the acceptance-rejection algorithm has the same distribution as X~$f(x)$?

Intuitionally it does, but how to prove it under the discrete and continuous cases respectively?

Answer 1

First, acceptance-rejection is absolutely nothing specific to Bayesian statistics since it applies to any simulation experiment.

Second, the question as stated is unclear since $X\sim f(x)$ does not spell out how $f$ is related with the acceptance-rejection algorithm. If $f$ is the target distribution density and $g$ the instrumental distribution density then indeed the accept-rejection is valid. As established in every simulation textbook.