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I have a question about acceptance rejection method and really appreciate your advice:

Suppose we want to generate random variable that has probability density function $f(x)$, since we're using acceptance-rejection method, we need another probability density function $g(x)$ and constant $M$ such that $f(x)/g(x)<=M$.

Our first step is: generate random variable $y$ from $g(y)$ and a random variable $v$ from standard uniform distribution $[0,1]$

Here is my doubt: is there one to one mapping between generated random variable $y$ and $v$? In other words, are they independent OR for each $y$, it is derived by cumulative distribution function $G^{-1}(v)$, be aware that we use $v$ in the following step

Our second step is: if $v<={f(x)}/{(M*g(X))}$, accept $x=y$

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They are independent.

The point is that $y$ is derived from your easily sampled distribution $g$ randomly. Now you have a random test (via $v$) that decides whether to accept $y$ or not as part of the random sample of the harder to sample $f$.

The procedure uses $M$ in the accept-reject method and whilst you can derive conservative estimates with $M$ quite high the number of rejected samples will be very high and so sampling will take a long time. Otherwise you can do some prior analysis to determine a supposed optimal underlying $g$ and low value of $M$ that will still generate a random sample with the distribution of $f$ but the number of rejected samples will be minimised.

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