Answer the following:

Consider an unnormalized density q(x), defined for a on a region D, and let f(x) denote its corresponding normalized density. Suppose there is a (normalized) trial distribution g(x) such that: (i) There is a constant M, such that Mag(x) 2 q(x) for a E D. (ii) We can sample from g(x). (a) Show that there is a constant M such that the ratio r(x) = q(x Mag(x) is equal to the rejection sampling acceptance ratio, i.e., r(x) = q(x) f(x) Mag(x) Mg(x) That is, rejection sampling can still be used to sample from f(x) even if we only know q(x). (b) Consider the unnormalized density q(x) = 2 for -1