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Consider the target p.d.f. f(x) =((2/π)√1−x2, −1 ≤ x ≤ 1, 0, otherwise. (a) (3 marks). Show that f(x) is a p.d.f. (b) (5 marks). Write down an algorithm for generating X ∼ f by the simple accept-reject method expressing your results in terms of i.i.d. U1,U2 ∼ U[0,1]. (c) (3 marks). On average, how many uniform random numbers Ui ∼ U[0,1] are needed to generate one X ∼ f by this method? (d) (3 marks). Using the free software R, generate a sample from X ∼ f using the simple accept-reject method. In particular, use the statement set.seed(12345) and 10,000 iterations. At the same time, verify part(b) by constructing a histogram with f(x) superimposed on it, and also verify part(c).
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