Answer to Question #102348 in Statistics and Probability for Peter Gemayel

Question

Consider the target p.d.f. f(x) = e−x2/2/√2π, −∞ < x < ∞. (a) (8 marks). Write down an algorithm for generating X ∼ f by the general accept-reject method using the standard Cauchy distribution with p.d.f. g(x) = 1/(π(1 + x2)) for −∞ < x < ∞ as the proposal distribution. Do this as efficiently as possible expressing your results in terms of i.i.d. U1,U2 ∼ U[0,1]. (b) (2 marks). On average, how many uniform random numbers Ui ∼ U[0,1] are needed to generate one X ∼ f by this method?
(c) (4 marks). Using the free software R, generate a sample from X ∼ f using the general accept-reject method with proposal p.d.f. g(x). In particular, use the statement set.seed(12345) and 10,000 iterations. At the same time, verify part(a) by constructing a histogram with f(x) and alpha*g(x) superimposed on it, and also verify part(b).

Answer 1

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