Consider the following technique, called the acceptreject method, for simulating values from a continuous distribution . Identify

Consider the following technique, called the accept–reject method, for simulating values from a continuous distribution ƒ. Identify a distribution g from which values can already be simulated and a constant c ≥ 1 such that ƒ(x) ≤ cg(x) for all x. Proceed as follows: (1) Generate Y ∼ g and, independently, U ∼ Unif[0, 1). (2) If u ≤ ƒ(y)/cg(y), then let x = y (i.e., “accept” the y value); otherwise, discard (“reject”) y. (3) Repeat steps (1) – (2) until the desired number of x values is obtained.

a. Show that the probability a y value is “accepted” equals 1/c. According to the algorithm, this occurs iff U ≤ ƒ(Y)/cg(Y). Compute the relevant double integral.
b. Argue that the average number of y values required to generate a single accepted x value is c.
c. Show that the accept–reject method does result in observations from f by showing that P(accepted value ≤ x) = F(x), where F is the cdf corresponding to ƒ. Let X denote the accepted value. Then P(X ≤ x) = P(Y ≤ x ∩  Y is accepted) = P(Y ≤ x  is acc.)/P(Y is acc.).