The general acceptance-rejection method of Sec. 8.2.4 has the following discrete analog. L...

The general acceptance-rejection method of Sec. 8.2.4 has the following discrete analog. Let X be discrete with probability mass function p(x_i) for i = 0, ±1, ±2, …, let the majorizing function be t(x_i) ≥ p(x_i) for all i, let , and let r(x_i) = t(x_i)/c for i = 0, ± 1, ± 2, …

1ʹ. Generate Y having probability mass function r.

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2ʹ. Generate U ~ U(0, 1), independent of Y.

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3ʹ. If U ≤ p(Y)/t(Y), return X = Y. Otherwise, go back to step 1ʹ and try again.

Show that this algorithm is valid by following steps similar to those in App. 8A. What considerations are important in choosing the function t(x_i)?