In Example 12.3.7, we simulated the values by first

Return to Exercise 10 in Sec. 12.2. Now that we know how to simulate exponential random variables, perform the simulation developed in that exercise as follows: a. Perform v0 = 2000 simulations and compute both the estimate of and its simulation standard error. b. Suppose that we want our estimator of to be within 0.01 of with probability 0.99. How many simulations should we perform?

Describe how to convert a random sample U1,...,Un from the uniform distribution on the interval [0, 1] to a random sample of size n from the uniform distribution on the interval [a, b]

Show how to use the probability integral transformation to simulate random variables with the two p.d.f.s in Eq. (12.3.3).

Show how to simulate Cauchy random variables using the probability integral transformation

Prove that the expected number of iterations of the acceptance/rejection method until the first acceptance is k. (Hint: Think of each iteration as a Bernoulli trial. What is the expected number of trials (not failures) until the first success?)

a. Show how to simulate a random variable having the Laplace distribution with parameters 0 and 1. The p.d.f. of the Laplace distribution with parameters and is given in Eq. (10.7.5). b. Show how to simulate a standard normal random variable by first simulating a Laplace random variable and then using acceptance/rejection. Hint: Maximize ex2/2/ex for x 0, and notice that both distributions are symmetric around 0.

Suppose that you have available as many i.i.d. standard normal pseudo-random numbers as you desire. Describe how you could simulate a pseudo-random number with an F distribution with four and seven degrees of freedom.

Let X and Y be independent random variables with X having the t distribution with five degrees of freedom and Y having the t distribution with seven degrees of freedom. We are interested in E(|X Y |). 816 Chapter 12 Simulation a. Simulate 1000 pairs of (Xi, Yi) each with the above joint distribution and estimate E(|X Y |). b. Use your 1000 simulated pairs to estimate the variance of |X Y | also. c. Based on your estimated variance, how many simulations would you need to be 99 percent confident that your estimator of E(|X Y |)is within 0.01 of the actual mean?

Show how to use acceptance/rejection to simulate random variables with the following p.d.f.: f (x) = 4 3 x if 0 < x 0.5, 2 3 if 0.5 < x 1.5, 8 3 4 3 x if 1.5 < x 2, 0 otherwise.

Implement the simulation in Example 12.2.3 for the clinical trial of Example 2.1.4 on page 57. Simulate 5000 parameter vectors. Use a prior distribution with 0 = 1 and 0 = 1. Estimate the probability that the imipramine group has the highest probability of no relapse. Calculate how many simulations you would need to be 95 percent confident that your estimator is within 0.01 of the true probability

In Example 12.3.7, we simulated the values by first simulating gamma random variables with parameters(n p)/2 and 1. Suppose that our statistical software allows us to simulate 2 random variables instead. Which 2 distribution should we use and how would we convert the simulated 2s to have the appropriate gamma distribution?

Use the blood pressure data in Table 9.2 that was described in Exercise 10 of Sec. 9.6. Suppose now that we are not confident that the variances are the same for the two treatment groups. Perform a simulation of the sort done in Example 12.3.8 to obtain a sample from the posterior distribution of the parameters when we allow the variances to be unequal. a. Draw a plot of the sample c.d.f. of the absolute value of the difference between the two group means. b. Draw a histogram of the logarithm of the ratio of the two variances to see how close together they seem to be.

Let F 1 be defined as in Eq. (12.3.7). Let U have the uniform distribution on the interval [0, 1]. Prove that F 1(U ) has the c.d.f. in Eq. (12.3.6).

Refer to the three curves in Fig. 12.4. Call those three sample c.d.f.s Gv,1, Gv,2, and Gv,3, and call the three c.d.f.s that they estimate G1, G2, and G3. Use the Glivenko-Cantelli lemma (Theorem 10.6.1) to show that Pr |Gv,i(x) Gi(x)| 0.0082, for all x and all i is about 0.9979 or larger. Hint: Use the Bonferroni inequality (Theorem 1.5.8).

Prove that the acceptance/rejection method works for discrete distributions. That is, let f and g be p.f.s rather than p.d.f.s, but let the rest of the acceptance/rejection method be exactly as stated. Hint: The proof can be translated by replacing integrals over x by sums. Integrals over u should be left as integrals.

Describe how to use the discrete version of the probability integral transformation to simulate a Poisson pseudo-random variable with mean .

Let f be a p.f., and assume that Eq. (12.3.8) holds, where each gi is another p.f. Assume that X is simulated using the method described immediately after Eq. (12.3.8). Prove that X has the p.f. f .

Use the alias method to simulate a random variable having the Poisson distribution with mean 5. Use the table of Poisson probabilities in the back of the book, and assume that 16 is the largest value that a Poisson random variable can equal. Assume that all of the probability not accounted for by the values 0,..., 15 is the value of the p.f. at k = 16.

. Let Y have the uniform distribution on the interval [0, 1]. Define I to be the greatest integer less than or equal to nY + 1, and define U = nY + 1 I . Prove that I and U are independent and that U has uniform distribution on the interval [0, 1].