Chapter 12: Q12.1-3E (page 791)
If \({\bf{X}}\) has the Cauchy distribution, the mean \({\bf{X}}\)does not exist. What would you expect to happen if you simulated a large number of Cauchy random variables and computed their average?
The average would increase or decrease depending on the observations.
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Use the data in table 11.19 on page 762.This time fits the model developed in Example 12.5.6.use the prior hyperparameters \(\,{{\bf{\lambda }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = }}{{\bf{\alpha }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}\,{\bf{ = 1,}}\,\,{{\bf{\beta }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = 0}}{\bf{.1}},{{\bf{\mu }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = 0}}{\bf{.001}}\)and \({{\bf{\psi }}_{\scriptstyle{\bf{0}}\atop\scriptstyle\,}}{\bf{ = 800}}\)obtain a sample of 10,000 from the posterior joint distribution of the parameters. Estimate the posterior mean of the three parameters \({{\bf{\mu }}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{\mu }}_{\scriptstyle{\bf{2}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{\mu }}_{\scriptstyle{\bf{3}}\atop\scriptstyle\,}}\)
Test the gamma pseudo-random number generator on your computer. Simulate 10,000 gamma pseudo-random variables with parameters a and 1 for \(a = 0.5,1,1.5,2,5,\) 10. Then draw gamma quantile plots
If \({\bf{X}}\)has the \({\bf{p}}.{\bf{d}}.{\bf{f}}.\)\({\bf{1/}}{{\bf{x}}^{\bf{2}}}\)for\({\bf{x > 1}}\), the mean of \({\bf{X}}\) is infinite. What would you expect to happen if you simulated a large number of random variables with this \({\bf{p}}.{\bf{d}}.{\bf{f}}.\) and computed their average?
Consider the power calculation done in Example 9.5.5.
a. Simulate \({v_0} = 1000\) i.i.d. noncentral t pseudo-random variables with 14 degrees of freedom and noncentrality parameter \(1.936.\)
b. Estimate the probability that a noncentral t random variable with 14 degrees of freedom and noncentrality parameter \(1.936\) is at least \(1.761.\) Also, compute the standard simulation error.
c. Suppose that we want our estimator of the noncentral t probability in part (b) to be closer than \(0.01\) the true value with probability \(0.99.\) How many noncentral t random variables do we need to simulate?
Let \({x_1},...,{x_n}\) be the observed values of a random sample \(X = \left( {{x_1},...,{x_n}} \right)\) . Let \({F_n}\)be the sample c.d.f. Let \(\,{j_1},...\,,{j_n}\) be a random sample with replacement from the numbers \(\left\{ {1,.....,n} \right\}\) Define\({x_i}^ * = x{j_i}\) for \(i = 1,..,n.\)ashow that \({x^ * } = \left( {{x_1}^ * ,...,{x_n}^ * } \right)\) is an i.i.d. sample from the distribution\({F_n}\)
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