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Chapter 12: Q12.1-2E (page 791)

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?

Short Answer

Expert verified

The mean would increase as number of observations increases.

Step by step solution

01

Definition for simulation

  • Simulation is model-based experimentation.
  • The model's behaviour imitates some salient aspect of the system under study's behaviour, and the user experiments with the model to infer this behaviour.
  • This broad framework has proven to be an effective tool for learning, problem solving, and design.
  • A probability density function (PDF) is a statistical expression that defines a probability distribution (the likelihood of an outcome) for a discrete random variable (for example, a stock or ETF) rather than a continuous random variable.
02

Determine the computation of average

  • Random variable\(X\)takes values which for large\(x\)get smaller.
  • By simulating large number,\(n\)of such variables, the average would get bigger as\(n\)increases.
  • This is because some values would be much larger then the others, which would appear more frequently when the number of observations increases. When the large observations appear, the mean increases.

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Most popular questions from this chapter

Let \({{\bf{z}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{z}}_{\scriptstyle{\bf{2}}\atop\scriptstyle\,}}....\) from a Markov chain, and assume that distribution of \({z_{\scriptstyle1\atop\scriptstyle\,}}\)is the stationary distribution. Show that the joint distribution \(\left( {{{\bf{z}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{z}}_{\scriptstyle{\bf{2}}\atop\scriptstyle\,}}} \right)\)of is the same as the joint distribution of \(\left( {{{\bf{z}}_{\scriptstyle{\bf{i}}\atop\scriptstyle\,}}{\bf{,}}{{\bf{z}}_{\scriptstyle{\bf{i + 1}}\atop\scriptstyle\,}}} \right)\) for all\(i > 1\) convenience, you may assume that the Markov chain has finite state space, but the result holds in general.

The \({\chi ^2}\) goodness-of-fit test (see Chapter 10) is based on an asymptotic approximation to the distribution of the test statistic. For small to medium samples, the asymptotic approximation might not be very good. Simulation can be used to assess how good the approximation is. Simulation can also be used to estimate the power function of a goodness-of-fit test. For this exercise, assume that we are performing the test that was done in Example 10.1.6. The idea illustrated in this exercise applies to all such problems.

a. Simulate \(v = 10,000\) samples of size \(n = 23\) from the normal distribution with a mean of 3.912 and variance of 0.25. For each sample, compute the \({\chi ^2}\) goodness of fit statistic Q using the same four intervals that were used in Example 10.1.6. Use the simulations to estimate the probability that Q is greater than or equal to the 0.9,0.95 and 0.99 quantiles of the \({\chi ^2}\) distribution with three degrees of freedom.

b. Suppose that we are interested in the power function of a \({\chi ^2}\) goodness-of-fit test when the actual distribution of the data is the normal distribution with a mean of 4.2 and variance of 0.8. Use simulation to estimate the power function of the level 0.1,0.05 and 0.01 tests at the alternative specified.

In Example 12.5.6, we modeled the parameters \({\tau _1}, \ldots {\tau _\pi }\) as i.i.d. having the gamma distribution with parameters \({\alpha _0}\) , and \({\beta _0}.\) We could have added a level to the hierarchical model that would allow the \({\tau _\iota }\) 's to come from a distribution with an unknown parameter. For example, suppose that we model the \({\tau _\iota }\) 's as conditionally independent, having the gamma distribution with parameters \({\alpha _0}\) and \(\beta \) given \(\beta \). Let \(\beta \) be independent of \(\psi \) and \({\mu _1}, \ldots ,{\mu _p}\) with \(\beta \) having the prior distributions as specified in Example 12.5.6.

a. Write the product of the likelihood and the prior as a function of the parameters \({\mu _1}, \ldots ,{\mu _p},{\tau _1}, \ldots ,{\tau _\pi },\psi ,\) \(\beta \).

b. Find the conditional distributions of each parameter given all of the others. Hint: For all the parameters besides \(\beta \), the distributions should be almost identical to those given in Example 12.5.6. It wherever \({\beta _0}.\) appears, of course, something will have to change.

c. Use a prior distribution in which and \({\psi _0} = 170.\) Fit the model to the hot dog calorie data from Example 11.6.2. Compute the posterior means of the four \({\mu _i}'s\) and

\(1/{\tau _i}^\prime s.\)

Let \(U\) have the uniform distribution on the interval\((0,1)\). Show that the random variable \(W\)defined in Eq. (12.4.6) has the p.d.f. \(h\)defined in Eq. (12.4.5).

\({{\bf{x}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}.....{{\bf{x}}_{\scriptstyle{\bf{n}}\atop\scriptstyle\,}}\) be uncorrelated, each with variance \({\sigma ^2}\) Let \({{\bf{y}}_{\scriptstyle{\bf{1}}\atop\scriptstyle\,}}.....{{\bf{y}}_{\scriptstyle{\bf{n}}\atop\scriptstyle\,}}\) be positively correlated. each with variance, prove that the variance of \(\overline x \)is smaller than the variance of \(\overline y \)
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