Chapter 8: Q8 E (page 469)
For the conditions of Exercise 5, how large a random sample must be taken so that\({{\bf{E}}_{\bf{p}}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}} - {\bf{p}}{{\bf{|}}^{\bf{2}}}} \right) \le {\bf{0}}{\bf{.01}}\)for every possible value ofp (0≤p≤1)?
The needed sample size is \(n \ge 25\)
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Question: Consider the calorie count data described in Example7.3.10 on page 400. Now assume that each observation has the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) given the parameter \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\). Use the normal-gamma conjugate prior distribution with prior hyper parameters
\({{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 60,}}{{\bf{\mu }}_{\bf{0}}}{\bf{ = 0}}\,\,{\bf{and}}\,\,{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 1}}\)The value of \({{\bf{s}}_{\bf{n}}}^{\bf{2}}\) is 2102.9.
a. Find the posterior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)
b. Compute \({\bf{Pr(\mu > 1|x)}}{\bf{.}}\)
Suppose that\({X_1},...,{X_n}\)form a random sample from the normal distribution with unknown mean μ and unknown variance\({\sigma ^2}\). Describe a method for constructing a confidence interval for\({\sigma ^2}\)with a specified confidence coefficient\(\gamma \left( {0 < \gamma < 1} \right)\) .
Suppose that X1,……,Xn form a random sample from a normal distribution for which the mean is known and the variance is unknown. Construct an efficient estimator that is not identically equal to a constant, and determine the expectation and the variance of this estimator.
Question: Consider the situation described in Exercise 7 of Sec. 8.5. Use a prior distribution from the normal-gamma family with values \({{\bf{\mu }}_{\bf{0}}}{\bf{ = 150,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 0}}{\bf{.5,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 4}}\)
a. Find the posterior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau = }}\frac{{\bf{1}}}{{{{\bf{\sigma }}^{\bf{2}}}}}\)
b. Find an interval (a, b) such that the posterior probability is 0.90 that a <\({\bf{\mu }}\)<b.
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