Chapter 8: Q 6E (page 485)
In Example 8.2.3, suppose that we will observe n = 20 cheese chunks with lactic acid concentrations \({X_1},...,{X_{20}}\) . Find a number c so that \(P\left( {{{\bar X}_{20}} \le \mu + c\sigma '} \right) = 0.95\)
The value of c is 0.387
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Suppose that \({X_1},...,{X_n}\) form a random sample from the normal distribution with unknown mean ฮผ (โโ < ฮผ < โ) and known precision \(\tau \) . Suppose also that the prior distribution of ฮผ is the normal distribution with mean \({\mu _0}\) and precision \({\lambda _0}\) . Show that the posterior distribution of ฮผ, given that \({X_i} = {x_i}\) (i = 1, . . . , n) is the normal distribution with mean
\(\frac{{{\lambda _0}{\mu _0} + n\tau {{\bar x}_n}}}{{{\lambda _0} + n\tau }}\)with precision \({\lambda _0} + n\tau \)
Suppose that X1,โฆโฆ.,Xn form a random sample from the Bernoulli distribution with unknown parameter p. Show that the variance of every unbiased estimator of (1-p)2 must be at least 4p(1-p)3/n.
In the June 1986 issue of Consumer Reports, some data on the calorie content of beef hot dogs is given. Here are the numbers of calories in 20 different hot dog brands:
186,181,176,149,184,190,158,139,175,148,
152,111,141,153,190,157,131,149,135,132.
Assume that these numbers are the observed values from a random sample of twenty independent standard random variables with meanฮผand variance \({{\bf{\sigma }}^{\bf{2}}}\), both unknown. Find a 90% confidence interval for the mean number of caloriesฮผ.
Suppose thatXhas the\({\chi ^{\bf{2}}}\)distribution with 200 degrees of freedom. Explain why the central limit theorem can be used to determine the approximate value of Pr(160<X<240)and find this approximate value.
Suppose that two random variables\({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)have the joint normal-gamma distribution with hyperparameters\({{\bf{\mu }}_{\bf{0}}}{\bf{ = 4,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 0}}{\bf{.5,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 8}}\)Find the values of (a)\({\bf{Pr}}\left( {{\bf{\mu > 0}}} \right)\)and (b)\({\bf{Pr}}\left( {{\bf{0}}{\bf{.736 < \mu < 15}}{\bf{.680}}} \right)\).
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