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Chapter 1: Q2SE (page 53)
If A, B, and D are three events such that,\({\rm P}\left( {{\rm A} \cup {\rm B} \cup D} \right) = 0.7\)what is the value of\({\rm P}\left( {{{\rm A}^c} \cap {{\rm B}^c} \cap {D^c}} \right)\)?
The value of \({\rm P}\left( {{{\rm A}^c} \cap {{\rm B}^c} \cap {D^c}} \right) = 0.3\)
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If three fair coins are tossed, what is the probability thatall three faces will be the same?
A box contains 24 light bulbs of which four are defective. If one person selects 10 bulbs from the box ina random manner, and a second person then takes theremaining 14 bulbs, what is the probability that all fourdefective bulbs will be obtained by the same person?
Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from a distribution for which the pdf. f (x|胃 ) is as follows:
\(\begin{array}{c}{\bf{f}}\left( {{\bf{x|\theta }}} \right){\bf{ = }}{{\bf{e}}^{{\bf{\theta - x}}}}\,\,{\bf{,\theta > 0}}\\{\bf{ = 0}}\,\,\,{\bf{otherwise}}\end{array}\)
Also, suppose that the value of 胃 is unknown (鈭掆垶 <胃< 鈭�).
a. Show that the M.L.E. of 胃 does not exist.
b. Determine another version of the pdf. of this same distribution for which the M.L.E. of 胃 will exist, and find this estimator.
Question:Let 胃 denote the proportion of registered voters in a large city who are in favor of a certain proposition. Suppose that the value of 胃 is unknown, and two statisticians A and B assign to 胃 the following different prior p.d.f.鈥檚\({{\bf{\xi }}_{\bf{A}}}\left( {\bf{\theta }} \right)\)and \({{\bf{\xi }}_{\bf{B}}}\left( {\bf{\theta }} \right)\) respectively:
\({{\bf{\xi }}_{\bf{A}}}\left( {\bf{\theta }} \right){\bf{ = 2\theta ,}}\,\,\,{\bf{0 < \theta < 1}}\)
\({{\bf{\xi }}_{\bf{B}}}\left( {\bf{\theta }} \right){\bf{ = 4}}{{\bf{\theta }}^{\bf{3}}}\,\,{\bf{,0 < \theta < 1}}\)
In a random sample of 1000 registered voters from the city, it is found that 710 are in favor of the proposition.
a. Find the posterior distribution that each statistician assigns to 胃.
b. Find the Bayes estimate for each statistician based on the squared error loss function.
c.Show that after the opinions of the 1000 registered voters in the random sample had been obtained, the Bayes estimates for the two statisticians could not possibly differ by more than 0.002, regardless of the number in the sample who were in favor of the proposition
Suppose that X has the lognormal distribution with parameters\({\bf{\mu }}\)and\({\sigma ^2}\) . Find the distribution of1/X.
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