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Chapter 1: Q23E (page 1)
Suppose that X has the log normal distribution with parameters 4.1 and 8. Find the distribution of3X1/2
3X1/2~ ln N (3.1486,2)
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Suppose that 100 mathematics students are dividedinto five classes, each containing 20 students, and thatawards are to be given to 10 of these students. If eachstudent is equally likely to receive an award, what is theprobability that exactly two students in each class willreceive awards?
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 a certain precinct contains 350 voters, of which 250 are Democrats and 100 are Republicans. If 30 voters are chosen at random from the precinct, what is the probability that exactly 18 Democrats will be selected?
Consider two events A and B with Pr(A) = 0.4 and Pr(B) = 0.7. Determine the maximum and minimum possible values of \(Pr\left( {A \cap B} \right)\) and the conditions under which each of these values is attained.
Assuming that A and B are independent events, prove that the events Ac and Bc are also independent.
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