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Chapter 1: Q1E (page 1)
Suppose that X has the uniform distribution on the interval [a, b]. Find the mean of X.
The mean of X is \(\frac{{b + a}}{2}\)
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Suppose that X is a random variable for which \({\bf{E}}\left( {\bf{X}} \right){\bf{ = 10}}\) ,\({\bf{P}}\left( {{\bf{X}} \le {\bf{7}}} \right){\bf{ = 0}}{\bf{.2}}\,\,{\bf{and}}\,{\bf{P}}\left( {{\bf{X}} \ge {\bf{13}}} \right){\bf{ = 0}}{\bf{.3}}\) .Prove that,\({\bf{Var}}\left( {\bf{X}} \right) \ge \frac{{\bf{9}}}{{\bf{2}}}\) .
Let S be a given sample space and let \({A_1},{A_2},.....\) bean infinite sequence of events. For \(n = 1,2,..........\), let \({B_n} = \bigcup\limits_{i = n}^\infty {{A_i}} \)and let \({C_n} = \bigcap\limits_{i = n}^\infty {{A_i}} \)
a. Show that \({B_1} \supset {B_2} \supset \ldots \ldots \) and that \({C_1} \subset {C_2} \subset \ldots \ldots \).
b. Show that an outcome in S belongs to the event\(\bigcap\limits_{n = 1}^\infty {{B_n}} \) if and only if it belongs toan infinite number of the events \({A_1},{A_2},.....\).
c. Show that an outcome in S belongs to the event \(\bigcup\limits_{n = 1}^\infty {{C_n}} \)if and only if it belongs to all the events\({A_1},{A_2},.....\) except possibly a finite number of those events.
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 two boys named Davis, three boys named Jones, and four boys named Smith are seated at random in a row containing nine seats. What is the probability that the Davis boys will occupy the first two seats in the row, the Jones boys will occupy the next three seats, and the Smith boys will occupy the last four seats?
Prove that for every two events A and B, the probability that exactly one of the two events will occur is given by the expression
\(\Pr \left( A \right) + \Pr \left( A \right) - 2\Pr \left( {A \cap B} \right)\)
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