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Chapter 1: Q6E (page 41)
Suppose that n people are seated in a random manner in a row of n theatre seats. What is the probability that two particular people A and B will be seated next to each other?
The probability of two people A and B will be seated next to each other in the theatre is\(\frac{2}{n}\).
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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,\({\bf{Var}}\left( {\bf{X}} \right) \ge \frac{{\bf{9}}}{{\bf{2}}}\) .
Suppose that one card is to be selected from a deck of 20 cards that contains 10 red cards numbered from 1 to 10 and 10 blue cards numbered from 1 to 10. Let A be the event that a card with an even number is selected, let B be the event that a blue card is selected, and let C be the event that a card with a number less than 5 is selected. Describe the sample space S and describe each of the following events both in words and as subsets of S:
a. \({\bf{A}} \cap {\bf{B}} \cap {\bf{C}}\)
b. \({\bf{B}} \cap {{\bf{C}}^{\bf{C}}}\)
c. \({\bf{A}} \cup {\bf{B}} \cup {\bf{C}}\)
d. \({\bf{A}} \cap {\bf{(B}} \cup {\bf{C)}}\)
e. \({{\bf{A}}^{\bf{c}}} \cap {{\bf{B}}^{\bf{c}}} \cap {{\bf{C}}^{\bf{c}}}\)
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\({\rm P}\left( {{{\rm A}^c} \cap {{\rm B}^c} \cap {D^c}} \right)\)?
Use the data on dishwasher shipment in Table \({\bf{11}}.{\bf{13}}\;\)on page \({\bf{744}}\) .suppose that we wish to fit a multiple linear regression model for predicting dishwasher shipment from time (the year 1960) and private residential investment. Suppose that the parameters have the improper prior proportional to \({\raise0.7ex\hbox{\(1\)} \!\mathord{\left/ {\vphantom {1 \tau }}\right.\\} \!\lower0.7ex\hbox{\(\tau \)}}\) the use of the Gibbs sampling algorithm to obtain a sample of size 10,000 from the joint posterior distribution of the parameters.
a. Let \({\beta _{1\,}}\) be the coefficient of time. Draw a plot of sample c.d.f of \(\left| {{\beta _{1\,}}} \right|\) using your posterior sample.
b. We are interested in the values of your posterior distribution 1986.
i. Draw a histogram of the values \({{\bf{\beta }}_{{\bf{0}}\,}}{\bf{ + 26}}{{\bf{\beta }}_{{\bf{1}}\,}}{\bf{ + 67}}{\bf{.2}}{{\bf{\beta }}_{{\bf{2}}\,}}\) from your posterior distribution.
ii. For each of your simulated parameters, simulate a dishwasher sales figure for
1986 (time = 26 and private residential investment = 67.2). compute a 90 percent prediction interval from the simulated values and compare it to the interval found in Example 11.5.7.
iii. Draw a histogram of the simulated 1986 sales figures, and compare it to the histogram in part I. Can you explain why one sample seems to have a more considerable variance than the other?
The United States Senate contains two senators from each of the 50 states. (a) If a committee of eight senators is selected at random, what is the probability that it will contain at least one of the two senators from a certainspecified state? (b) What is the probability that a group of 50 senators selected at random will contain one senator from each state?
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