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Chapter 1: Q2E (page 1)
Suppose that X is a random variable for which\({\bf{P}}\left( {{\bf{X}} \ge {\bf{0}}} \right){\bf{ = 1}}\,{\bf{and}}\,{\bf{P}}\left( {{\bf{X}} \ge {\bf{10}}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{5}}}\) .
Prove that \({\bf{E}}\left( {\bf{X}} \right) \ge {\bf{2}}\) .
\(E\left( X \right) \ge 2\).
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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\({\rm P}\left( {{{\rm A}^c} \cap {{\rm B}^c} \cap {D^c}} \right)\)?
Prove that for all positive integers n and k \(\left( {n \ge k} \right)\) ,
\(\left( {^n{C_k}} \right) + \left( {^n{C_{k - 1}}} \right) = \left( {^{n + 1}{C_k}} \right)\)
Consider a state lottery game in which each winning combination and each ticket consists of one set of k numbers chosen from the numbers 1 to n without replacement. We shall compute the probability that the winning combination contains at least one pair of consecutive numbers.
a. Prove that if\({\bf{n < 2k - 1}}\), then every winning combination has at least one pair of consecutive numbers. For the rest of the problem, assume that\({\bf{n}} \le {\bf{2k - 1}}\).
b. Let\({{\bf{i}}_{\bf{1}}}{\bf{ < }}...{\bf{ < }}{{\bf{i}}_{\bf{k}}}\)be an arbitrary possible winning combination arranged in order from smallest to largest. For\({\bf{s = 1,}}...{\bf{,k}}\), let\({{\bf{j}}_{\bf{s}}}{\bf{ = }}{{\bf{i}}_{\bf{s}}}{\bf{ - }}\left( {{\bf{s - 1}}} \right)\). That is,
\(\begin{array}{c}{{\bf{j}}_{\bf{1}}}{\bf{ = }}{{\bf{i}}_{\bf{1}}}\\{{\bf{j}}_{\bf{2}}}{\bf{ = }}{{\bf{i}}_{\bf{2}}}{\bf{ - 1}}\\{\bf{.}}\\{\bf{.}}\\{\bf{.}}\\{{\bf{j}}_{\bf{k}}}{\bf{ = }}{{\bf{i}}_{\bf{k}}}{\bf{ - }}\left( {{\bf{k - 1}}} \right)\end{array}\)
Prove that\(\left( {{{\bf{i}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{i}}_{\bf{k}}}} \right)\)contains at least one pair of consecutive numbers if and only if\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)contains repeated numbers.
c. Prove that\({\bf{1}} \le {{\bf{j}}_{\bf{1}}} \le ... \le {{\bf{j}}_{\bf{k}}} \le {\bf{n - k + 1}}\)and that the number of\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)sets with no repeats is\(\left( {\begin{array}{*{20}{c}}{{\bf{n - k + 1}}}\\{\bf{k}}\end{array}} \right)\)
d. Find the probability that there is no pair of consecutive numbers in the winning combination.
e. Find the probability of at least one pair of consecutive numbers in the winning combination
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?
If two balanced dice are rolled, what is the probability that the difference between the two numbers that appear will be less than 3?
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