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Chapter 4: Q4.1-9E (page 216)
Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Find the expected value of the size of the longer piece.
The expected value of the length of the longer piece is \(\frac{3}{4}\)
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Prove that if\({\bf{Var}}\left( {\bf{X}} \right){\bf{ < }}\infty \)and\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ < }}\infty \), then\({\bf{Cov}}\left( {{\bf{X,Y}}} \right)\)is finite. Hint:By considering the relation
\({\left( {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right){\bf{ \pm }}\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right)^{\bf{2}}} \ge {\bf{0}}\), show that
\(\left| {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right)\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right| \le \frac{{\bf{1}}}{{\bf{2}}}{\left( {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right){\bf{ \pm }}\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right)^{\bf{2}}}\).
Let\({\bf{\alpha > 0}}\). A decision-maker has a utility function for money of the form
\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{{\bf{x}}^{\bf{\alpha }}}}&{{\bf{if}}\,{\bf{x > 0,}}}\\{\bf{x}}&{{\bf{if}}\,{\bf{x}} \le {\bf{0}}{\bf{.}}}\end{align}} \right.\)
Suppose that this decision maker is trying to decide whether or not to buy a lottery ticket for \(1. The lottery ticket pays \)500 with a probability of 0.001, and it pays $0 with a probability of 0.999. What would the values of 伪 have to be for this decision-maker to prefer buying the ticket to not buying it?
Suppose that\({\bf{X}}\)and\({\bf{Y}}\)are random variables such that
\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = 9}}\),\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ = 4}}\),and\({\bf{\rho }}\left( {{\bf{X,Y}}} \right){\bf{ = - }}\frac{{\bf{1}}}{{\bf{6}}}\).Determine
(a)\({\bf{Var}}\left( {{\bf{X + Y}}} \right)\)and(b)\({\bf{Var}}\left( {{\bf{X - 3Y + 4}}} \right)\).
Consider a coin for which the probability of obtaining a head on each given toss is 0.3. Suppose that the coin is to be tossed 15 times, and let X denote the number of heads that will be obtained.
Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is as follows:
\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{x + y}}}&{{\bf{for}}\,\,{\bf{0}} \le {\bf{x}} \le {\bf{1}}\,\,{\bf{and}}\,\,{\bf{0}} \le {\bf{y}} \le {\bf{1,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\).
Find\({\bf{E}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\)and\({\bf{Var}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\).
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