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Chapter 4: Q 4E (page 271)
Determine which of the three gambles in Exercise 2 would be preferred by a person whose utility function has the form\({\bf{U}}\left( {\bf{x}} \right){\bf{ = ax + b}}\)where a and b are constants\(\left( {{\bf{a > 0}}} \right)\).
All three gambles are equally preferred.
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Consider a utility function U for which\({\bf{U}}\left( {\bf{0}} \right){\bf{ = 0}}\)and\({\bf{U}}\left( {{\bf{100}}} \right){\bf{ = 1}}\). Suppose that a person who has this utility function is indifferent to accepting a gamble from which his gain will be 0 dollars with probability\({\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{3}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{3}}\)}}\)or 100 dollars with probability\({\raise0.7ex\hbox{\({\bf{2}}\)} \!\mathord{\left/ {\vphantom {{\bf{2}} {\bf{3}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{3}}\)}}\)or accepting 50 dollars as a sure thing. What is the value\({\bf{U}}\left( {{\bf{50}}} \right)\)?
Suppose that 20 percent of the students who took acertain test were from schoolAand that the arithmetic average of their scores on the test was 80. Suppose alsothat 30 percent of the students were from school B and that the arithmetic average of their scores was 76. Suppose, finally, that the other 50 percent of the students were from schoolCand that the arithmetic average of their scores was 84. If a student is selected at random from the entiregroup that took the test, what is the expected value of herscore?
A person is given m dollars, which he must allocate between an event A and its complement\({{\bf{A}}^{\bf{c}}}\). Suppose that he allocates a dollars to A and m-a dollars to\({{\bf{A}}^{\bf{c}}}\). The person鈥檚 gain is then determined as follows: If A occurs, his gain is\({g_1}a\); if\({A^c}\)occurs, his gain is\({{\bf{g}}_{\bf{2}}}\left( {{\bf{m - a}}} \right)\)Here,\({{\bf{g}}_{\bf{1}}}\,{{\bf{g}}_{\bf{2}}}\)are given positive constants. Suppose also that\({\bf{{\rm P}}}\left( {\bf{A}} \right){\bf{ = p}}\)and the person鈥檚 utility function is\({\bf{U}}\left( {\bf{x}} \right){\bf{ = logx}}\)for x>0. Determine the amount a that will maximize the person鈥檚 expected utility, and show that this amount does not depend on the values of g1 and g2.
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
\(\left| {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right)\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right| \le \frac{{\bf{1}}}{{\bf{2}}}{\left( {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right){\bf{ \pm }}\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right)^{\bf{2}}}\).
Suppose that one word is to be selected at random from the sentence the girl put on her beautiful red hat. If X denotes the number of letters in the word that is selected, what is the value of E(X)?
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