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Chapter 4: Q4.4-1E (page 207)
IfXhas the uniform distribution on the interval [a, b], what is the value of the fifth central moment ofX?
The fifth central moment of X is 0.
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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.
Suppose that a person must accept a gamble X of the following form:
\({\bf{Pr}}\left( {{\bf{X = a}}} \right){\bf{ = p}}\,\,\,{\bf{and}}\,\,\,{\bf{Pr}}\left( {{\bf{X = 1}} - {\bf{a}}} \right){\bf{ = 1}} - {\bf{p}}\),
where p is a given number such that\({\bf{0 < p < 1}}\), suppose also that the person can choose and fix the value of a\(\left( {{\bf{0}} \le {\bf{a}} \le {\bf{1}}} \right)\)to be used in this gamble. Determine the value of a that the person would choose if his utility function was\({\bf{U}}\left( {\bf{x}} \right){\bf{ = logx}}\)for\({\bf{x > 0}}\).
Consider a utility function U for which\({\bf{U}}\left( {\bf{0}} \right){\bf{ = 5}}\), \({\bf{U}}\left( {\bf{1}} \right){\bf{ = 8}}\), and \({\bf{U}}\left( {\bf{2}} \right){\bf{ = 10}}\). Suppose that a person who has this utility function is indifferent to either of two gambles X and Y, for which the probability distributions of the gains are as follows:
\(\begin{align}{\bf{Pr}}\left( {{\bf{X = - 1}}} \right){\bf{ = 0}}{\bf{.6,}}\,{\bf{Pr}}\left( {{\bf{X = 0}}} \right){\bf{ = 0}}{\bf{.2,}}\,{\bf{Pr}}\left( {{\bf{X = 2}}} \right){\bf{ = 0}}{\bf{.2;}}\\{\bf{Pr}}\left( {{\bf{Y = 0}}} \right){\bf{ = 0}}{\bf{.9,}}\,\,\,\,{\bf{Pr}}\left( {{\bf{Y = 1}}} \right){\bf{ = 0}}{\bf{.1}}\end{align}\)
What is the value\({\bf{U}}\left( { - {\bf{1}}} \right)\)?
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?
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.
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