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Chapter 4: Q 9E (page 271)

Determine the value of a that a person would choose in Exercise 7 if his utility function was\({\bf{U}}\left( {\bf{x}} \right){\bf{ = x}}\)for\({\bf{x}} \ge {\bf{0}}\).

Short Answer

Expert verified

If \(p = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\ } \!\lower0.7ex\hbox{$2$}}\), then \(E\left( {U\left( X \right)} \right) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\ } \!\lower0.7ex\hbox{$2$}}\) for all values of a

Step by step solution

01

Given information

\(\Pr \left( {X = a} \right) = p\,\,\,and\,\,\,\Pr \left( {X = 1 - a} \right) = 1 - p\)

The utility function is, given by,

\(U\left( x \right) = x\)

02

Finding the value of a

For any given value of a,

\(E\left( {U\left( X \right)} \right) = pa + \left( {1 - p} \right)\left( {1 - a} \right)\)

This is the linear function of a.

If \(p = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\ } \!\lower0.7ex\hbox{$2$}}\), then \(E\left( {U\left( X \right)} \right) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\ } \!\lower0.7ex\hbox{$2$}}\) for all values of a

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Most popular questions from this chapter

Suppose that a class contains 10 boys and 15 girls, and suppose that eight students are to be selected randomly from the class without replacement. Let X denote the number of boys selected, and let Y denote the number of girls selected. Find E(X 鈭� Y ).

Suppose thatX1, . . . ,Xnare random variables suchthat thevariance of each variable is 1 and the correlationbetween each pair of different variables is 1/4. DetermineVar(X1+. . .+Xn).

Suppose that a person has a lottery ticket from which she will win X dollars, where X has the uniform distribution on the interval\(\left( {{\bf{0,4}}} \right)\). Suppose also that the person鈥檚 utility function is\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{\alpha }}}\)for\({\bf{x}} \ge {\bf{0}}\), where 伪 is a given positive constant. For how many dollars\({{\bf{x}}_{\bf{0}}}\)would the person be willing to sell this lottery ticket?

LetYbe a discrete random variable whose p.f. is the

functionfin Example 4.1.4. LetX= |Y|. Prove that the

distribution ofXhas the p.d.f. in Example 4.1.5

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{2}}\left| {\bf{x}} \right|\left( {\left| {\bf{x}} \right|{\bf{ + 1}}} \right)}}{\bf{,x = \pm 1, \pm 2 \ldots ,}}\\{\bf{0,Otherwise}}\end{array} \right.\)

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{x}}\left( {{\bf{x + 1}}} \right)}}{\bf{,x = 1,2,3 \ldots ,}}\\{\bf{0,Otherwise}}\end{array} \right.\)

Prove the following extension of Theorem 4.5.1. Let m be the pquantile of the random variable X. (See Definition 3.3.2) If r is a strictly increasing function, then \(r\left( m \right)\) is the pquantile of \(r\left( X \right)\).
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