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Chapter 4: Q 14E (page 255)

Suppose that\({\bf{X}}\),\({\bf{Y}}\),and\({\bf{Z}}\)are three random variables such that\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = 1}}\),\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ = 4}}\),\({\bf{Var}}\left( {\bf{Z}} \right){\bf{ = 8}}\),\({\bf{Cov}}\left( {{\bf{X,Y}}} \right){\bf{ = 1}}\),\({\bf{Cov}}\left( {{\bf{X,Z}}} \right){\bf{ = - 1}}\),and\({\bf{Cov}}\left( {{\bf{Y,Z}}} \right){\bf{ = 2}}\). Determine (a)\({\bf{Var}}\left( {{\bf{X + Y + Z}}} \right)\)and (b)\({\bf{Var}}\left( {{\bf{3X - Y - 2Z + 1}}} \right)\).

Short Answer

Expert verified

a.\(Var\left( {X + Y + Z} \right) = 17\)

b.\(Var\left( {3X - Y - 2Z + 1} \right) = 59\)

Step by step solution

01

Given information

\(X\),\(Y\),\(Z\)are random variables.

\(\begin{align}Var\left( X \right) &= 1\\Var\left( Y \right) &= 4\\Var\left( Z \right) &= 8\end{align}\)

\(\begin{align}Cov\left( {X,Y} \right) &= 1\\Cov\left( {X,Z} \right) &= - 1\\Cov\left( {Y,Z} \right) &= 2\end{align}\)

02

Calculate \(Var\left( {X + Y + Z} \right)\)

The variance of \(\left( {X + Y + Z} \right)\) is given by,

\(\begin{align}Var\left( {X + Y + Z} \right) &= Var\left( X \right) + Var\left( Y \right) + Var\left( Z \right) + 2Cov\left( {X,Y} \right) + 2Cov\left( {X,Z} \right) + 2Cov\left( {Y,Z} \right)\\ &= 1 + 4 + 8 + \left( {2 \times 1} \right) + 2\left( { - 1} \right) + \left( {2 \times 2} \right)\\ &= 13 + 2 - 2 + 4\\ &= 17\end{align}\)

Hence,\(Var\left( {X + Y + Z} \right) = 17\).

03

Calculate \({\bf{Var}}\left( {{\bf{3X - Y - 2Z + 1}}} \right)\)

The variance of \(\left( {3X - Y - 2Z + 1} \right)\)is given by

\(\begin{array}{c}Var\left( {3X - Y - 2Z + 1} \right) = \left( \begin{array}{l}\left( {{{\left( 3 \right)}^2}Var\left( X \right)} \right) + \left( {{{\left( { - 1} \right)}^2}Var\left( Y \right)} \right) + \left( {{{\left( { - 2} \right)}^2}Var\left( Z \right)} \right) + \left( {2 \times 3 \times \left( { - 1} \right)Cov\left( {X,Y} \right)} \right)\\ + \left( {2 \times 3 \times \left( { - 2} \right)Cov\left( {X,Z} \right)} \right) + \left( {2 \times \left( { - 1} \right) \times \left( { - 2} \right)Cov\left( {Y,Z} \right)} \right)\end{array} \right)\\ = \left( {9 \times 1} \right) + \left( {1 \times 4} \right) + \left( {4 \times 8} \right) + \left( {\left( { - 6} \right) \times 1} \right) + \left( {\left( { - 12} \right) \times \left( { - 1} \right)} \right) + \left( {4 \times 2} \right)\\ = 9 + 4 + 32 - 6 + 12 + 8\\ = 59\end{array}\)

Hence,\(Var\left( {3X - Y - 2Z + 1} \right) = 59\).

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Most popular questions from this chapter

Consider three gambles, X, Y, and Z, for which the probability distributions of the gains are as follows:

\(\begin{align}{}{\bf{Pr}}\left( {{\bf{X = 5}}} \right){\bf{ = Pr}}\left( {{\bf{X = 25}}} \right){\bf{ = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{2}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{2}}\)}}{\bf{,}}\\{\bf{Pr}}\left( {{\bf{Y = 10}}} \right){\bf{ = Pr}}\left( {{\bf{Y = 20}}} \right){\bf{ = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{2}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{2}}\)}}{\bf{,}}\\{\bf{Pr}}\left( {{\bf{Z = 15}}} \right){\bf{ = 1}}{\bf{.}}\end{align}\)

Suppose that a person鈥檚 utility function has the form\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}\)for\({\bf{x > 0}}\). Which of the three gambles would she prefer?

Suppose that a person has a given fortune\({\bf{A > 0}}\)and can bet any amount b of this fortune in a certain game\(\left( {{\bf{0}} \le {\bf{b}} \le {\bf{A}}} \right)\). If he wins the bet, then his fortune becomes\({\bf{A + b}}\); if he loses the bet, then his fortune becomes\({\bf{A - b}}\). In general, let X denote his fortune after he has won or lost. Assume that the probability of his winning is p\(\left( {{\bf{0 < p < 1}}} \right)\)and the probability of his losing is\({\bf{1 - p}}\). Assume also that his utility function, as a function of his final fortune x, is\({\bf{U}}\left( {\bf{x}} \right){\bf{ = logx}}\)for\({\bf{x > 0}}\). If the person wishes to bet an amount b for which the expected utility of his fortune\({\bf{E}}\left( {{\bf{U}}\left( {\bf{x}} \right)} \right)\)will be a maximum, what amount b should he bet?

Show that two random variablesXandYcannot possibly have the following properties:\(E\left( X \right) = 3\),\(E\left( Y \right) = 2\),\(E\left( {{X^2}} \right) = 10\),\(E\left( {{Y^2}} \right) = 29\), and\(E\left( {XY} \right) = 0\).

Suppose that an observed value of X is equally likely to come from a continuous distribution for which the pdf is for from one for which the pdf is g. Suppose that \(f\left( x \right) > 0\) for \(0 < x < 1\) and \(f\left( x \right) = 0\) otherwise, and suppose also that \(g\left( x \right) > 0\) for \(2 < x < 4\) and \(g\left( x \right) = 0\) otherwise. Determine:

  1. the mean and
  2. the median of the distribution of X.

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.
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