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Chapter 4: Q13E (page 247)
Suppose that the distribution of X is symmetric around a point m. Prove that m is a median of X.
If the distribution of X is symmetric around a point m, then m is a median of X.
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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\(E\left( {XY} \right) = 0\).
Consider the example efficient portfolio at the end of Example 4.3.7. Suppose that \({R_i}\) has the uniform distribution on the interval \(\left( {{a_i},{b_i}} \right)\) for \(i = 1,2\).
a. Find the two intervals \(\left( {{a_1},{b_1}} \right)\) and \(\left( {{a_2},{b_2}} \right)\). Hint: The intervals are determined by the means and variances.
b. Find the value at risk (VaR) for the example portfolio at probability level 0.97. Hint: Review Example 3.9.5 to see how to find the p.d.f. of the sum of two uniform random variables.
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\(\left( {{\bf{a > 0}}} \right)\).
Suppose thatXandYhave a continuous joint distribution
for which the joint p.d.f. is as follows:
\(f\left( {x,y} \right) = \left\{ {\begin{align}{}{\frac{1}{3}\left( {x + y} \right)}&{0 \le x \le 1,0 \le y \le 2}\\0&{otherwise}\end{align}} \right.\)
Determine the value of Var(2X鈭�3Y+8).
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?
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