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Chapter 4: Q7SE (page 207)

Suppose that an automobile dealer pays an amount X (in thousands of dollars) for a used car and then sells it for an amount Y. Suppose that the random variables X and Y have the following joint p.d.f. Determine the dealer鈥檚 expected gain from the sale.

\(f\left( {x,y} \right) = \left\{ \begin{aligned}{}\frac{1}{{36}}x\;\;\;\;\;\;\;\;\;\;for0 < x < y < 6\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise.\end{aligned} \right.\)

Short Answer

Expert verified

The dealer鈥檚 expected gain is\(\frac{3}{2}\)

Step by step solution

01

Given information

An automobile dealer pays an amount X for a used car and sells it for an amount Y.

02

Calculating the dealer’s expected to gain 

Dealer鈥檚 expected gain is given by\(E\left( {Y - X} \right)\)

\(\begin{aligned}{}E\left( {Y - X} \right) = \frac{1}{{36}}\int\limits_0^6 {\int\limits_0^y {\left( {y - x} \right)xdxdy} } \\ = \frac{1}{{36}}\int\limits_0^6 {\frac{{{y^3}}}{6}dy} \\ = \frac{1}{{216}}\int\limits_0^6 {{y^3}dy} \end{aligned}\)

\(\begin{aligned}{}E\left( {Y - X} \right) = \frac{1}{{216}}\left( {\frac{{{y^4}}}{4}} \right)_0^6\\ = \frac{1}{{216}}\frac{{1296}}{4}\\ = \frac{{324}}{{216}}\end{aligned}\)

\(E\left( {Y - X} \right)\)\( = \frac{3}{2}\)

Hence, the dealer鈥檚 expected gain is \(\frac{3}{2}\)

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Most popular questions from this chapter

Suppose that the random variables \({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}....{{\bf{X}}_{\bf{n}}}\)form a random sample of sizenfrom the uniform distribution on the interval [0,1].

Let \({{\bf{Y}}_{\bf{1}}}{\bf{ = min\{ }}{{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}....{{\bf{X}}_{\bf{n}}}{\bf{\} }}\), and let \({{\bf{Y}}_{\bf{n}}}{\bf{ = max\{ }}{{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}....{{\bf{X}}_{\bf{n}}}{\bf{\} }}\)

Find \({\bf{E(}}{{\bf{Y}}_{\bf{1}}}{\bf{)}}\)and \({\bf{E(}}{{\bf{Y}}_{\bf{n}}}{\bf{)}}\)

Let X be a random variable with c.d.f F. Suppose that\(a < b\)are numbers such that both a and b are medians of X.

  1. Prove that\(F\left( a \right) = \frac{1}{2}\).
  2. Prove that there exists a smallest \(c \le a\)and a largest\(d \ge b\)such that every number in the closed interval\(\left( {c,d} \right)\) is a median of X.
  3. If X has a discrete distribution, prove that \(F\left( d \right) > \frac{1}{2}\).

Consider the box of red and blue balls in Examples 4.2.4 and 4.2.5. Suppose that we sample n>1 balls with replacement, and let X be the number of red balls in the sample. Then we sample n balls without replacement, and we let Y be the number of red balls in the sample. Prove that\({\bf{{\rm P}}}\left( {{\bf{X = n}}} \right){\bf{ > {\rm P}}}\left( {{\bf{Y = n}}} \right)\)

Suppose that a person鈥檚 utility function is\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}\)for\({\bf{x}} \ge {\bf{0}}\). Show that the person will always prefer to take a gamble in which she will receive a random gain of X dollars rather than receive the amount E(X) with certainty, where\({\bf{{\rm P}}}\left( {{\bf{X}} \ge {\bf{0}}} \right){\bf{ = 1}}\)and\({\bf{E}}\left( {\bf{X}} \right){\bf{ < }}\infty \)

Suppose that the distribution of a random variable Xis symmetric with respect to the point \(x = 0\) and that \(E\left( {{X^4}} \right) < \infty \).Show that \(E\left( {{{\left( {X - d} \right)}^4}} \right)\)is minimized by the value \(d = 0\).
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