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Chapter 4: Q 13E (page 255)

Suppose that\({\bf{X}}\)and\({\bf{Y}}\)are random variables such that

\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = 9}}\),\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ = 4}}\),and\({\bf{\rho }}\left( {{\bf{X,Y}}} \right){\bf{ = - }}\frac{{\bf{1}}}{{\bf{6}}}\).Determine

(a)\({\bf{Var}}\left( {{\bf{X + Y}}} \right)\)and(b)\({\bf{Var}}\left( {{\bf{X - 3Y + 4}}} \right)\).

Short Answer

Expert verified

a.\(Var\left( {X + Y} \right) = \)\(11\)

b.\(Var\left( {X - 3Y + 4} \right) = \)\(51\)

Step by step solution

01

Given information

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

\(\begin{align}Var\left( X \right) &= 9\\Var\left( Y \right) &= 4\\\rho \left( {X,Y} \right) &= - \frac{1}{6}\end{align}\)

02

Calculate \({\bf{Cov}}\left( {{\bf{X,Y}}} \right)\)

Referring Definition 4.6.2 for the equation.

The Correlation of\(X\),\(Y\)is given by,

\(\rho \left( {X,Y} \right) = \frac{{Cov\left( {X,Y} \right)}}{{\rho \left( X \right)\rho \left( Y \right)}}\)

The Covariance of\(X\),\(Y\)is

\(\begin{align}Cov\left( {X,Y} \right) &= \rho \left( {X,Y} \right)\rho \left( X \right)\rho \left( Y \right)\\ &= \left( { - \frac{1}{6}} \right)\sqrt 9 \sqrt 4 \\ &= - \frac{1}{6} \times 3 \times 2\\ &= - 1\end{align}\)

03

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

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

\(\begin{align}Var\left( {X + Y} \right) &= Var\left( X \right) + Var\left( Y \right) + 2Cov\left( {X,Y} \right)\\ &= 9 + 4 + 2\left( { - 1} \right)\\ &= 11\end{align}\)

Hence,\(Var\left( {X + Y} \right) = 11\).

04

Calculate \({\bf{Var}}\left( {{\bf{X - 3Y + 4}}} \right)\)

The variance of \(\left( {X - 3Y + 4} \right)\) is given by,

\(\begin{align}Var\left( {X - 3Y + 4} \right) &= Var\left( X \right) + {\left( { - 3} \right)^2}Var\left( Y \right) + 2\left( { - 3} \right)Cov\left( {X,Y} \right)\\ &= 9 + \left( {9 \times 4} \right) + 2\left( { - 3} \right)\left( { - 1} \right)\\ &= 9 + 36 + 6\\ &= 51\end{align}\)

Hence,\(Var\left( {X - 3Y + 4} \right) = 51\).

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Most popular questions from this chapter

Suppose that a random variable X has a continuous distribution for which the pdf is as follows:

\(f\left( x \right) = \left\{ {\begin{aligned}{{}{}}{{e^{ - x}}}&{{\rm{for }}x > 0}\\0&{{\rm{otherwise}}}\end{aligned}} \right.\)

Determine all the medians of this distribution.

Prove that if\({\bf{Var}}\left( {\bf{X}} \right){\bf{ < }}\infty \)and\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ < }}\infty \), then\({\bf{Cov}}\left( {{\bf{X,Y}}} \right)\)is finite. Hint:By considering the relation

\({\left( {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right){\bf{ \pm }}\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right)^{\bf{2}}} \ge {\bf{0}}\), show that

\(\left| {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right)\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right| \le \frac{{\bf{1}}}{{\bf{2}}}{\left( {\left( {{\bf{X - }}{{\bf{\mu }}_{\bf{X}}}} \right){\bf{ \pm }}\left( {{\bf{Y - }}{{\bf{\mu }}_{\bf{Y}}}} \right)} \right)^{\bf{2}}}\).

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 \(d = 0\).

Suppose that three random variables\({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}{{\bf{X}}_{\bf{3}}}\)froma random sample from a distribution for which the meanis 5. Determine the value of

\({\bf{E}}\left( {{\bf{2}}{{\bf{X}}_{\bf{1}}}{\bf{ - 3}}{{\bf{X}}_{\bf{2}}}{\bf{ + }}{{\bf{X}}_{\bf{3}}}{\bf{ - 4}}} \right)\).

Suppose thatXandYhave a continuous joint distributionfor which the joint p.d.f. is as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{12}}{{\bf{y}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{y}} \le {\bf{x}} \le {\bf{1}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

Find the value ofE(XY).
See all solutions

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