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Chapter 4: Q 7E (page 255)

Let\({\bf{X}}\),\({\bf{Y}}\), and\({\bf{Z}}\)be three random variables such that\({\bf{Cov}}\left( {{\bf{X,Z}}} \right)\)and\({\bf{Cov}}\left( {{\bf{Y,Z}}} \right)\)exist, and let\({\bf{a}}\),\({\bf{b}}\)and\({\bf{c}}\)be arbitrary given constants. Show that\({\bf{Cov}}\left( {{\bf{aX + bY + c,Z}}} \right){\bf{ = aCov}}\left( {{\bf{X,Z}}} \right){\bf{ + bCov}}\left( {{\bf{Y,Z}}} \right)\).

Short Answer

Expert verified

If X, Y, and Z are random variables such that\(Cov\left( {X,Z} \right)\)and\(Cov\left( {Y,Z} \right)\)exist and\(a,b,c\)are arbitrary constants, then

\(Cov\left( {aX + bY + c,Z} \right) = aCov\left( {X,Z} \right) + bCov\left( {Y,Z} \right)\).

Step by step solution

01

Given information

X, Y, and Z are random variables and \(a,b,c\)are arbitrary constants

\(Cov\left( {X,Z} \right)\) and\(Cov\left( {Y,Z} \right)\) exists.

02

Determine the Covariance

The expectation of \(aX + bY + c\) is

\(\begin{align}E\left( {aX + bY + c} \right) &= aE\left( X \right) + bE\left( Y \right) + c\\ &= a{\mu _X} + b{\mu _Y} + c\end{align}\)

Then, the covariance of\(aX + bY + c\)and\(Z\)is

\(\begin{align}Cov\left( {aX + bY + c,Z} \right) &= E\left( {\left( {aX + bY + c - a{\mu _x} - b{\mu _Y} - c} \right)\left( {Z - {\mu _z}} \right)} \right)\\ &= E\left( {\left( {a\left( {X - {\mu _x}} \right) + b\left( {Y - {\mu _Y}} \right)} \right)\left( {Z - {\mu _z}} \right)} \right)\\ &= aE\left( {\left( {X - {\mu _x}} \right)\left( {Z - {\mu _z}} \right)} \right) + bE\left( {\left( {Y - {\mu _Y}} \right)\left( {Z - {\mu _z}} \right)} \right)\\ &= aCov\left( {X,Z} \right) + bCov\left( {Y,Z} \right)\end{align}\)

Hence, if X, Y, and Z are random variables such that\(Cov\left( {X,Z} \right)\)and\(Cov\left( {Y,Z} \right)\)exists and\(a,b,c\)are arbitrary constants, then

\(Cov\left( {aX + bY + c,Z} \right) = aCov\left( {X,Z} \right) + bCov\left( {Y,Z} \right)\).

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