91Ó°ÊÓ

A linear combination is an expression made up of a series of terms multiplied by a constant and then added together. Linear combinations are a fundamental idea in linear algebra and related branches of mathematics.

a)

The following holds

\(\begin{array}{c}{\rm{V(aX + Y) = }}{{\rm{a}}^{\rm{2}}}{\rm{V(X) + V(Y) + 2aCov(X,Y)}}\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ + 2a}}{{\rm{\sigma }}_{\rm{X}}}{{\rm{\sigma }}_{\rm{Y}}}{\rm{\rho }}\end{array}\)

(1): see the hint in the exercise.

When

\({\rm{a = }}\frac{{{{\rm{\sigma }}_{\rm{Y}}}}}{{{{\rm{\sigma }}_{\rm{X}}}}}\)

the following is true

\(\begin{array}{c}{\rm{0£ V(aX + Y) = }}{{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ + 2a}}{{\rm{\sigma }}_{\rm{X}}}{{\rm{\sigma }}_{\rm{Y}}}{\rm{\rho }}\\{\rm{ = }}{\left( {\frac{{{{\rm{\sigma }}_{\rm{Y}}}}}{{{{\rm{\sigma }}_{\rm{X}}}}}} \right)^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ + 2}}\frac{{{{\rm{\sigma }}_{\rm{Y}}}}}{{{{\rm{\sigma }}_{\rm{X}}}}}{{\rm{\sigma }}_{\rm{X}}}{{\rm{\sigma }}_{\rm{Y}}}{\rm{\rho }}\\{\rm{ = \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ + 2\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{\rho }}\\{\rm{ = 2\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{(1 + \rho )}}\end{array}\)

hence,

\[{{\text{2\sigma }}_{\text{Y}}^{\text{2}}{\text{(1 + \rho )0}}}\]

\[{{\text{\sigma }}_{\text{Y}}^{\text{2}}{\text{0}}}\]

implies that

\[{\text{1 + \rho 0}}\]

or equally

\(\rho \ge - 1\)

(b):

The following holds

\(\begin{array}{c}{\rm{V(aX - Y) = }}{{\rm{a}}^{\rm{2}}}{\rm{V(X) + ( - 1}}{{\rm{)}}^{\rm{2}}}{\rm{V(Y) - 2aCov(X,Y)}}\\\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ - 2a}}{{\rm{\sigma }}_{\rm{X}}}{{\rm{\sigma }}_{\rm{Y}}}{\rm{\rho }}\end{array}\)

(1): see the hint in the exercise.

When

\({\rm{a = }}\frac{{{{\rm{\sigma }}_{\rm{Y}}}}}{{{{\rm{\sigma }}_{\rm{X}}}}}\)

the following is true

\(\begin{array}{c}{\rm{0£ V(aX - Y) = }}{{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ - 2a}}{{\rm{\sigma }}_{\rm{X}}}{{\rm{\sigma }}_{\rm{Y}}}{\rm{\rho }}\\{\rm{ = }}{\left( {\frac{{{{\rm{\sigma }}_{\rm{Y}}}}}{{{{\rm{\sigma }}_{\rm{X}}}}}} \right)^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ - 2}}\frac{{{{\rm{\sigma }}_{\rm{Y}}}}}{{{{\rm{\sigma }}_{\rm{X}}}}}{{\rm{\sigma }}_{\rm{X}}}{{\rm{\sigma }}_{\rm{Y}}}{\rm{\rho }}\\{\rm{ = \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ - 2\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{\rho }}\\{\rm{ = 2\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{(1 - \rho )}}\end{array}\)

hence,

\(\begin{array}{*{20}{r}}{2\sigma _Y^2(1 - \rho ) \ge 0}\\{\sigma _Y^2 \ge 0}\end{array}\)

implies that

\(1 - \rho \ge 0\)

or equally

\(\rho \le 1.\)

(c):

Assume that\({\rm{\rho = 1}}\), then from

\(\begin{array}{c}{\rm{V(aX - Y) = 2}}\\{\rm{\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{(1 - \rho ) = 2}}\\{\rm{\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{(1 - 1) = 0}}\end{array}\)

for any random variables \({\rm{X}}\)and\({\rm{Y}}\). From implication

\[V(W) = 0W = b,\quad b - {\text{ constant }}\]

the following holds

\[V(aX - Y) = 0aX - Y = b,\quad b - {\text{ constant }}\]

or equally

\({\rm{Y = aX + b}}\)